Gödel's incompleteness theorems

3154:, a hierarchical, self-referential structure existing within an axiomatic formal system. He argues that this is the same kind of structure that gives rise to consciousness, the sense of "I", in the human mind. While the self-reference in Gödel's theorem comes from the Gödel sentence asserting its unprovability within the formal system of Principia Mathematica, the self-reference in the human mind comes from how the brain abstracts and categorises stimuli into "symbols", or groups of neurons which respond to concepts, in what is effectively also a formal system, eventually giving rise to symbols modeling the concept of the very entity doing the perception. Hofstadter argues that a strange loop in a sufficiently complex formal system can give rise to a "downward" or "upside-down" causality, a situation in which the normal hierarchy of cause-and-effect is flipped upside-down. In the case of Gödel's theorem, this manifests, in short, as the following: 4330:. . . he found the translation "not quite so good" as he had expected . . . agreed to its publication" (ibid). (In a footnote Dawson states that "he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints" (ibid)). Dawson states that "The translation that Gödel favored was that by Jean van Heijenoort" (ibid). For the serious student another version exists as a set of lecture notes recorded by Stephen Kleene and J. B. Rosser "during lectures given by Gödel at to the Institute for Advanced Study during the spring of 1934" (cf commentary by 995:, a particular system of arithmetic, but a parallel demonstration could be given for any effective system of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness. In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal system. The terminology used to state these conditions was not yet developed in 1931 when Gödel published his results. 8159: 3546: 1056:) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true formal theories of arithmetic (theories whose axioms are all true statements about natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem. 3420:

felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized provability and had only a superficial resemblance to Gödel's work. Gödel read the paper but found it deeply flawed, and his response to Finsler laid out concerns about the lack of formalization. Finsler continued to argue for his philosophy of mathematics, which eschewed formalization, for the remainder of his career.

2020:. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem only applies to theories with the additional property that all their axioms are true in the standard model of the natural numbers. Gödel's incompleteness theorem is distinguished by its applicability to consistent theories that nonetheless include false statements in the standard model; these theories are known as 3437:. However, Zermelo did not relent and published his criticisms in print with "a rather scathing paragraph on his young competitor". Gödel decided that pursuing the matter further was pointless, and Carnap agreed. Much of Zermelo's subsequent work was related to logic stronger than first-order logic, with which he hoped to show both the consistency and categoricity of mathematical theories. 3532: 901:, and the truth of the Gödel sentence follows from the fact that no standard natural number has the property in question. Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" – it must contain elements that do not correspond to any standard natural number ( 416:) has studied some weak families of arithmetic systems which allow enough arithmetic as relations to formalise Gödel numbering, but which are not strong enough to have multiplication as a function, and so fail to prove the second incompleteness theorem; that is to say, these systems are consistent and capable of proving their own consistency (see 2132:, in such a way that it is possible to mechanically convert back and forth between formulas and Gödel numbers. The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that such numbers can be constructed. A simple example is how English can be stored as a 725:, p. 141). As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form. In particular, it can be expressed as a formula in the language of arithmetic consisting of a number of leading universal quantifiers followed by a quantifier-free body (these formulas are at level 3419:

to construct an expression that was false but unprovable in a particular, informal framework he had developed. Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p. 9). Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler

452:

The first incompleteness theorem shows that, in formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent statement is added as an axiom, there are other true statements that still cannot be proved, even

3351:

Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for a conversation. Later that year, working independently with knowledge of the first

3117:

suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented

1327:

There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert–Bernays conditions. Peano arithmetic, however, is strong enough to verify these conditions, as are all theories stronger than

709:

of sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the

290:

seems consistent. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem. Thus by the first incompleteness theorem, Peano Arithmetic is not complete. The theorem gives

405:

consists of a set of axioms for the natural numbers with just the addition operation (multiplication is omitted). Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs

222:

consists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent and complete, and contains a sufficient amount of arithmetic. However, it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the

3379:

Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church, Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used).

3511:

argue that Wittgenstein had a more complete understanding of the incompleteness theorem than was previously assumed. They are particularly concerned with the interpretation of a Gödel sentence for an ω-inconsistent system as saying "I am not provable", since the system has no models in which the

3325:

was a joint meeting of three academic societies, with many of the key logicians of the time in attendance. Carnap, Heyting, and von Neumann delivered one-hour addresses on the mathematical philosophies of logicism, intuitionism, and formalism, respectively. The conference also included Hilbert's

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was able to correct the proof for a system of arithmetic without any axioms of induction. By 1928, Ackermann had communicated a modified proof to Bernays; this modified proof led Hilbert to announce his belief in 1929 that the consistency of arithmetic had been demonstrated and that a consistent

1734:

the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proved from ZFC.

1702:

of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth

1162:

of PA" is consistent. But, because PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent". What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest consistent subset of PA" is

3163:

In the case of the mind, a far more complex formal system, this "downward causality" manifests, in Hofstadter's view, as the ineffable human instinct that the causality of our minds lies on the high level of desires, concepts, personalities, thoughts, and ideas, rather than on the low level of

2046:

Statements in the system can be represented by natural numbers (known as Gödel numbers). The significance of this is that properties of statements—such as their truth and falsehood—will be equivalent to determining whether their Gödel numbers have certain properties, and that properties of the

1122:

itself. This theorem is stronger than the first incompleteness theorem because the statement constructed in the first incompleteness theorem does not directly express the consistency of the system. The proof of the second incompleteness theorem is obtained by formalizing the proof of the first

448:

The pattern illustrated in the previous sections with Peano arithmetic, ZFC, and ZFC + "there exists an inaccessible cardinal" cannot generally be broken. Here ZFC + "there exists an inaccessible cardinal" cannot from itself, be proved consistent. It is also not complete, as illustrated by the

4411:

Martin Davis editor, 1965, ibid. "On Undecidable Propositions of Formal Mathematical Systems." A copy with Gödel's corrections of errata and Gödel's added notes begins on page 41, preceded by two pages of Davis's commentary. Until Davis included this in his volume this lecture existed only as

3485:

have provided textual readings arguing that most commentary misunderstands Wittgenstein. On their release, Bernays, Dummett, and Kreisel wrote separate reviews on Wittgenstein's remarks, all of which were extremely negative. The unanimity of this criticism caused Wittgenstein's remarks on the

3158:

Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a

985:

Compared to the theorems stated in Gödel's 1931 paper, many contemporary statements of the incompleteness theorems are more general in two ways. These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions.

1530:

This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the consistency of Peano arithmetic using any finitistic means that can be formalized in a system the consistency of which is provable in Peano arithmetic (PA). For example, the system of

3352:

incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930. Gödel had independently obtained the second incompleteness theorem and included it in his submitted manuscript, which was received by

3077:, which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided (see " 3380:

Rosser proved in 1936 that the hypothesis of ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency if the Gödel sentence was changed appropriately. These developments left the incompleteness theorems in essentially their modern form.

190:

There are several properties that a formal system may have, including completeness, consistency, and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties.

3432:

wrote to Gödel to announce what he described as an "essential gap" in Gödel's argument. In October, Gödel replied with a 10-page letter, where he pointed out that Zermelo mistakenly assumed that the notion of truth in a system is definable in that system; it is not true in general by

1885:

eventually halts when run with a particular given input. Kleene showed that the existence of a complete effective system of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction. This method of proof has also been presented by

374:. Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems. 175:. In general, a formal system is a deductive apparatus that consists of a particular set of axioms along with rules of symbolic manipulation (or rules of inference) that allow for the derivation of new theorems from the axioms. One example of such a system is first-order 3065:

argue that it is not a problem for logicism because the incompleteness theorems apply equally to first-order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.

428:

In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any incorrect results. For example, we could imagine a set of true axioms which allow us to prove every true arithmetical claim about the natural numbers

3498:

understand (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).

2270:

is true if and only if it can be proved (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as

2594:

This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is obtained as a result, and thus this sentence indirectly asserts its own unprovability. The proof of the diagonal lemma employs a similar method.

4408:(pbk). van Heijenoort did the translation. He states that "Professor Gödel approved the translation, which in many places was accommodated to his wishes." (p. 595). Gödel's paper begins on p. 595; van Heijenoort's commentary begins on p. 592. 1791:

These are natural mathematical equivalents of the Gödel "true but undecidable" sentence. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic.

2077:

Within the formal system this statement permits a demonstration that it is neither provable nor disprovable in the system, and therefore the system cannot in fact be ω-consistent. Hence the original assumption that the proposed system met the criteria is

3168:

There is thus a curious upside-downness to our normal human way of perceiving the world: we are built to perceive “big stuff” rather than “small stuff”, even though the domain of the tiny seems to be where the actual motors driving reality reside.

453:

with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent. It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized.

385:

is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers. A similar example is the theory of

893:, p. 135). That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false. As described earlier, the Gödel sentence of a system 4313:

None of the following agree in all translated words and in typography. The typography is a serious matter, because Gödel expressly wished to emphasize "those metamathematical notions that had been defined in their usual sense before . . ."

3297:

proof of analysis would likely soon follow. After the publication of the incompleteness theorems showed that Ackermann's modified proof must be erroneous, von Neumann produced a concrete example showing that its main technique was unsound.

211:. This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and 2171:

In principle, proving a statement true or false can be shown to be equivalent to proving that the number matching the statement does or does not have a given property. Because the formal system is strong enough to support reasoning about

1539:, which aimed to justify the use of "ideal" (infinitistic) mathematical principles in the proofs of "real" (finitistic) mathematical statements by giving a finitistic proof that the ideal principles are consistent, cannot be carried out. 4354:(pbk.) This contains a useful translation of Gödel's German abbreviations on pp. 33–34. As noted above, typography, translation and commentary is suspect. Unfortunately, this translation was reprinted with all its suspect content by 2196:

Having shown that in principle the system can indirectly make statements about provability, by analyzing properties of those numbers representing statements it is now possible to show how to create a statement that actually does this.

2827:. This gives the first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic. 259:

In a system of mathematics, thinkers such as Hilbert believed that it was just a matter of time to find such an axiomatization that would allow one to either prove or disprove (by proving its negation) every mathematical formula.

3212:

Appeals and analogies are sometimes made to the incompleteness of theorems in support of arguments that go beyond mathematics and logic. Several authors have commented negatively on such extensions and interpretations, including

3300:

In the course of his research, Gödel discovered that although a sentence, asserting its falsehood leads to paradox, a sentence that asserts its non-provability does not. In particular, Gödel was aware of the result now called

2322:

The notion of provability itself can also be encoded by Gödel numbers, in the following way: since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for every statement

3486:

incompleteness theorems to have little impact on the logic community. In 1972, Gödel stated: "Has Wittgenstein lost his mind? Does he mean it seriously? He intentionally utters trivially nonsensical statements", and wrote to

2581:

states that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement. But when this calculation is performed, the resulting Gödel number turns out to be the Gödel number of

3364: 1103: 926: 3336:, and, in my opinion, not at all for natural science either. ... The true reason why has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish 2471:

The next step in the proof is to obtain a statement which, indirectly, asserts its own unprovability. Although Gödel constructed this statement directly, the existence of at least one such statement follows from the

3387:

for first-order arithmetic in 1936. Hilbert accepted this proof as "finitary" although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is being proved consistent.

667:

The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in

471: 1829:, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system ATR 243:

if, for any statement in the axioms' language, that statement or its negation is provable from the axioms. This is the notion relevant for Gödel's first Incompleteness theorem. It is not to be confused with

933:

is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence

3326:

retirement address, as he was leaving his position at the University of Göttingen. Hilbert used the speech to argue his belief that all mathematical problems can be solved. He ended his address by saying,

1535:(PRA), which is widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. Thus PRA cannot prove the consistency of PA. This fact is generally seen to imply that 765:, the Gödel sentence can be re-written as a statement that a particular polynomial in many variables with integer coefficients never takes the value zero when integers are substituted for its variables ( 2906:

The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within a system

263:

A formal system might be syntactically incomplete by design, as logics generally are. Or it may be incomplete simply because not all the necessary axioms have been discovered or included. For example,

3399:), along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full published proof of the second incompleteness theorem. 3196:) argues that replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show that naive mathematics is inconsistent, and uses this as evidence for 717:, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a 256:

complete. But it is not syntactically complete, since there are sentences expressible in the language of first-order logic that can be neither proved nor disproved from the axioms of logic alone.

367:

The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. One sufficient collection is the set of theorems of

5004: 482:. The resulting theorem (incorporating Rosser's improvement) may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated. 3466:

dominated the circle's thinking. There has been some controversy about whether Wittgenstein misunderstood the incompleteness theorem or just expressed himself unclearly. Writings in Gödel's

3736: 705:

To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on

3454:, particularly, one section sometimes called the "notorious paragraph" where he seems to confuse the notions of "true" and "provable" in Russell's system. Gödel was a member of the 1948:= 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation 897:

is an arithmetical statement which claims that no number exists with a particular property. The incompleteness theorem shows that this claim will be independent of the system

2841:

The incompleteness theorems are among a relatively small number of nontrivial theorems that have been transformed into formalized theorems that can be completely verified by

167:

that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent and effectively axiomatized. Particularly in the context of

755: 4318:, p. 595). Three translations exist. Of the first John Dawson states that: "The Meltzer translation was seriously deficient and received a devastating review in the 3391:

The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume of

5541: 3322: 3306: 2708:

is not provable (from the previous paragraph). Thus on one hand the system proves there is a number with a certain property (that it is the Gödel number of the proof of

796:. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true ( 3103:

have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a

2033: 462: 314:

Peano arithmetic is provably consistent from ZFC, but not from within itself. Similarly, ZFC is not provably consistent from within itself, but ZFC + "there exists an

2103:, which could easily give rise to an infinite regress. Gödel's technique is to show that statements can be matched with numbers (often called the arithmetization of 271:

is incomplete, because some statements in the language (such as the parallel postulate itself) can not be proved from the remaining axioms. Similarly, the theory of

2916:

for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the system

2266: 433:, p. 2). In the standard system of first-order logic, an inconsistent set of axioms will prove every statement in its language (this is sometimes called the 348:

as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication. However, it is not consistent.

3305:, although he never published it. Gödel announced his first incompleteness theorem to Carnap, Feigel, and Waismann on August 26, 1930; all four would attend the 4689:" English translation of a lecture delivered before the International Congress of Mathematicians at Paris, containing Hilbert's statement of his Second Problem. 3013:: to prove the 2nd Incompleteness Theorem, we obtain a contradiction with the 1st Incompleteness Theorem which can do only by showing that the theorem holds in 4334:, p. 39 and beginning on p. 41); this version is titled "On Undecidable Propositions of Formal Mathematical Systems". In their order of publication: 1964:

were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of

524:, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and any 291:

an explicit example of a statement of arithmetic that is neither provable nor disprovable in Peano's arithmetic. Moreover, this statement is true in the usual

4477: 2734:

is not provable only requires the assumption that the system is consistent. The stronger assumption of ω-consistency is required to show that the negation of

830:

may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as

1960:

is ω-consistent. In that case, it will never prove that a particular polynomial equation has a solution when there is no solution in the integers. Thus, if

17: 2115:. This allows a self-referential formula to be constructed in a way that avoids any infinite regress of definitions. The same technique was later used by 2047:

statements can therefore be demonstrated by examining their Gödel numbers. This part culminates in the construction of a formula expressing the idea that

1044:

exists while denying that it has any specific value. The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency.

123:. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. 8183: 6538: 1542:

The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would provide no interesting information if a system

3078: 1823:, a statement about sequences of natural numbers somewhat simpler than the Paris–Harrington principle, is also undecidable in Peano arithmetic. 4326:, p. 216). "Fortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthology 6183: 4693: 2128:

In simple terms, a method can be devised so that every formula or statement that can be formulated in the system gets a unique number, called its

183:, only some sentences of the formal system express statements about the natural numbers. The incompleteness theorems are about formal provability 1562:

would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a system

133:, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by 7213: 3009:" is not provable") is what we construct to be unprovable. Notice that this is why we require formalizing the first Incompleteness Theorem in 2927:

stand for the undecidable sentence constructed above, and assume for purposes of obtaining a contradiction that the consistency of the system

283:

that is not provable within ZFC, so ZFC is not complete. In this case, there is no obvious candidate for a new axiom that resolves the issue.

3159:

mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false.

973:, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, 1680:, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no 7296: 6437: 4428: 3348:", were used as Hilbert's epitaph in 1943). Although Gödel was likely in attendance for Hilbert's address, the two never met face to face. 3164:

interactions between neurons or even fundamental particles, even though according to physics the latter seems to possess the causal power.

3507:

in 2000, a series of papers in philosophy have sought to evaluate whether the original criticism of Wittgenstein's remarks was justified.

3001:

is not provable. But this is a contradiction since by the 1st Incompleteness Theorem, this sentence (ie. what is implied in the sentence

1546:

proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of

356: 2901: 2395:, the German word for "provable"; this name was originally used by Gödel to denote the provability formula just described. Note that " 1186: 2006:

can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are

998:

Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent but

493:

within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of

3450: 42: 4252: 2339:, the potential Gödel number of its proof, is an arithmetical relation between two numbers. Therefore, there is a statement form 398:. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory. 7610: 3111:, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it. 2233:

statement, and it is then either provable in the system, or not. For certain formulas one can show that for every natural number

1873:

presented a proof of Gödel's incompleteness theorem using basic results of computability theory. One such result shows that the

5564: 2777:" as axioms. But then the definition of "being a Gödel number of a proof" of a statement changes. which means that the formula 1698:

Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the

1076:. This formula expresses the property that "there does not exist a natural number coding a formal derivation within the system 3624: 7768: 6090: 6020: 5976: 5837: 5771: 5674: 5578: 5423: 5400: 5334: 5224: 5103: 4856: 4541: 1659: 1143:. There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons( 819:). However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence 126:

The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

7623: 6946: 1080:

whose conclusion is a syntactic contradiction." The syntactic contradiction is often taken to be "0=1", in which case Cons(

5600: 1163:

meant here to be the largest consistent initial segment of the axioms of PA under some particular effective enumeration.)

6286: 6171: 6158: 6146: 3560: 2013: 1760: 1613:

The second incompleteness theorem does not rule out altogether the possibility of proving the consistency of some theory

1135:

There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of

3200:. The cause of this inconsistency is the inclusion of a truth predicate for a system within the language of the system. 7628: 7618: 7355: 7208: 6561: 5506: 4301: 4278: 3619: 3434: 3302: 3178:

Although Gödel's theorems are usually studied in the context of classical logic, they also have a role in the study of

2716:, we can prove that it does not have this property. This is impossible in an ω-consistent system. Thus the negation of 1664:

There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the

1158:, and some may even be provable. For example, first-order Peano arithmetic (PA) can prove that "the largest consistent 970: 134: 6552: 6349: 4866:

Willard, Dan E. (2001). "Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles".

3577: 352: 249: 63: 7764: 6394: 6056: 6002: 5954: 5793: 5641: 5550: 5452: 5353: 5262: 5184: 5111: 5080: 5072: 5041: 5014: 4980: 4619: 4464: 4405: 4373: 4351: 3679: 3132:

is interpreted by modern standards, namely via computational complexity, the Gödel phenomena are very much with us."

1854: 1692: 7106: 2278:

Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form

1668:

sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified

7861: 7605: 6430: 6239: 5843:

Rodych, Victor (2003). "Misunderstanding Gödel: New Arguments about Wittgenstein and New Remarks by Wittgenstein".

4474: 3704: 3599: 3367:"). As the title implies, Gödel originally planned to publish a second part of the paper in the next volume of the 2087:

The main problem in fleshing out the proof described above is that it seems at first that to construct a statement

248:

completeness, which means that the set of axioms proves all the semantic tautologies of the given language. In his

6222: 3090: 2647:. Thus the system would be inconsistent, proving both a statement and its negation. This contradiction shows that 7166: 6859: 4786: 3683: 3363:

in 1931 under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("

3125: 2845:

software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in

1842: 280: 212: 8213: 6600: 6379: 5458:

Floyd, Juliet; Putnam, Hilary (2000). "A Note on Wittgenstein's "Notorious Paragraph" about the Godel Theorem".

3257:'s invocation of the theorem in the context of sociology; Debray has defended this use as metaphorical (ibid.). 275:

is not complete, but becomes complete with an extra axiom stating that there are no endpoints in the order. The

8122: 7824: 7587: 7582: 7407: 6828: 6512: 5318: 5297: 5282: 4634: 3463: 3384: 2769:

If one tries to "add the missing axioms" to avoid the incompleteness of the system, then one has to add either

2066:

and essentially says that it (i.e. the statement itself) is unprovable. This is done using a technique called "

1756: 1643: 1532: 831: 437:), and is thus automatically complete. A set of axioms that is both complete and consistent, however, proves a 3582: 550:

as an additional axiom. This will not result in a complete system, because Gödel's theorem will also apply to

8117: 7900: 7817: 7530: 7461: 7338: 6580: 6232: 5880: 2071: 252:(not to be confused with the incompleteness theorems described here), Gödel proved that first-order logic is 130: 80:

that are concerned with the limits of provability in formal axiomatic theories. These results, published by

8042: 7868: 7554: 7188: 6787: 4551: 4383: 4251:

Kurt Gödel, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I",

2003: 1804: 1625:

proved the consistency of Peano arithmetic in a different system that includes an axiom asserting that the

718: 710:

arithmetical properties of numbers themselves, which would be decidable by the system if it were complete.

6192: 3593: 3288:

had published a flawed consistency proof for analysis in 1925, in which he attempted to use the method of

8188: 7920: 7915: 7525: 7264: 7193: 6522: 6423: 6404: 6227: 5257:. Cambridge tracts in theoretical computer science. Vol. 38. Cambridge: Cambridge University Press. 3629: 3344:

This speech quickly became known as a summary of Hilbert's beliefs on mathematics (its final six words, "

3274: 3074: 2406:" is merely an abbreviation that represents a particular, very long, formula in the original language of 2062:

In the formal system it is possible to construct a number whose matching statement, when interpreted, is

307:

if there is no statement such that both the statement and its negation are provable from the axioms, and

6180: 5697:"A machine-assisted proof of Gödel's incompleteness theorems for the theory of hereditarily finite sets" 4831:—, 1939, "An Informal Exposition of proofs of Gödel's Theorem and Church's Theorem", Reprinted from the 4697: 2590:", when preceded by itself in quotes, is unprovable.", when preceded by itself in quotes, is unprovable. 7849: 7439: 6833: 6801: 6492: 6399: 6337: 6126: 5214: 4533:

The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions

4388:

The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable problems and Computable Functions

2554:

defined by this roughly states that its own Gödel number is the Gödel number of an unprovable formula.

1796: 1711: 531:

Each effectively generated system has its own Gödel sentence. It is possible to define a larger system

378: 4575: 3969: 3448:

wrote several passages about the incompleteness theorems that were published posthumously in his 1953

2792:, different from the previous one, which will be undecidable in the new system if it is ω-consistent. 8139: 8088: 7985: 7483: 7444: 6921: 6566: 5524: 5460: 4390:, Raven Press, New York, no ISBN. Gödel's paper begins on page 5, preceded by one page of commentary. 4265:—, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", in 3459: 3030: 2836: 1909: 1905: 1826: 1704: 961:

It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "

295:. In addition, no effectively axiomatized, consistent extension of Peano arithmetic can be complete. 85: 38: 6595: 6165: 3490:

that Wittgenstein's comments demonstrate a misunderstanding of the incompleteness theorems writing:

179:, a system in which all variables are intended to denote natural numbers. In other systems, such as 7980: 7910: 7449: 7301: 7284: 7007: 6487: 6322: 4456: 3332: 3108: 3096: 2811:

to construct a true but unprovable formula. A similar proof method was independently discovered by

2788:

is now different. Thus when we apply the diagonal lemma to this new Bew, we obtain a new statement

382: 3993: 2727:

is undecidable in our axiomatic system: it can neither be proved nor disproved within the system.

728: 474:

I". The hypotheses of the theorem were improved shortly thereafter by J. Barkley Rosser (

7812: 7789: 7750: 7636: 7577: 7223: 7143: 6987: 6931: 6544: 6342: 6279: 5378: 3278: 2885: 2007: 1816: 925:

as semantical analogues to his syntactical incompleteness result in the introductory section of "

878: 417: 311:

otherwise. That is to say, a consistent axiomatic system is one that is free from contradiction.

1980:

cannot be ω-consistent and complete. Moreover, for each consistent effectively generated system

1714:

has given two concrete examples of undecidable statements (in the first sense of the term): The

8102: 7829: 7807: 7774: 7667: 7513: 7498: 7471: 7422: 7306: 7241: 7066: 7032: 7027: 6901: 6732: 6709: 6374: 4686: 3520:

explores the relationship between Wittgenstein's writing and theories of paraconsistent logic.

3242: 3058: 3045:

The incompleteness theorem is sometimes thought to have severe consequences for the program of

2039: 1820: 1171:

The standard proof of the second incompleteness theorem assumes that the provability predicate

758: 434: 208: 6030: 5063: 3566: 3281:

were known as "analysis", while theories of natural numbers alone were known as "arithmetic".

3140: 2042:

has three essential parts. To begin, choose a formal system that meets the proposed criteria:

2021: 1345:

satisfying the technical conditions outlined above cannot prove the consistency of any system

1114:

within which a certain amount of elementary arithmetic can be carried out, the consistency of

882: 449:

continuum hypothesis, which is unresolvable in ZFC + "there exists an inaccessible cardinal".

8208: 8203: 8198: 8193: 8032: 7885: 7677: 7395: 7131: 7037: 6896: 6881: 6762: 6737: 6256: 6067:. Handbook of the Philosophy of Science. Vol. 5. Amsterdam: Elsevier. pp. 411–447. 5828: 4958: 4607: 4281:. The original German with a facing English translation, preceded by an introductory note by 2816: 2757:

If the system is consistent, it may have the same situation, or it may prove the negation of

2017: 1976:

has no solution" is found, in contradiction to Matiyasevich's theorem. Hence it follows that

1768: 1106:". In the following statement, the term "formalized system" also includes an assumption that 991: 402: 315: 6153: 4769: 4288:—, 1951, "Some basic theorems on the foundations of mathematics and their implications", in 969:

of a false formula" cannot be represented as a formula of arithmetic. This result, known as

8005: 7967: 7844: 7648: 7488: 7412: 7390: 7218: 7176: 7075: 7042: 6906: 6694: 6605: 6369: 6364: 6316: 5964: 5944: 5903: 5363: 5310: 5234: 5191: 5158: 5118: 5051: 4731: 4376:. Gödel's paper appears starting on p. 1097, with Hawking's commentary starting on p. 1089. 3416: 3266: 3179: 2852:

Computer-verified proofs of versions of the first incompleteness theorem were announced by

2121: 1952:= 0 does have a solution in the integers then any sufficiently strong system of arithmetic 1944:

with integer coefficients, determines whether there is an integer solution to the equation

1715: 1673: 1536: 918: 886: 334: 276: 143: 89: 5207: 5087: 5022: 3790: 2242: 158: 8: 8134: 8025: 8010: 7990: 7947: 7834: 7784: 7710: 7655: 7592: 7385: 7380: 7328: 7096: 7085: 6757: 6657: 6585: 6576: 6572: 6507: 6502: 6310: 6044: 5823: 5762:

Priest, Graham (2004). "Wittgenstein's Remarks on Gödel's Theorem". In Max Kölbel (ed.).

5627: 5568: 5341: 5094: 4817: 4739: 4629: 4501:

Charlesworth, Arthur (1981). "A Proof of Godel's Theorem in Terms of Computer Programs".

4393: 4282: 3445: 3146: 1838: 1763:

states that for any system that can represent enough arithmetic, there is an upper bound

1695:

is sometimes used instead of undecidable for the "neither provable nor refutable" sense.

1685: 1681: 1084:) states "there is no natural number that codes a derivation of '0=1' from the axioms of 368: 112: 4900:"The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program" 3587: 942:

makes a similar assertion to the liar sentence, but with truth replaced by provability:

8163: 7932: 7895: 7880: 7873: 7856: 7660: 7642: 7508: 7434: 7417: 7370: 7183: 7092: 6926: 6911: 6871: 6823: 6808: 6796: 6752: 6727: 6497: 6446: 6272: 6104: 6082: 6068: 5865: 5856: 5726: 5708: 5680: 5652: 5560: 5485: 5058: 4972: 4942: 4916: 4883: 4806: 4671: 4595: 4518: 4486: 3551: 3135: 2804: 2133: 1780: 672:. However, the sequence of steps is such that the constructed sentence turns out to be 516:

referred to by the theorem is often referred to as "the Gödel sentence" for the system

395: 391: 272: 268: 264: 77: 7116: 6253: 5696: 4755: 4646:(1981). "How to Gödel a Frege-Russell: Gödel's Incompleteness Theorems and Logicism". 1992:= 0 has no solutions over the integers, but the lack of solutions cannot be proved in 8158: 8098: 7905: 7715: 7705: 7597: 7478: 7313: 7289: 7070: 7054: 6959: 6936: 6813: 6782: 6747: 6642: 6477: 6243: 6096: 6086: 6052: 6016: 5998: 5972: 5950: 5833: 5789: 5767: 5670: 5637: 5574: 5546: 5502: 5477: 5448: 5440: 5429: 5419: 5396: 5349: 5330: 5314: 5293: 5278: 5258: 5220: 5180: 5107: 5099: 5076: 5068: 5037: 5010: 4976: 4934: 4899: 4852: 4663: 4615: 4599: 4537: 4460: 4401: 4369: 4347: 4339: 4297: 4274: 3609: 3545: 3537: 3285: 3238: 3204:

gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism.

2853: 2823:

of true sentences of arithmetic, another sentence which is true but not contained in

1913: 1742: 1053: 1045: 479: 387: 168: 6215: 5730: 5684: 5647:

O'Connor, Russell (2005). "Essential Incompleteness of Arithmetic Verified by Coq".

5619: 5584: 5029: 4946: 4810: 4694:

On formally undecidable propositions of Principia Mathematica and related systems I.

1147:) from the second incompleteness theorem is a particular expression of consistency. 877:

Although the Gödel sentence of a consistent theory is true as a statement about the

8112: 8107: 8000: 7957: 7779: 7740: 7735: 7720: 7546: 7503: 7400: 7198: 7148: 6722: 6684: 6332: 6141: 6078: 5986: 5923: 5889: 5861: 5852: 5750: 5718: 5692: 5662: 5615: 5469: 5271: 5018: 4968: 4926: 4875: 4798: 4751: 4717: 4655: 4643: 4587: 4555: 4510: 4491: 4308: 4289: 4266: 4256: 3365:

On Formally Undecidable Propositions in Principia Mathematica and Related Systems I

3293: 3289: 3069:

Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to

3054: 2881: 2846: 2599: 2476:, which says that for any sufficiently strong formal system and any statement form 1864: 1812: 1677: 1669: 1647: 1104:

On Formally Undecidable Propositions in Principia Mathematica and Related Systems I

1000: 927:

On Formally Undecidable Propositions in Principia Mathematica and Related Systems I

345: 287: 176: 62:

For the earlier theory about the correspondence between truth and provability, see

6108: 6063:

Zach, Richard (2007). "Hilbert's Program Then and Now". In Jacquette, Dale (ed.).

5141: 5136: 3572: 8093: 8083: 8037: 8020: 7975: 7937: 7839: 7759: 7566: 7493: 7466: 7454: 7360: 7274: 7248: 7203: 7171: 6972: 6774: 6717: 6667: 6632: 6590: 6299: 6247: 6187: 5911: 5899: 5360: 5324: 5307: 5230: 5204: 5188: 5172: 5115: 5084: 5047: 4727: 4531: 4481: 4361: 4344:

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

2842: 1874: 1860: 1834: 1752: 1727: 1622: 1094:

shows that, under general assumptions, this canonical consistency statement Cons(

1068:

containing basic arithmetic, it is possible to canonically define a formula Cons(

525: 472:

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

240: 219: 152: 6264: 6209: 4998:

There's Something about Gödel: The Complete Guide to the Incompleteness Theorem

4077: 3516:

argues that their interpretation of Wittgenstein is not historically justified.

3371:; the prompt acceptance of the first paper was one reason he changed his plans. 3254: 2350:

that uses this arithmetical relation to state that a Gödel number of a proof of

2129: 1833:

codifying the principles acceptable based on a philosophy of mathematics called

966: 706: 8078: 8057: 8015: 7995: 7890: 7745: 7343: 7333: 7323: 7318: 7252: 7126: 7002: 6891: 6886: 6864: 6465: 6389: 5990: 5927: 5871: 5536: 4722: 4705: 4137: 3124:

has proposed that the concept of mathematical "knowability" should be based on

3104: 3062: 3034: 2865: 2473: 2067: 2063: 1877:

is undecidable: no computer program can correctly determine, given any program

1808: 1800: 1626: 1416:

has the decidable property of not being a code for a proof of contradiction in

1110:

is effectively axiomatized. This theorem states that for any consistent system

1052:) strengthened the incompleteness theorem by finding a variation of the proof ( 713:

Thus, although the Gödel sentence refers indirectly to sentences of the system

120: 5722: 4930: 4820:, 1936, "Extensions of some theorems of Gödel and Church", reprinted from the 4495: 2880:). A computer-verified proof of both incompleteness theorems was announced by 1703:

value can never be known or is ill-specified, is a controversial point in the

1672:. The second sense, which will not be discussed here, is used in relation to 8177: 8052: 7730: 7237: 7022: 7012: 6982: 6967: 6637: 6137: 6100: 6026: 5939: 5803: 5781: 5738: 5532: 5515: 5481: 4938: 4682: 4667: 4448: 4423: 4366:

God Created the Integers: The Mathematical Breakthroughs That Changed History

3701:

is a consistent formalized system which contains elementary arithmetic. Then

3455: 3429: 3100: 3070: 3050: 1482:), then it would itself be inconsistent. This reasoning can be formalized in 974: 442: 164: 138: 6122: 5433: 3310: 2803:

sketches an alternative proof of the first incompleteness theorem that uses

2191: 88:. The theorems are widely, but not universally, interpreted as showing that 81: 7952: 7799: 7700: 7692: 7572: 7520: 7429: 7365: 7348: 7279: 7138: 6997: 6699: 6482: 5519: 5415: 5168: 5123: 4954: 4895: 4177: 3604: 3270: 3246: 2808: 2331:

is the Gödel number of its proof. The relation between the Gödel number of

1776: 1746: 1665: 1320: 1008:

if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate

958:

is a formalized version of the analysis of the truth of the liar sentence.

930: 922: 762: 292: 4221: 8062: 7942: 7121: 7111: 7058: 6742: 6662: 6647: 6527: 6472: 6384: 6327: 5807: 5376:

Berto, Francesco (2009). "The Gödel Paradox and Wittgenstein's Reasons".

5196: 4844: 4802: 3487: 3207: 3197: 3040: 2812: 2116: 1699: 1639: 438: 409: 338: 304: 148: 104: 97: 5666: 3037:, which use a single system of formal logic to define their principles. 2165:, which can be converted into the number 120061121032061062032121061120. 1916:

proved that there is no algorithm that, given a multivariate polynomial

1749:

is undecidable, in the first sense of the term, in standard set theory.

6992: 6847: 6818: 6624: 5935: 5894: 5875: 5754: 5651:. Lecture Notes in Computer Science. Vol. 3603. pp. 245–260. 5489: 5148: 4887: 4675: 4591: 4522: 4260: 3277:. At the time, theories of natural numbers and real numbers similar to 3184: 2451:

would have a corresponding Gödel number, the existence of which causes

1912:

can be used to obtain a proof to Gödel's first incompleteness theorem.

1859:

The incompleteness theorem is closely related to several results about

1723: 180: 159:

Formal systems: completeness, consistency, and effective axiomatization

4957:(2005). "Kurt Gödel, paper on the incompleteness theorems (1931)". In 3269:

as his doctoral thesis in 1929, he turned to a second problem for his

2016:

gives a different method of producing independent sentences, based on

8144: 8047: 7100: 7017: 6977: 6941: 6877: 6689: 6679: 6652: 6415: 6359: 6295: 6073: 4962: 4921: 3838: 3021:

is consistent. And the 2nd Incompleteness Theorem statement follows.

2873: 2229: 1730:

can neither be proved nor refuted in ZF (which is all the ZFC axioms

187:

these systems, rather than about "provability" in an informal sense.

116: 5473: 4879: 4659: 4514: 1786: 1102:. The theorem first appeared as "Theorem XI" in Gödel's 1931 paper " 1028:, and yet the system also proves that there exists a natural number 8129: 7927: 7375: 7080: 6674: 5876:"Infinite Abelian groups, Whitehead problem and some constructions" 5713: 5657: 4907: 4787:"On the philosophical relevance of Gödel's incompleteness theorems" 4398:

From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931

3467: 3046: 2765:") which is false but provable, and the system is not ω-consistent. 2180:

as well. Crucially, because the system can support reasoning about

1984:, it is possible to effectively generate a multivariate polynomial 1518:

does not prove its consistency, it cannot prove the consistency of

5971:. Natick, Mass.: Association for Symbolic Logic (published 2001). 3474: 3284:

Gödel was not the only person working on the consistency problem.

2227:

is replaced by a specific number, the statement form turns into a

1767:

such that no specific number can be proved in that system to have

520:. The proof constructs a particular Gödel sentence for the system 341:

of ZFC, and a theory is consistent if and only if it has a model.

7725: 6517: 5201:

Infinity and the Mind: The Science and Philosophy of the Infinite

3614: 3473:

Multiple commentators have read Wittgenstein as misunderstanding

1336:

Gödel's second incompleteness theorem also implies that a system

73: 6199: 6038:

Kurt Gödel and the Foundations of Mathematics: Horizons of Truth

5131: 4706:"On Formalization of Model-Theoretic Proofs of Gödel's Theorems" 4309:

Translations, during his lifetime, of Gödel's paper into English

406:

the theory to encode not just addition but also multiplication.

5034:

Gödel's theorem : an incomplete guide to its use and abuse

4153: 3057:, which aimed to define the natural numbers in terms of logic. 2104: 1159: 777:

The first incompleteness theorem shows that the Gödel sentence

6013:

Lectures in Logic and Set Theory, Volume 1, Mathematical Logic

2586:

itself. This is similar to the following sentence in English:

1691:

Because of the two meanings of the word undecidable, the term

1684:

that correctly answers every question in the problem set (see

7269: 6615: 6460: 4101: 4025: 3822: 3494:

It is clear from the passages you cite that Wittgenstein did

2857: 2162: 2149: 1610:

by the above corollary of the second incompleteness theorem.

108: 93: 84:

in 1931, are important both in mathematical logic and in the

5238: 3929: 2935:

itself. This is equivalent to proving the statement "System

351:

Additional examples of inconsistent theories arise from the

5542:

Introduction to Automata Theory, Languages, and Computation

4654:(4 - Special Issue on Philosophy of Mathematics): 451–468. 4426:(1989). "A New Proof of the Gödel Incompleteness Theorem". 3307:

Second Conference on the Epistemology of the Exact Sciences

1759:

and proved another incompleteness theorem in that setting.

119:) is capable of proving all truths about the arithmetic of 6258:

Gödel's incompleteness theorems formalised in Isabelle/HOL

4696:." An English translation of Gödel's paper. Archived from 3878: 3150:, cites Gödel's theorems as an example of what he calls a 588:, no contradiction is presented by its provability within 6132: 4322:; "Gödel also complained about Braithwaite's commentary ( 3919: 3917: 3512:

provability predicate corresponds to actual provability.

3273:. His original goal was to obtain a positive solution to 2379:

is the Gödel number of a proof of the formula encoded by

2161:

is encoded as 120-061-121-032-061-062-032-121-061-120 in

2152:, which can be converted into the number 104101108108111. 1719: 592:. However, because the incompleteness theorem applies to 4843:

Smoryński, C. (1977). "The incompleteness theorems". In

3470:

express the belief that Wittgenstein misread his ideas.

3128:

rather than logical decidability. He writes that "when

2308:) is not relevant to the assignment of the Gödel number 1509:

is consistent. Since, by second incompleteness theorem,

1040:). That is, the system says that a number with property 980: 881:

of arithmetic, the Gödel sentence will be false in some

5326:

The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs

4915:(1). Springer Science and Business Media LLC: 211–259. 989:

Gödel demonstrated the incompleteness of the system of

662: 4851:. Amsterdam: North-Holland Pub. Co. pp. 821–866. 3914: 3208:

Appeals to the incompleteness theorems in other fields

3041:

Consequences for logicism and Hilbert's second problem

2993:

is consistent (ie. the statement in the hypothesis of

2895: 1855:

Halting problem § Gödel's incompleteness theorems

1331: 470:

first appeared as "Theorem VI" in Gödel's 1931 paper "

6167:

Paraconsistent Logic § Arithmetic and Gödel's Theorem

4346:, Dover Publications, New York (Dover edition 1992), 3902: 3866: 3854: 3810: 3741: 3707: 3648: 2414:" itself is not claimed to be part of this language. 2245: 2034:

Proof sketch for Gödel's first incompleteness theorem

2027: 1653: 1594:= primitive recursive arithmetic, the consistency of 912: 862:

is a canonical sentence asserting the consistency of

731: 463:

Proof sketch for Gödel's first incompleteness theorem

4835:, v. 4 (1939) pp. 53–60, in Martin Davis 1965, 4824:, v. 1 (1936) pp. 87–91, in Martin Davis 1965, 4273:. Oxford University Press, pp. 144–195. 4197: 4165: 4125: 4113: 4065: 4053: 4041: 4013: 3945: 3527: 3241:'s comments on the disparity between Gödel's avowed 2746:

If the system is ω-consistent, it can prove neither

2712:), but on the other hand, for every specific number 2677:

was constructed to be equivalent to the negation of

6031:"The Gödel Phenomena in Mathematics: A Modern View" 5499:

Handbook of practical logic and automated reasoning

5445:

Incompleteness: the Proof and Paradox of Kurt Gödel

5285:- puzzles based on undecidability in formal systems 4632:(1967). "Gödel's Theorem". In Edwards, Paul (ed.). 4296:, Oxford University Press, pp. 304–323. 4209: 4089: 3957: 3625:

Theory of everything#Gödel's incompleteness theorem

2989:Observe then, that if we can prove that the system 2815:. Boolos's proof proceeds by constructing, for any 2606:be the statement obtained in the previous section. 2300:. The choice of the free variable used in the form 362: 151:'s theorem that there is no algorithm to solve the 4964:Landmark Writings in Western Mathematics 1640-1940 4612:Landmark Writings in Western Mathematics 1640-1940 3890: 3775:(2009). Perspectives in Logic, ISBN 9780521884396. 3730: 3660: 3292:originally developed by Hilbert. Later that year, 2260: 1848: 1558:is consistent; no doubts about the consistency of 749: 6294: 5412:Logical dilemmas: The life and work of Kurt Gödel 5393:Logical dilemmas: The life and work of Kurt Gödel 4744:Transactions of the American Mathematical Society 4475:The Scope of Gödel's First Incompleteness Theorem 3778: 3753: 1787:Undecidable statements provable in larger systems 1166: 954:." The analysis of the truth and provability of 804:, pp. 28–33). For this reason, the sentence 8175: 5786:In Contradiction: A Study of the Transconsistent 4742:(1943). "Recursive predicates and quantifiers". 4606: 4159: 4147: 3458:during the period in which Wittgenstein's early 3374: 3091:Mechanism (philosophy) § Gödelian arguments 2184:, the results are equivalent to reasoning about 1646:. Gentzen's theorem spurred the development of 1621:itself can prove to be consistent. For example, 1059: 103:The first incompleteness theorem states that no 6210:World's shortest explanation of Gödel's theorem 5985: 5219:. Cambridge, U.K.: Cambridge University Press. 4704:Kikuchi, Makoto; Tanaka, Kazuyuki (July 1994). 4003: 3987: 3340:, our credo avers: We must know. We shall know! 3234: 3226: 2761:. In the later case, we have a statement ("not 2447:is also provable. This is because any proof of 2192:Construction of a statement about "provability" 2136:and then combined into a single larger number: 2113:"testing whether a number has a given property" 694:indirectly states its own unprovability within 456: 344:If one takes all statements in the language of 6218:about/including Gödel's Incompleteness theorem 6200:What is Mathematics:Gödel's Theorem and Around 5531: 4628: 4315: 4107: 4083: 3804: 3079:Modern viewpoints on the status of the problem 3024: 2178:numbers that represent formulae and statements 1956:will prove this. Moreover, suppose the system 1895: 1815:, but can be proved in the stronger system of 1478:is consistent (that is, that there is no such 772: 6431: 6280: 5934: 5608:Bulletin of the American Mathematical Society 4703: 4400:, Harvard University Press, Cambridge Mass., 4007: 3999: 3983: 3731:{\displaystyle F\not \vdash {\text{Cons}}(F)} 3250: 3230: 3222: 2099:would somehow have to contain a reference to 871: 497:which can neither be proved nor disproved in 5947:: Postmodern Intellectuals' Abuse of Science 5914:(2002). "Incompleteness and Inconsistency". 5829:A Concise Introduction to Mathematical Logic 5646: 5255:Metamathematics, machines, and Gödel's proof 5064:Gödel, Escher, Bach: An Eternal Golden Braid 4500: 4429:Notices of the American Mathematical Society 3800: 3182:and of inherently contradictory statements ( 2939:is consistent". Now consider the statement 2869: 2795: 2082: 1891: 1012:such that for every specific natural number 815:is often said to be "true but unprovable." ( 6049:A Logical Journey: From Gödel to Philosophy 5457: 5369: 5346:A Logical Journey: From Gödel to Philosophy 4990: 4784: 4767: 4638:. Vol. 3. Macmillan. pp. 348–357. 4227: 4187: 3979: 3694: 3508: 3478: 3249:uses to which his ideas are sometimes put. 3218: 2830: 2700:cannot be the Gödel number of the proof of 902: 816: 194: 23: 8184:Theorems in the foundations of mathematics 6623: 6438: 6424: 6287: 6273: 5963: 5559: 5163:Gödel´s Theorem: A Very Short Introduction 4576:"Formale Beweise und die Entscheidbarkeit" 3923: 3796: 3765: 3684:Continuum hypothesis#Independence from ZFC 2289:can be assigned a Gödel number denoted by 2186:provability of their equivalent statements 1887: 1775:. While Gödel's theorem is related to the 1399:is in fact consistent. For the claim that 357:axiom schema of unrestricted comprehension 24: 6072: 6025: 5893: 5816:Reprinted in Anderson, A. R., ed., 1964. 5712: 5656: 5501:. Cambridge: Cambridge University Press. 4920: 4842: 4721: 3908: 3844: 3451:Remarks on the Foundations of Mathematics 3121: 2598:Now, assume that the axiomatic system is 1999: 1988:over the integers such that the equation 1570:that is in some sense less doubtful than 1130: 1123:incompleteness theorem within the system 867: 797: 92:to find a complete and consistent set of 5496: 5165:. Oxford University Press, Oxford, 2022. 3503:Since the publication of Wittgenstein's 2971:is not provable", (or identically, "not 2955:is not provable". The proof of sentence 2877: 2742:is constructed for a particular system: 2628:would be provable, as argued above. But 2211:that contains exactly one free variable 2055:(which can be applied to any statement " 1578:. For many naturally occurring theories 1408:is consistent has form "for all numbers 1197:represent the Gödel number of a formula 683:itself. In this way, the Gödel sentence 59:Limitative results in mathematical logic 32:This is an accepted version of this page 6212:using a printing machine as an example. 5910: 5691: 5252: 5028: 4865: 4642: 4573: 4095: 3975: 3963: 3884: 3816: 3747: 3654: 3412: 3396: 3265:After Gödel published his proof of the 3214: 3201: 3173: 2889: 2861: 2070:" (so-called because of its origins as 1901: 1150:Other formalizations of the claim that 906: 890: 801: 766: 584:states only that it is not provable in 558:also cannot be complete. In this case, 413: 137:on the formal undefinability of truth, 14: 8176: 6445: 5870: 5842: 5802: 5780: 5766:. Psychology Press. pp. 207–227. 5761: 5741:(1984). "Logic of Paradox Revisited". 5737: 5649:Theorem Proving in Higher Order Logics 5626: 5409: 5390: 4738: 4447: 4439: 4422: 4323: 4255:, v. 38 n. 1, pp. 173–198. 4231: 4191: 4183: 4171: 4143: 4131: 4119: 4071: 4059: 4047: 4019: 3951: 3939: 3935: 3896: 3872: 3860: 3848: 3784: 3759: 3513: 3482: 3193: 3189: 3114: 3029:The incompleteness results affect the 2902:Hilbert–Bernays provability conditions 2800: 1870: 1738: 1187:Hilbert–Bernays provability conditions 1049: 596:, there will be a new Gödel statement 475: 318:" proves ZFC is consistent because if 6419: 6268: 5598: 5375: 5212: 4772:. Stanford Encyclopedia of Philosophy 4761: 4550: 4529: 4416: 4331: 4253:Monatshefte für Mathematik und Physik 4235: 4203: 3832: 3828: 3773:Subsystems of Second-Order Arithmetic 3666: 3517: 3346:Wir müssen wissen. Wir werden wissen! 3084: 3017:. So we cannot prove that the system 2931:can be proved from within the system 2872:) and by John Harrison in 2009 using 2864:), by Russell O'Connor in 2003 using 2692:). However, for each specific number 2375:is the Gödel number of a formula and 2148:is encoded as 104-101-108-108-111 in 1718:can neither be proved nor refuted in 1676:and applies not to statements but to 1660:List of statements independent of ZFC 1154:is consistent may be inequivalent in 1092:Gödel's second incompleteness theorem 981:Extensions of Gödel's original result 722: 699: 430: 390:, which is essentially equivalent to 163:The incompleteness theorems apply to 6062: 6015:, Cambridge University Press, 2003. 5340: 4953: 4894: 4294:Kurt Gödel Collected works, Vol. III 4245: 4215: 4035: 4031: 2959:can be formalized within the system 2417:An important feature of the formula 1819:. Kirby and Paris later showed that 1554:would give us no clue as to whether 1167:The Hilbert–Bernays conditions 663:Syntactic form of the Gödel sentence 468:Gödel's first incompleteness theorem 423: 6172:Stanford Encyclopedia of Philosophy 6159:Stanford Encyclopedia of Philosophy 6147:Stanford Encyclopedia of Philosophy 5764:Wittgenstein's lasting significance 5601:"Undecidable Diophantine Equations" 5565:"Chapter 12. On Downward Causality" 5216:An introduction to Gödel's Theorems 5175:, Douglas Hofstadter, 2002 (1958). 4791:Revue Internationale de Philosophie 3359:Gödel's paper was published in the 2896:Proof sketch for the second theorem 2134:sequence of numbers for each letter 1755:produced undecidable statements in 1332:Implications for consistency proofs 111:whose theorems can be listed by an 56: 18:Gödel's incompleteness theorem 6156:entry by Panu Raatikainen in the 5866:10.1111/j.1746-8361.2003.tb00272.x 5857:10.1111/j.1746-8361.2003.tb00272.x 5545:. Reading, Mass.: Addison-Wesley. 5304:Diagonalization and Self-Reference 4710:Notre Dame Journal of Formal Logic 4271:Kurt Gödel Collected works, Vol. I 3330:For the mathematician there is no 3095:Authors including the philosopher 2466: 2028:Proof sketch for the first theorem 1811:, is undecidable in (first-order) 1654:Examples of undecidable statements 1590:= Zermelo–Fraenkel set theory and 1451:is the code of a contradiction in 1201:, the provability conditions say: 913:Relationship with the liar paradox 733: 573:, because it is an axiom. Because 279:is a statement in the language of 57: 8225: 6154:"Gödel's Incompleteness Theorems" 6116: 5806:(1960). "Minds and Machines". In 5009:. Bern: Peter Lang. 142 S. 1990. 4770:"Gödel's Incompleteness Theorems" 4756:10.1090/S0002-9947-1943-0007371-8 2176:, it can support reasoning about 1779:, Chaitin's result is related to 1617:, only doing so in a theory that 1574:itself, for example, weaker than 535: that contains the whole of 322:is the least such cardinal, then 171:, formal systems are also called 8157: 6083:10.1016/b978-044451541-4/50014-2 5155:. Clarendon Press, Oxford, 1970. 4973:10.1016/b978-044450871-3/50152-2 4490:, v. 8, pp. 499–552. 4396:editor, 1967, 3rd edition 1967. 3600:Non-standard model of arithmetic 3561:Chaitin's incompleteness theorem 3544: 3530: 2014:Chaitin's incompleteness theorem 1837:. The related but more general 1805:Paris–Harrington principle 1761:Chaitin's incompleteness theorem 1722:(the standard axiomatization of 1606:cannot prove the consistency of 1434:were in fact inconsistent, then 1363:. This is because such a system 1139:as a formula in the language of 1072:) expressing the consistency of 883:nonstandard models of arithmetic 788:of an appropriate formal theory 489:: "Any consistent formal system 363:Systems which contain arithmetic 6355:Gödel's incompleteness theorems 6123:Godel's Incompleteness Theorems 6051:, The MIT Press, Cambridge MA, 5812:Dimensions of Mind: A Symposium 5620:10.1090/S0273-0979-1980-14832-6 5290:Godel's Incompleteness Theorems 5145:, Lecture Notes in Logic v. 10. 4368:, Running Press, Philadelphia, 3620:Tarski's undefinability theorem 3440: 3435:Tarski's undefinability theorem 3316: 3303:Tarski's indefinability theorem 2982:") can be proved in the system 2432:is provable in the system then 2327:, one may ask whether a number 1849:Relationship with computability 1843:computational complexity theory 1710:The combined work of Gödel and 1354:that proves the consistency of 971:Tarski's undefinability theorem 834:, which proves the implication 226: 135:Tarski's undefinability theorem 70:Gödel's incompleteness theorems 6223:"Gödel incompleteness theorem" 5743:Journal of Philosophical Logic 5447:, W. W. Norton & Company. 4967:. Elsevier. pp. 917–925. 4849:Handbook of mathematical logic 3725: 3719: 3688: 3672: 3464:Tractatus Logico-Philosophicus 2963:, and therefore the statement 2255: 2249: 1757:algorithmic information theory 1533:primitive recursive arithmetic 950:is not provable in the system 885:, as a consequence of Gödel's 832:primitive recursive arithmetic 298: 199:A formal system is said to be 13: 1: 8118:History of mathematical logic 6216:October 2011 RadioLab episode 6040:. Cambridge University Press. 5881:Israel Journal of Mathematics 5036:. Wellesley, MA: A K Peters. 4828:(loc. cit.) pp. 230–235. 4338:B. Meltzer (translation) and 3636: 3402: 3375:Generalization and acceptance 3235:Stangroom & Benson (2006) 3227:Stangroom & Benson (2006) 1060:Second incompleteness theorem 8043:Primitive recursive function 6350:Gödel's completeness theorem 5814:. New York University Press. 5414:. Wellesley, Massachusetts: 5410:Dawson, John W. Jr. (1997). 5391:Dawson, John W. Jr. (1996). 4839:(loc. cit.) pp. 223–230 4556:"The Incompleteness Theorem" 3641: 3578:Gödel's completeness theorem 2997:), then we have proved that 2849:intended for human readers. 2484:such that the system proves 2004:recursively inseparable sets 1896:Hopcroft & Ullman (1979) 1841:(2003) has consequences for 1807:, a version of the infinite 750:{\displaystyle \Pi _{1}^{0}} 487:First Incompleteness Theorem 457:First incompleteness theorem 303:A set of axioms is (simply) 207:) if its set of theorems is 64:Gödel's completeness theorem 7: 6228:Encyclopedia of Mathematics 5788:. Oxford University Press. 5128:The drama of the quantities 4004:Stangroom & Benson 2006 3988:Stangroom & Benson 2006 3630:Typographical Number Theory 3523: 3251:Sokal & Bricmont (1999) 3231:Sokal & Bricmont (1999) 3223:Sokal & Bricmont (1999) 3033:, particularly versions of 3025:Discussion and implications 2673:would be provable (because 2002:shows how the existence of 1644:Gentzen's consistency proof 773:Truth of the Gödel sentence 379:algebraically closed fields 213:Zermelo–Fraenkel set theory 10: 8230: 7107:Schröder–Bernstein theorem 6834:Monadic predicate calculus 6493:Foundations of mathematics 6338:Foundations of mathematics 5636:Reprinted by Dover, 2002. 5000:John Wiley and Sons. 2010. 4785:Raatikainen, Panu (2005). 4768:Raatikainen, Panu (2020). 4635:Encyclopedia of Philosophy 3805:Hopcroft & Ullman 1979 3423: 3407: 3354:Monatshefte für Mathematik 3260: 3237:, for example, quote from 3088: 2899: 2834: 2738:is not provable. Thus, if 2561:is not literally equal to 2072:Cantor's diagonal argument 2053:is provable in the system" 2031: 1852: 1657: 1381:proves the consistency of 1098:) will not be provable in 460: 359:is assumed in set theory. 286:The theory of first-order 61: 8153: 8140:Philosophy of mathematics 8089:Automated theorem proving 8071: 7966: 7798: 7691: 7543: 7260: 7236: 7214:Von Neumann–Bernays–Gödel 7159: 7053: 6957: 6855: 6846: 6773: 6708: 6614: 6536: 6453: 6306: 5723:10.1017/S1755020314000112 5525:Grundlagen der Mathematik 5461:The Journal of Philosophy 5203:. Princeton Univ. Press. 5142:Aspects of Incompleteness 4868:Journal of Symbolic Logic 4833:Journal of Symbolic Logic 4822:Journal of Symbolic Logic 4580:Mathematische Zeitschrift 4496:10.1007/s11787-014-0107-3 4320:Journal of Symbolic Logic 4086:, page 328, footnote 68a. 4008:Sokal & Bricmont 1999 4000:Sokal & Bricmont 1999 3984:Sokal & Bricmont 1999 3594:Minds, Machines and Gödel 3509:Floyd & Putnam (2000) 3479:Floyd & Putnam (2000) 3460:ideal language philosophy 3393:Grundlagen der Mathematik 3031:philosophy of mathematics 2910:using a formal predicate 2837:Automated theorem proving 2796:Proof via Berry's paradox 2750:nor its negation, and so 2083:Arithmetization of syntax 1705:philosophy of mathematics 917:Gödel specifically cites 872:Kikuchi & Tanaka 1994 505:The unprovable statement 223:incompleteness theorems. 86:philosophy of mathematics 6380:Löwenheim–Skolem theorem 5928:10.1093/mind/111.444.817 5701:Review of Symbolic Logic 5599:Jones, James P. (1980). 5395:. Taylor & Francis. 5370:Miscellaneous references 5003:Norbert Domeisen, 1990. 4991:Books about the theorems 4723:10.1305/ndjfl/1040511346 4457:Harvard University Press 3583:Gödel's speed-up theorem 3275:Hilbert's second problem 3126:computational complexity 2831:Computer verified proofs 2632:asserts the negation of 2525:, we obtain the theorem 800:, p. 825; also see 195:Effective axiomatization 141:'s proof that Hilbert's 39:latest accepted revision 7790:Self-verifying theories 7611:Tarski's axiomatization 6562:Tarski's undefinability 6557:incompleteness theorems 6405:Use–mention distinction 6240:How Gödel's Proof Works 5379:Philosophia Mathematica 5329:. Courier Corporation. 5153:The Freedom of the Will 4931:10.1023/a:1026247421383 4453:Logic, logic, and logic 4228:Floyd & Putnam 2000 4188:Floyd & Putnam 2000 3279:second-order arithmetic 2428:is that if a statement 2091:that is equivalent to " 2008:essentially undecidable 1906:Matiyasevich's solution 1817:second-order arithmetic 1064:For each formal system 879:intended interpretation 569:is indeed a theorem in 418:self-verifying theories 201:effectively axiomatized 8164:Mathematics portal 7775:Proof of impossibility 7423:propositional variable 6733:Propositional calculus 6400:Type–token distinction 6206:. An online free book. 6177:MacTutor biographies: 5832:, 3rd. ed., Springer, 5561:Hofstadter, Douglas R. 5468:(11). JSTOR: 624–632. 5306:. Oxford Univ. Press. 5292:. Oxford Univ. Press. 4959:Grattan-Guinness, Ivor 4692:Martin Hirzel, 2000, " 4687:Mathematical Problems. 4608:Grattan-Guinness, Ivor 4574:Finsler, Paul (1926). 4530:Davis, Martin (1965). 4342:(Introduction), 1962. 3732: 3501: 3383:Gentzen published his 3356:on November 17, 1930. 3342: 3309:, a key conference in 3171: 3161: 2730:In fact, to show that 2262: 2155:The logical statement 2040:proof by contradiction 1910:Hilbert's 10th problem 1827:Kruskal's tree theorem 1131:Expressing consistency 759:arithmetical hierarchy 751: 503: 435:principle of explosion 209:recursively enumerable 8033:Kolmogorov complexity 7986:Computably enumerable 7886:Model complete theory 7678:Principia Mathematica 6738:Propositional formula 6567:Banach–Tarski paradox 5965:Shoenfield, Joseph R. 5497:Harrison, J. (2009). 5213:Smith, Peter (2007). 4160:Grattan-Guinness 2005 4148:Grattan-Guinness 2005 3733: 3492: 3328: 3323:Königsberg conference 3166: 3156: 3118:by a Turing machine. 2817:computably enumerable 2480:there is a statement 2263: 2182:properties of numbers 2111:can be replaced with 2109:"proving a statement" 2107:) in such a way that 2018:Kolmogorov complexity 1769:Kolmogorov complexity 1443:would prove for some 1046:J. Barkley Rosser 992:Principia Mathematica 752: 501:." (Raatikainen 2020) 484: 403:Presburger arithmetic 355:that result when the 316:inaccessible cardinal 205:effectively generated 7981:Church–Turing thesis 7968:Computability theory 7177:continuum hypothesis 6695:Square of opposition 6553:Gödel's completeness 6323:Church–Turing thesis 6317:Entscheidungsproblem 6162:, November 11, 2013. 5945:Fashionable Nonsense 5820:. Prentice-Hall: 77. 5628:Kleene, Stephen Cole 5253:Shankar, N. (1994). 5159:Adrian William Moore 5006:Logik der Antinomien 4803:10.3917/rip.234.0513 4630:van Heijenoort, Jean 4503:Mathematics Magazine 4473:Bernd Buldt, 2014, " 3705: 3678:in technical terms: 3313:the following week. 3267:completeness theorem 3180:paraconsistent logic 3174:Paraconsistent logic 3109:Church–Turing thesis 2951:is consistent, then 2658:were provable, then 2651:cannot be provable. 2613:were provable, then 2261:{\displaystyle F(n)} 2243: 2122:Entscheidungsproblem 1972:has a solution" or " 1716:continuum hypothesis 1674:computability theory 1500:is consistent, then 1319: (analogue of 1237:proves 1.; that is, 1118:cannot be proved in 887:completeness theorem 729: 616:is also incomplete. 335:von Neumann universe 277:continuum hypothesis 250:completeness theorem 231:A set of axioms is ( 218:The theory known as 144:Entscheidungsproblem 8214:Works by Kurt Gödel 8135:Mathematical object 8026:P versus NP problem 7991:Computable function 7785:Reverse mathematics 7711:Logical consequence 7588:primitive recursive 7583:elementary function 7356:Free/bound variable 7209:Tarski–Grothendieck 6728:Logical connectives 6658:Logical equivalence 6508:Logical consequence 6065:Philosophy of logic 5824:Wolfgang Rautenberg 5667:10.1007/11541868_16 5570:I Am a Strange Loop 5095:I Am a Strange Loop 4818:John Barkley Rosser 4442:, pp. 383–388) 4412:mimeographed notes. 4394:Jean van Heijenoort 4316:van Heijenoort 1967 4283:Stephen Cole Kleene 4150:, pp. 512–513. 4108:van Heijenoort 1967 4084:van Heijenoort 1967 3887:, pp. 451–468. 3567:Gödel, Escher, Bach 3446:Ludwig Wittgenstein 3428:In September 1931, 3147:I Am a Strange Loop 3141:Gödel, Escher, Bach 2723:Thus the statement 2654:If the negation of 2514:be the negation of 2119:in his work on the 2095:cannot be proved", 1892:Charlesworth (1981) 1839:graph minor theorem 1821:Goodstein's theorem 1686:undecidable problem 1682:computable function 746: 719:primitive recursive 655:, rather than 369:Robinson arithmetic 333:sitting inside the 273:dense linear orders 147:is unsolvable, and 113:effective procedure 29:Page version status 8189:Mathematical logic 7933:Transfer principle 7896:Semantics of logic 7881:Categorical theory 7857:Non-standard model 7371:Logical connective 6498:Information theory 6447:Mathematical logic 6186:2005-10-13 at the 6011:George Tourlakis, 5969:Mathematical logic 5895:10.1007/BF02757281 5818:Minds and Machines 5755:10.1007/BF00453020 5632:Mathematical Logic 5528:, Springer-Verlag. 5059:Douglas Hofstadter 4764:, pp. 255–287 4592:10.1007/bf01283861 4563:Notices of the AMS 4487:Logica Universalis 4480:2016-03-06 at the 4417:Articles by others 4261:10.1007/BF01700692 3728: 3552:Mathematics portal 3415:used a version of 3219:Raatikainen (2005) 3136:Douglas Hofstadter 3085:Minds and machines 2258: 2174:numbers in general 1881:as input, whether 1372:can prove that if 1328:Peano arithmetic. 1016:the system proves 747: 732: 396:Euclidean geometry 388:real closed fields 269:parallel postulate 265:Euclidean geometry 78:mathematical logic 35: 8171: 8170: 8103:Abstract category 7906:Theories of truth 7716:Rule of inference 7706:Natural deduction 7687: 7686: 7232: 7231: 6937:Cartesian product 6842: 6841: 6748:Many-valued logic 6723:Boolean functions 6606:Russell's paradox 6581:diagonal argument 6478:First-order logic 6413: 6412: 6244:Natalie Wolchover 6092:978-0-444-51541-4 6021:978-0-521-75373-9 5995:Why Truth Matters 5987:Stangroom, Jeremy 5978:978-1-56881-135-2 5838:978-1-4419-1220-6 5773:978-1-134-40617-3 5693:Paulson, Lawrence 5676:978-3-540-28372-0 5580:978-0-465-03078-1 5533:Hopcroft, John E. 5441:Rebecca Goldstein 5425:978-1-56881-256-4 5402:978-1-56881-025-6 5335:978-0-486-49705-1 5276:Forever Undecided 5226:978-0-521-67453-9 5104:978-0-465-03078-1 5067:. Vintage Books. 4996:Francesco Berto. 4858:978-0-444-86388-1 4700:. Sept. 16, 2004. 4644:Hellman, Geoffrey 4543:978-0-911216-01-1 4340:R. B. Braithwaite 4246:Articles by Gödel 3801:Charlesworth 1981 3717: 3610:Provability logic 3538:Philosophy portal 3417:Richard's paradox 3385:consistency proof 3239:Rebecca Goldstein 2947:= "If the system 2854:Natarajan Shankar 2720:is not provable. 2463:to be satisfied. 2059:" in the system). 1888:Shoenfield (1967) 1743:Whitehead problem 1678:decision problems 1650:in proof theory. 1537:Hilbert's program 1469:also proved that 919:Richard's paradox 792:is unprovable in 628:will differ from 424:Conflicting goals 169:first-order logic 131:diagonal argument 105:consistent system 90:Hilbert's program 26: 16:(Redirected from 8221: 8162: 8161: 8113:History of logic 8108:Category of sets 8001:Decision problem 7780:Ordinal analysis 7721:Sequent calculus 7619:Boolean algebras 7559: 7558: 7533: 7504:logical/constant 7258: 7257: 7244: 7167:Zermelo–Fraenkel 6918:Set operations: 6853: 6852: 6790: 6621: 6620: 6601:Löwenheim–Skolem 6488:Formal semantics 6440: 6433: 6426: 6417: 6416: 6333:Effective method 6311:Cantor's theorem 6289: 6282: 6275: 6266: 6265: 6250:, July 14, 2020. 6236: 6193:Gerhard Gentzen. 6142:Juliette Kennedy 6112: 6076: 6041: 6035: 6008: 5982: 5960: 5931: 5912:Shapiro, Stewart 5907: 5897: 5860: 5815: 5799: 5777: 5758: 5734: 5716: 5688: 5660: 5635: 5623: 5605: 5595: 5593: 5592: 5583:. Archived from 5556: 5512: 5493: 5437: 5406: 5387: 5359: 5272:Raymond Smullyan 5268: 5249: 5247: 5246: 5237:. Archived from 5173:James Roy Newman 5132:Real View Books. 5075:. 1999 reprint: 5055: 4986: 4950: 4924: 4904: 4891: 4862: 4814: 4781: 4779: 4777: 4759: 4735: 4725: 4679: 4639: 4625: 4603: 4570: 4560: 4547: 4526: 4470: 4444: 4436:: 388–390, 676. 4290:Solomon Feferman 4267:Solomon Feferman 4239: 4225: 4219: 4213: 4207: 4201: 4195: 4181: 4175: 4169: 4163: 4157: 4151: 4141: 4135: 4129: 4123: 4117: 4111: 4105: 4099: 4093: 4087: 4081: 4075: 4069: 4063: 4057: 4051: 4045: 4039: 4029: 4023: 4017: 4011: 3997: 3991: 3980:Raatikainen 2005 3973: 3967: 3961: 3955: 3949: 3943: 3933: 3927: 3921: 3912: 3906: 3900: 3894: 3888: 3882: 3876: 3870: 3864: 3858: 3852: 3842: 3836: 3826: 3820: 3814: 3808: 3794: 3788: 3782: 3776: 3769: 3763: 3757: 3751: 3745: 3739: 3737: 3735: 3734: 3729: 3718: 3715: 3700: 3697: : "Assume 3695:Raatikainen 2020 3692: 3686: 3676: 3670: 3664: 3658: 3652: 3554: 3549: 3548: 3540: 3535: 3534: 3533: 3188:). Priest ( 3122:Wigderson (2010) 3055:Bertrand Russell 3020: 3016: 3012: 3008: 3004: 3000: 2996: 2992: 2985: 2981: 2970: 2966: 2962: 2958: 2954: 2950: 2946: 2942: 2938: 2934: 2930: 2926: 2919: 2915: 2909: 2882:Lawrence Paulson 2847:natural language 2826: 2822: 2807:rather than the 2791: 2787: 2776: 2772: 2764: 2760: 2753: 2749: 2741: 2737: 2733: 2726: 2719: 2715: 2711: 2707: 2703: 2699: 2695: 2691: 2676: 2672: 2657: 2650: 2646: 2631: 2627: 2612: 2605: 2585: 2580: 2576: 2560: 2553: 2546: 2524: 2513: 2505: 2483: 2479: 2462: 2450: 2446: 2431: 2427: 2413: 2409: 2405: 2382: 2378: 2374: 2370: 2353: 2349: 2338: 2334: 2330: 2326: 2318: 2307: 2303: 2299: 2288: 2274: 2269: 2267: 2265: 2264: 2259: 2236: 2226: 2214: 2210: 2159: 2146: 2102: 2098: 2094: 2090: 2064:self-referential 2058: 2052: 2000:Smoryński (1977) 1995: 1991: 1987: 1983: 1979: 1975: 1971: 1967: 1963: 1959: 1955: 1951: 1947: 1943: 1884: 1880: 1865:recursion theory 1861:undecidable sets 1813:Peano arithmetic 1803:proved that the 1774: 1766: 1741:showed that the 1670:deductive system 1648:ordinal analysis 1637: 1620: 1616: 1609: 1605: 1601: 1597: 1593: 1589: 1585: 1581: 1577: 1573: 1569: 1565: 1561: 1557: 1553: 1549: 1545: 1526: 1517: 1508: 1499: 1491:to show that if 1490: 1481: 1477: 1468: 1459: 1450: 1446: 1442: 1433: 1424: 1415: 1411: 1407: 1398: 1389: 1380: 1371: 1362: 1353: 1344: 1318: 1278: 1272: 1240: 1236: 1230: 1216: 1212: 1208: 1200: 1196: 1184: 1157: 1153: 1146: 1142: 1138: 1126: 1109: 1101: 1097: 1087: 1083: 1079: 1075: 1071: 1067: 1043: 1039: 1035: 1031: 1027: 1015: 1011: 964: 957: 953: 949: 945: 941: 937: 909:, p. 135). 903:Raatikainen 2020 900: 896: 874:, p. 403). 865: 861: 850: 829: 817:Raatikainen 2020 814: 795: 791: 787: 756: 754: 753: 748: 745: 740: 716: 702:, p. 135). 697: 693: 682: 671: 658: 654: 650: 638: 627: 615: 611: 607: 595: 591: 587: 583: 572: 568: 557: 553: 549: 538: 534: 523: 519: 515: 500: 496: 492: 373: 346:Peano arithmetic 332: 330: 321: 288:Peano arithmetic 177:Peano arithmetic 21: 8229: 8228: 8224: 8223: 8222: 8220: 8219: 8218: 8174: 8173: 8172: 8167: 8156: 8149: 8094:Category theory 8084:Algebraic logic 8067: 8038:Lambda calculus 7976:Church encoding 7962: 7938:Truth predicate 7794: 7760:Complete theory 7683: 7552: 7548: 7544: 7539: 7531: 7251: and 7247: 7242: 7228: 7204:New Foundations 7172:axiom of choice 7155: 7117:Gödel numbering 7057: and 7049: 6953: 6838: 6788: 6769: 6718:Boolean algebra 6704: 6668:Equiconsistency 6633:Classical logic 6610: 6591:Halting problem 6579: and 6555: and 6543: and 6542: 6537:Theorems ( 6532: 6449: 6444: 6414: 6409: 6302: 6300:metamathematics 6293: 6248:Quanta Magazine 6221: 6204:Karlis Podnieks 6188:Wayback Machine 6150:, July 5, 2011. 6119: 6093: 6033: 6005: 5991:Benson, Ophelia 5979: 5957: 5922:(444): 817–32. 5872:Shelah, Saharon 5796: 5774: 5677: 5603: 5590: 5588: 5581: 5573:. Basic Books. 5553: 5537:Ullman, Jeffrey 5509: 5474:10.2307/2678455 5426: 5403: 5372: 5356: 5265: 5244: 5242: 5227: 5199:, 1995 (1982). 5179:, revised ed. 5098:. Basic Books. 5092: —, 2007. 5044: 5030:Franzén, Torkel 4993: 4983: 4902: 4880:10.2307/2695030 4859: 4837:The Undecidable 4826:The Undecidable 4775: 4773: 4660:10.2307/2214847 4622: 4558: 4544: 4536:. Raven Press. 4515:10.2307/2689794 4482:Wayback Machine 4467: 4459:. p. 443. 4419: 4362:Stephen Hawking 4328:The Undecidable 4311: 4248: 4243: 4242: 4226: 4222: 4214: 4210: 4202: 4198: 4182: 4178: 4170: 4166: 4162:, pp. 513. 4158: 4154: 4142: 4138: 4130: 4126: 4118: 4114: 4106: 4102: 4094: 4090: 4082: 4078: 4070: 4066: 4058: 4054: 4046: 4042: 4034:, p. 418; 4030: 4026: 4018: 4014: 3998: 3994: 3974: 3970: 3962: 3958: 3950: 3946: 3934: 3930: 3924:Hofstadter 2007 3922: 3915: 3907: 3903: 3895: 3891: 3883: 3879: 3871: 3867: 3859: 3855: 3847:, p. 842; 3843: 3839: 3831:, p. 416; 3827: 3823: 3815: 3811: 3799:, p. 132; 3797:Shoenfield 1967 3795: 3791: 3783: 3779: 3771:S. G. Simpson, 3770: 3766: 3758: 3754: 3746: 3742: 3714: 3706: 3703: 3702: 3698: 3693: 3689: 3677: 3673: 3665: 3661: 3653: 3649: 3644: 3639: 3634: 3550: 3543: 3536: 3531: 3529: 3526: 3443: 3426: 3410: 3405: 3377: 3319: 3263: 3210: 3176: 3138:, in his books 3093: 3087: 3043: 3027: 3018: 3014: 3010: 3006: 3002: 2998: 2994: 2990: 2983: 2972: 2968: 2964: 2960: 2956: 2952: 2948: 2944: 2940: 2936: 2932: 2928: 2924: 2917: 2911: 2907: 2904: 2898: 2843:proof assistant 2839: 2833: 2824: 2820: 2805:Berry's paradox 2798: 2789: 2778: 2774: 2770: 2762: 2758: 2754:is undecidable. 2751: 2747: 2739: 2735: 2731: 2724: 2717: 2713: 2709: 2705: 2701: 2697: 2693: 2678: 2674: 2659: 2655: 2648: 2633: 2629: 2614: 2610: 2603: 2583: 2578: 2562: 2558: 2551: 2529: 2515: 2511: 2488: 2481: 2477: 2469: 2467:Diagonalization 2452: 2448: 2433: 2429: 2418: 2411: 2407: 2396: 2380: 2376: 2372: 2358: 2351: 2340: 2336: 2332: 2328: 2324: 2309: 2305: 2301: 2290: 2279: 2272: 2244: 2241: 2240: 2238: 2234: 2224: 2212: 2201: 2194: 2157: 2144: 2100: 2096: 2092: 2088: 2085: 2068:diagonalization 2056: 2050: 2036: 2030: 1993: 1989: 1985: 1981: 1977: 1973: 1969: 1965: 1961: 1957: 1953: 1949: 1945: 1941: 1934: 1927: 1917: 1882: 1878: 1875:halting problem 1857: 1851: 1832: 1789: 1781:Berry's paradox 1772: 1764: 1753:Gregory Chaitin 1728:axiom of choice 1666:proof-theoretic 1662: 1656: 1636: 1630: 1623:Gerhard Gentzen 1618: 1614: 1607: 1603: 1599: 1598:is provable in 1595: 1591: 1587: 1583: 1579: 1575: 1571: 1567: 1566:in some system 1563: 1559: 1555: 1551: 1547: 1543: 1525: 1519: 1516: 1510: 1507: 1501: 1498: 1492: 1489: 1483: 1479: 1476: 1470: 1467: 1461: 1458: 1452: 1448: 1444: 1441: 1435: 1432: 1426: 1423: 1417: 1413: 1409: 1406: 1400: 1397: 1391: 1388: 1382: 1379: 1373: 1370: 1364: 1361: 1355: 1352: 1346: 1343: 1337: 1334: 1312: 1301: 1286: 1280: 1276: 1266: 1259: 1248: 1242: 1238: 1234: 1224: 1218: 1214: 1210: 1206: 1198: 1190: 1178: 1172: 1169: 1155: 1151: 1144: 1140: 1136: 1133: 1124: 1107: 1099: 1095: 1085: 1081: 1077: 1073: 1069: 1065: 1062: 1041: 1037: 1033: 1029: 1017: 1013: 1009: 983: 962: 955: 951: 947: 943: 939: 935: 915: 898: 894: 870:, p. 840, 863: 852: 849: 835: 828: 820: 813: 805: 793: 789: 786: 778: 775: 769:, p. 71). 741: 736: 730: 727: 726: 714: 695: 692: 684: 681: 673: 669: 665: 656: 652: 649: 640: 637: 629: 626: 617: 613: 612:, showing that 609: 606: 597: 593: 589: 585: 582: 574: 570: 567: 559: 555: 551: 548: 540: 536: 532: 526:logically valid 521: 517: 514: 506: 498: 494: 490: 465: 459: 426: 410:Dan Willard 392:Tarski's axioms 371: 365: 331: 328: 323: 319: 301: 229: 220:true arithmetic 197: 173:formal theories 161: 153:halting problem 121:natural numbers 100:is impossible. 67: 60: 55: 54: 53: 52: 51: 50: 34: 22: 15: 12: 11: 5: 8227: 8217: 8216: 8211: 8206: 8201: 8196: 8191: 8186: 8169: 8168: 8154: 8151: 8150: 8148: 8147: 8142: 8137: 8132: 8127: 8126: 8125: 8115: 8110: 8105: 8096: 8091: 8086: 8081: 8079:Abstract logic 8075: 8073: 8069: 8068: 8066: 8065: 8060: 8058:Turing machine 8055: 8050: 8045: 8040: 8035: 8030: 8029: 8028: 8023: 8018: 8013: 8008: 7998: 7996:Computable set 7993: 7988: 7983: 7978: 7972: 7970: 7964: 7963: 7961: 7960: 7955: 7950: 7945: 7940: 7935: 7930: 7925: 7924: 7923: 7918: 7913: 7903: 7898: 7893: 7891:Satisfiability 7888: 7883: 7878: 7877: 7876: 7866: 7865: 7864: 7854: 7853: 7852: 7847: 7842: 7837: 7832: 7822: 7821: 7820: 7815: 7808:Interpretation 7804: 7802: 7796: 7795: 7793: 7792: 7787: 7782: 7777: 7772: 7762: 7757: 7756: 7755: 7754: 7753: 7743: 7738: 7728: 7723: 7718: 7713: 7708: 7703: 7697: 7695: 7689: 7688: 7685: 7684: 7682: 7681: 7673: 7672: 7671: 7670: 7665: 7664: 7663: 7658: 7653: 7633: 7632: 7631: 7629:minimal axioms 7626: 7615: 7614: 7613: 7602: 7601: 7600: 7595: 7590: 7585: 7580: 7575: 7562: 7560: 7541: 7540: 7538: 7537: 7536: 7535: 7523: 7518: 7517: 7516: 7511: 7506: 7501: 7491: 7486: 7481: 7476: 7475: 7474: 7469: 7459: 7458: 7457: 7452: 7447: 7442: 7432: 7427: 7426: 7425: 7420: 7415: 7405: 7404: 7403: 7398: 7393: 7388: 7383: 7378: 7368: 7363: 7358: 7353: 7352: 7351: 7346: 7341: 7336: 7326: 7321: 7319:Formation rule 7316: 7311: 7310: 7309: 7304: 7294: 7293: 7292: 7282: 7277: 7272: 7267: 7261: 7255: 7238:Formal systems 7234: 7233: 7230: 7229: 7227: 7226: 7221: 7216: 7211: 7206: 7201: 7196: 7191: 7186: 7181: 7180: 7179: 7174: 7163: 7161: 7157: 7156: 7154: 7153: 7152: 7151: 7141: 7136: 7135: 7134: 7127:Large cardinal 7124: 7119: 7114: 7109: 7104: 7090: 7089: 7088: 7083: 7078: 7063: 7061: 7051: 7050: 7048: 7047: 7046: 7045: 7040: 7035: 7025: 7020: 7015: 7010: 7005: 7000: 6995: 6990: 6985: 6980: 6975: 6970: 6964: 6962: 6955: 6954: 6952: 6951: 6950: 6949: 6944: 6939: 6934: 6929: 6924: 6916: 6915: 6914: 6909: 6899: 6894: 6892:Extensionality 6889: 6887:Ordinal number 6884: 6874: 6869: 6868: 6867: 6856: 6850: 6844: 6843: 6840: 6839: 6837: 6836: 6831: 6826: 6821: 6816: 6811: 6806: 6805: 6804: 6794: 6793: 6792: 6779: 6777: 6771: 6770: 6768: 6767: 6766: 6765: 6760: 6755: 6745: 6740: 6735: 6730: 6725: 6720: 6714: 6712: 6706: 6705: 6703: 6702: 6697: 6692: 6687: 6682: 6677: 6672: 6671: 6670: 6660: 6655: 6650: 6645: 6640: 6635: 6629: 6627: 6618: 6612: 6611: 6609: 6608: 6603: 6598: 6593: 6588: 6583: 6571:Cantor's 6569: 6564: 6559: 6549: 6547: 6534: 6533: 6531: 6530: 6525: 6520: 6515: 6510: 6505: 6500: 6495: 6490: 6485: 6480: 6475: 6470: 6469: 6468: 6457: 6455: 6451: 6450: 6443: 6442: 6435: 6428: 6420: 6411: 6410: 6408: 6407: 6402: 6397: 6392: 6390:Satisfiability 6387: 6382: 6377: 6375:Interpretation 6372: 6367: 6362: 6357: 6352: 6347: 6346: 6345: 6335: 6330: 6325: 6320: 6313: 6307: 6304: 6303: 6292: 6291: 6284: 6277: 6269: 6260: 6259: 6251: 6237: 6219: 6213: 6207: 6197: 6196: 6195: 6190: 6175: 6163: 6151: 6135: 6118: 6117:External links 6115: 6114: 6113: 6091: 6060: 6042: 6027:Wigderson, Avi 6023: 6009: 6003: 5983: 5977: 5961: 5955: 5940:Bricmont, Jean 5932: 5908: 5888:(3): 243–256. 5868: 5851:(3): 279–313. 5840: 5821: 5804:Putnam, Hilary 5800: 5794: 5782:Priest, Graham 5778: 5772: 5759: 5749:(2): 153–179. 5739:Priest, Graham 5735: 5707:(3): 484–498. 5689: 5675: 5644: 5624: 5614:(2): 859–862. 5596: 5579: 5557: 5551: 5529: 5513: 5508:978-0521899574 5507: 5494: 5455: 5438: 5424: 5407: 5401: 5388: 5371: 5368: 5367: 5366: 5354: 5338: 5321: 5300: 5286: 5269: 5263: 5250: 5225: 5210: 5194: 5166: 5156: 5146: 5134: 5121: 5090: 5056: 5042: 5026: 5001: 4992: 4989: 4988: 4987: 4981: 4951: 4892: 4874:(2): 536–596. 4863: 4857: 4840: 4829: 4815: 4797:(4): 513–534. 4782: 4765: 4736: 4716:(3): 403–412. 4701: 4690: 4680: 4640: 4626: 4620: 4610:, ed. (2005). 4604: 4571: 4548: 4542: 4527: 4509:(3): 109–121. 4498: 4471: 4465: 4449:Boolos, George 4445: 4424:Boolos, George 4418: 4415: 4414: 4413: 4409: 4391: 4386:editor, 1965. 4380: 4379: 4378: 4377: 4364:editor, 2005. 4356: 4355: 4310: 4307: 4306: 4305: 4302:978-0195147223 4286: 4279:978-0195147209 4263: 4247: 4244: 4241: 4240: 4220: 4218:, p. 179. 4208: 4206:, p. 208. 4196: 4176: 4164: 4152: 4146:, p. 76; 4136: 4124: 4112: 4110:, p. 328. 4100: 4088: 4076: 4064: 4052: 4040: 4024: 4012: 4010:, p. 187. 4006:, p. 10; 3992: 3968: 3956: 3944: 3928: 3913: 3909:Wigderson 2010 3901: 3889: 3877: 3875:, p. 388. 3865: 3863:, p. 383. 3853: 3851:, p. 274. 3845:Smoryński 1977 3837: 3821: 3809: 3789: 3777: 3764: 3752: 3750:, p. 106. 3740: 3727: 3724: 3721: 3713: 3710: 3687: 3671: 3659: 3657:, p. 112. 3646: 3645: 3643: 3640: 3638: 3635: 3633: 3632: 3627: 3622: 3617: 3612: 3607: 3602: 3597: 3590: 3585: 3580: 3575: 3570: 3563: 3557: 3556: 3555: 3541: 3525: 3522: 3442: 3439: 3425: 3422: 3413:Finsler (1926) 3409: 3406: 3404: 3401: 3376: 3373: 3318: 3315: 3290:ε-substitution 3262: 3259: 3215:Franzén (2005) 3209: 3206: 3202:Shapiro (2002) 3175: 3172: 3105:Turing machine 3099:and physicist 3089:Main article: 3086: 3083: 3075:second problem 3063:Crispin Wright 3042: 3039: 3026: 3023: 2897: 2894: 2884:in 2013 using 2856:in 1986 using 2832: 2829: 2797: 2794: 2767: 2766: 2755: 2592: 2591: 2557:The statement 2548: 2547: 2508: 2507: 2474:diagonal lemma 2468: 2465: 2410:; the string " 2385: 2384: 2257: 2254: 2251: 2248: 2217:statement form 2193: 2190: 2169: 2168: 2167: 2166: 2153: 2084: 2081: 2080: 2079: 2075: 2060: 2032:Main article: 2029: 2026: 2022:ω-inconsistent 1968:until either " 1939: 1932: 1925: 1902:Franzén (2005) 1850: 1847: 1830: 1809:Ramsey theorem 1788: 1785: 1655: 1652: 1634: 1523: 1514: 1505: 1496: 1487: 1474: 1465: 1456: 1439: 1430: 1421: 1404: 1395: 1386: 1377: 1368: 1359: 1350: 1341: 1333: 1330: 1325: 1324: 1310: 1299: 1284: 1274: 1264: 1257: 1246: 1232: 1222: 1185:satisfies the 1176: 1168: 1165: 1132: 1129: 1061: 1058: 1054:Rosser's trick 1004:. A system is 982: 979: 914: 911: 868:Smoryński 1977 847: 824: 809: 798:Smoryński 1977 782: 774: 771: 744: 739: 735: 688: 677: 664: 661: 651:will refer to 644: 633: 621: 601: 578: 563: 544: 510: 480:Rosser's trick 458: 455: 425: 422: 401:The system of 383:characteristic 377:The theory of 364: 361: 327: 300: 297: 228: 225: 196: 193: 165:formal systems 160: 157: 58: 36: 30: 27: 25: 9: 6: 4: 3: 2: 8226: 8215: 8212: 8210: 8207: 8205: 8202: 8200: 8197: 8195: 8192: 8190: 8187: 8185: 8182: 8181: 8179: 8166: 8165: 8160: 8152: 8146: 8143: 8141: 8138: 8136: 8133: 8131: 8128: 8124: 8121: 8120: 8119: 8116: 8114: 8111: 8109: 8106: 8104: 8100: 8097: 8095: 8092: 8090: 8087: 8085: 8082: 8080: 8077: 8076: 8074: 8070: 8064: 8061: 8059: 8056: 8054: 8053:Recursive set 8051: 8049: 8046: 8044: 8041: 8039: 8036: 8034: 8031: 8027: 8024: 8022: 8019: 8017: 8014: 8012: 8009: 8007: 8004: 8003: 8002: 7999: 7997: 7994: 7992: 7989: 7987: 7984: 7982: 7979: 7977: 7974: 7973: 7971: 7969: 7965: 7959: 7956: 7954: 7951: 7949: 7946: 7944: 7941: 7939: 7936: 7934: 7931: 7929: 7926: 7922: 7919: 7917: 7914: 7912: 7909: 7908: 7907: 7904: 7902: 7899: 7897: 7894: 7892: 7889: 7887: 7884: 7882: 7879: 7875: 7872: 7871: 7870: 7867: 7863: 7862:of arithmetic 7860: 7859: 7858: 7855: 7851: 7848: 7846: 7843: 7841: 7838: 7836: 7833: 7831: 7828: 7827: 7826: 7823: 7819: 7816: 7814: 7811: 7810: 7809: 7806: 7805: 7803: 7801: 7797: 7791: 7788: 7786: 7783: 7781: 7778: 7776: 7773: 7770: 7769:from ZFC 7766: 7763: 7761: 7758: 7752: 7749: 7748: 7747: 7744: 7742: 7739: 7737: 7734: 7733: 7732: 7729: 7727: 7724: 7722: 7719: 7717: 7714: 7712: 7709: 7707: 7704: 7702: 7699: 7698: 7696: 7694: 7690: 7680: 7679: 7675: 7674: 7669: 7668:non-Euclidean 7666: 7662: 7659: 7657: 7654: 7652: 7651: 7647: 7646: 7644: 7641: 7640: 7638: 7634: 7630: 7627: 7625: 7622: 7621: 7620: 7616: 7612: 7609: 7608: 7607: 7603: 7599: 7596: 7594: 7591: 7589: 7586: 7584: 7581: 7579: 7576: 7574: 7571: 7570: 7568: 7564: 7563: 7561: 7556: 7550: 7545:Example 7542: 7534: 7529: 7528: 7527: 7524: 7522: 7519: 7515: 7512: 7510: 7507: 7505: 7502: 7500: 7497: 7496: 7495: 7492: 7490: 7487: 7485: 7482: 7480: 7477: 7473: 7470: 7468: 7465: 7464: 7463: 7460: 7456: 7453: 7451: 7448: 7446: 7443: 7441: 7438: 7437: 7436: 7433: 7431: 7428: 7424: 7421: 7419: 7416: 7414: 7411: 7410: 7409: 7406: 7402: 7399: 7397: 7394: 7392: 7389: 7387: 7384: 7382: 7379: 7377: 7374: 7373: 7372: 7369: 7367: 7364: 7362: 7359: 7357: 7354: 7350: 7347: 7345: 7342: 7340: 7337: 7335: 7332: 7331: 7330: 7327: 7325: 7322: 7320: 7317: 7315: 7312: 7308: 7305: 7303: 7302:by definition 7300: 7299: 7298: 7295: 7291: 7288: 7287: 7286: 7283: 7281: 7278: 7276: 7273: 7271: 7268: 7266: 7263: 7262: 7259: 7256: 7254: 7250: 7245: 7239: 7235: 7225: 7222: 7220: 7217: 7215: 7212: 7210: 7207: 7205: 7202: 7200: 7197: 7195: 7192: 7190: 7189:Kripke–Platek 7187: 7185: 7182: 7178: 7175: 7173: 7170: 7169: 7168: 7165: 7164: 7162: 7158: 7150: 7147: 7146: 7145: 7142: 7140: 7137: 7133: 7130: 7129: 7128: 7125: 7123: 7120: 7118: 7115: 7113: 7110: 7108: 7105: 7102: 7098: 7094: 7091: 7087: 7084: 7082: 7079: 7077: 7074: 7073: 7072: 7068: 7065: 7064: 7062: 7060: 7056: 7052: 7044: 7041: 7039: 7036: 7034: 7033:constructible 7031: 7030: 7029: 7026: 7024: 7021: 7019: 7016: 7014: 7011: 7009: 7006: 7004: 7001: 6999: 6996: 6994: 6991: 6989: 6986: 6984: 6981: 6979: 6976: 6974: 6971: 6969: 6966: 6965: 6963: 6961: 6956: 6948: 6945: 6943: 6940: 6938: 6935: 6933: 6930: 6928: 6925: 6923: 6920: 6919: 6917: 6913: 6910: 6908: 6905: 6904: 6903: 6900: 6898: 6895: 6893: 6890: 6888: 6885: 6883: 6879: 6875: 6873: 6870: 6866: 6863: 6862: 6861: 6858: 6857: 6854: 6851: 6849: 6845: 6835: 6832: 6830: 6827: 6825: 6822: 6820: 6817: 6815: 6812: 6810: 6807: 6803: 6800: 6799: 6798: 6795: 6791: 6786: 6785: 6784: 6781: 6780: 6778: 6776: 6772: 6764: 6761: 6759: 6756: 6754: 6751: 6750: 6749: 6746: 6744: 6741: 6739: 6736: 6734: 6731: 6729: 6726: 6724: 6721: 6719: 6716: 6715: 6713: 6711: 6710:Propositional 6707: 6701: 6698: 6696: 6693: 6691: 6688: 6686: 6683: 6681: 6678: 6676: 6673: 6669: 6666: 6665: 6664: 6661: 6659: 6656: 6654: 6651: 6649: 6646: 6644: 6641: 6639: 6638:Logical truth 6636: 6634: 6631: 6630: 6628: 6626: 6622: 6619: 6617: 6613: 6607: 6604: 6602: 6599: 6597: 6594: 6592: 6589: 6587: 6584: 6582: 6578: 6574: 6570: 6568: 6565: 6563: 6560: 6558: 6554: 6551: 6550: 6548: 6546: 6540: 6535: 6529: 6526: 6524: 6521: 6519: 6516: 6514: 6511: 6509: 6506: 6504: 6501: 6499: 6496: 6494: 6491: 6489: 6486: 6484: 6481: 6479: 6476: 6474: 6471: 6467: 6464: 6463: 6462: 6459: 6458: 6456: 6452: 6448: 6441: 6436: 6434: 6429: 6427: 6422: 6421: 6418: 6406: 6403: 6401: 6398: 6396: 6393: 6391: 6388: 6386: 6383: 6381: 6378: 6376: 6373: 6371: 6368: 6366: 6363: 6361: 6358: 6356: 6353: 6351: 6348: 6344: 6341: 6340: 6339: 6336: 6334: 6331: 6329: 6326: 6324: 6321: 6319: 6318: 6314: 6312: 6309: 6308: 6305: 6301: 6297: 6290: 6285: 6283: 6278: 6276: 6271: 6270: 6267: 6263: 6257: 6254: 6252: 6249: 6245: 6241: 6238: 6234: 6230: 6229: 6224: 6220: 6217: 6214: 6211: 6208: 6205: 6201: 6198: 6194: 6191: 6189: 6185: 6182: 6179: 6178: 6176: 6173: 6170:entry in the 6169: 6168: 6164: 6161: 6160: 6155: 6152: 6149: 6148: 6143: 6139: 6136: 6134: 6130: 6129: 6124: 6121: 6120: 6110: 6106: 6102: 6098: 6094: 6088: 6084: 6080: 6075: 6070: 6066: 6061: 6058: 6057:0-262-23189-1 6054: 6050: 6046: 6043: 6039: 6032: 6028: 6024: 6022: 6018: 6014: 6010: 6006: 6004:0-8264-9528-1 6000: 5997:. Continuum. 5996: 5992: 5988: 5984: 5980: 5974: 5970: 5966: 5962: 5958: 5956:0-312-20407-8 5952: 5948: 5946: 5941: 5937: 5933: 5929: 5925: 5921: 5917: 5913: 5909: 5905: 5901: 5896: 5891: 5887: 5883: 5882: 5877: 5873: 5869: 5867: 5863: 5858: 5854: 5850: 5846: 5841: 5839: 5835: 5831: 5830: 5825: 5822: 5819: 5813: 5809: 5805: 5801: 5797: 5795:0-19-926329-9 5791: 5787: 5783: 5779: 5775: 5769: 5765: 5760: 5756: 5752: 5748: 5744: 5740: 5736: 5732: 5728: 5724: 5720: 5715: 5710: 5706: 5702: 5698: 5694: 5690: 5686: 5682: 5678: 5672: 5668: 5664: 5659: 5654: 5650: 5645: 5643: 5642:0-486-42533-9 5639: 5633: 5629: 5625: 5621: 5617: 5613: 5609: 5602: 5597: 5587:on 2019-05-08 5586: 5582: 5576: 5572: 5571: 5566: 5562: 5558: 5554: 5552:0-201-02988-X 5548: 5544: 5543: 5538: 5534: 5530: 5527: 5526: 5521: 5517: 5516:David Hilbert 5514: 5510: 5504: 5500: 5495: 5491: 5487: 5483: 5479: 5475: 5471: 5467: 5463: 5462: 5456: 5454: 5453:0-393-05169-2 5450: 5446: 5442: 5439: 5435: 5431: 5427: 5421: 5417: 5413: 5408: 5404: 5398: 5394: 5389: 5385: 5381: 5380: 5374: 5373: 5365: 5362: 5357: 5355:0-262-23189-1 5351: 5348:. MIT Press. 5347: 5343: 5339: 5336: 5332: 5328: 5327: 5322: 5320: 5316: 5312: 5309: 5305: 5301: 5299: 5295: 5291: 5287: 5284: 5280: 5277: 5273: 5270: 5266: 5264:0-521-58533-3 5260: 5256: 5251: 5241:on 2005-10-23 5240: 5236: 5232: 5228: 5222: 5218: 5217: 5211: 5209: 5206: 5202: 5198: 5195: 5193: 5190: 5186: 5185:0-8147-5816-9 5182: 5178: 5177:Gödel's Proof 5174: 5170: 5167: 5164: 5160: 5157: 5154: 5151:, FBA, 1970. 5150: 5147: 5144: 5143: 5138: 5137:Per Lindström 5135: 5133: 5129: 5126:, OSB, 2005. 5125: 5122: 5120: 5117: 5113: 5112:0-465-03078-5 5109: 5105: 5101: 5097: 5096: 5091: 5089: 5086: 5082: 5081:0-465-02656-7 5078: 5074: 5073:0-465-02685-0 5070: 5066: 5065: 5060: 5057: 5053: 5049: 5045: 5043:1-56881-238-8 5039: 5035: 5031: 5027: 5024: 5020: 5016: 5015:3-261-04214-1 5012: 5008: 5007: 5002: 4999: 4995: 4994: 4984: 4982:9780444508713 4978: 4974: 4970: 4966: 4965: 4960: 4956: 4955:Zach, Richard 4952: 4948: 4944: 4940: 4936: 4932: 4928: 4923: 4918: 4914: 4910: 4909: 4901: 4897: 4896:Zach, Richard 4893: 4889: 4885: 4881: 4877: 4873: 4869: 4864: 4860: 4854: 4850: 4846: 4841: 4838: 4834: 4830: 4827: 4823: 4819: 4816: 4812: 4808: 4804: 4800: 4796: 4792: 4788: 4783: 4771: 4766: 4763: 4760:Reprinted in 4757: 4753: 4749: 4745: 4741: 4740:Kleene, S. C. 4737: 4733: 4729: 4724: 4719: 4715: 4711: 4707: 4702: 4699: 4695: 4691: 4688: 4684: 4683:David Hilbert 4681: 4677: 4673: 4669: 4665: 4661: 4657: 4653: 4649: 4645: 4641: 4637: 4636: 4631: 4627: 4623: 4621:9780444508713 4617: 4613: 4609: 4605: 4601: 4597: 4593: 4589: 4585: 4581: 4577: 4572: 4568: 4564: 4557: 4553: 4552:Davis, Martin 4549: 4545: 4539: 4535: 4534: 4528: 4524: 4520: 4516: 4512: 4508: 4504: 4499: 4497: 4493: 4489: 4488: 4483: 4479: 4476: 4472: 4468: 4466:0-674-53766-1 4462: 4458: 4454: 4450: 4446: 4443: 4441: 4438:reprinted in 4435: 4431: 4430: 4425: 4421: 4420: 4410: 4407: 4406:0-674-32449-8 4403: 4399: 4395: 4392: 4389: 4385: 4382: 4381: 4375: 4374:0-7624-1922-9 4371: 4367: 4363: 4360: 4359: 4358: 4357: 4353: 4352:0-486-66980-7 4349: 4345: 4341: 4337: 4336: 4335: 4333: 4329: 4325: 4321: 4317: 4303: 4299: 4295: 4292:, ed., 1995. 4291: 4287: 4284: 4280: 4276: 4272: 4269:, ed., 1986. 4268: 4264: 4262: 4258: 4254: 4250: 4249: 4237: 4233: 4229: 4224: 4217: 4212: 4205: 4200: 4193: 4189: 4185: 4180: 4174:, p. 77. 4173: 4168: 4161: 4156: 4149: 4145: 4140: 4134:, p. 76. 4133: 4128: 4122:, p. 89. 4121: 4116: 4109: 4104: 4097: 4092: 4085: 4080: 4074:, p. 70. 4073: 4068: 4062:, p. 72. 4061: 4056: 4050:, p. 69. 4049: 4044: 4038:, p. 33. 4037: 4033: 4028: 4022:, p. 63. 4021: 4016: 4009: 4005: 4001: 3996: 3989: 3985: 3981: 3977: 3972: 3965: 3960: 3954:, p. 47. 3953: 3948: 3941: 3937: 3932: 3925: 3920: 3918: 3910: 3905: 3898: 3893: 3886: 3881: 3874: 3869: 3862: 3857: 3850: 3846: 3841: 3834: 3830: 3825: 3819:, p. 73. 3818: 3813: 3806: 3802: 3798: 3793: 3786: 3781: 3774: 3768: 3761: 3756: 3749: 3744: 3722: 3711: 3708: 3696: 3691: 3685: 3681: 3675: 3669:, p. 24. 3668: 3663: 3656: 3651: 3647: 3631: 3628: 3626: 3623: 3621: 3618: 3616: 3613: 3611: 3608: 3606: 3603: 3601: 3598: 3596: 3595: 3591: 3589: 3588:Löb's Theorem 3586: 3584: 3581: 3579: 3576: 3574: 3573:Gödel machine 3571: 3569: 3568: 3564: 3562: 3559: 3558: 3553: 3547: 3542: 3539: 3528: 3521: 3519: 3515: 3514:Rodych (2003) 3510: 3506: 3500: 3497: 3491: 3489: 3484: 3483:Priest (2004) 3480: 3476: 3471: 3469: 3465: 3461: 3457: 3456:Vienna Circle 3453: 3452: 3447: 3438: 3436: 3431: 3430:Ernst Zermelo 3421: 3418: 3414: 3400: 3398: 3394: 3389: 3386: 3381: 3372: 3370: 3366: 3362: 3357: 3355: 3349: 3347: 3341: 3339: 3335: 3334: 3327: 3324: 3314: 3312: 3308: 3304: 3298: 3295: 3291: 3287: 3282: 3280: 3276: 3272: 3268: 3258: 3256: 3252: 3248: 3244: 3240: 3236: 3232: 3228: 3224: 3220: 3216: 3205: 3203: 3199: 3195: 3191: 3187: 3186: 3181: 3170: 3165: 3160: 3155: 3153: 3149: 3148: 3143: 3142: 3137: 3133: 3131: 3127: 3123: 3119: 3116: 3115:Putnam (1960) 3112: 3110: 3106: 3102: 3101:Roger Penrose 3098: 3092: 3082: 3080: 3076: 3072: 3071:David Hilbert 3067: 3064: 3060: 3056: 3052: 3051:Gottlob Frege 3048: 3038: 3036: 3032: 3022: 2987: 2979: 2975: 2921: 2914: 2903: 2893: 2891: 2887: 2883: 2879: 2878:Harrison 2009 2875: 2871: 2870:O'Connor 2005 2867: 2863: 2859: 2855: 2850: 2848: 2844: 2838: 2828: 2818: 2814: 2810: 2806: 2802: 2801:Boolos (1989) 2793: 2785: 2781: 2756: 2745: 2744: 2743: 2728: 2721: 2689: 2685: 2681: 2670: 2666: 2662: 2652: 2644: 2640: 2636: 2625: 2621: 2617: 2607: 2601: 2596: 2589: 2588: 2587: 2574: 2570: 2566: 2555: 2544: 2540: 2536: 2532: 2528: 2527: 2526: 2522: 2518: 2503: 2499: 2495: 2491: 2487: 2486: 2485: 2475: 2464: 2460: 2456: 2444: 2440: 2436: 2425: 2421: 2415: 2403: 2399: 2394: 2391:is short for 2390: 2369: 2365: 2361: 2357: 2356: 2355: 2347: 2343: 2320: 2316: 2312: 2297: 2293: 2286: 2282: 2276: 2252: 2246: 2232: 2231: 2223:. As soon as 2222: 2218: 2208: 2204: 2198: 2189: 2187: 2183: 2179: 2175: 2164: 2160: 2158:x=y => y=x 2154: 2151: 2147: 2141: 2140: 2139: 2138: 2137: 2135: 2131: 2126: 2124: 2123: 2118: 2114: 2110: 2106: 2076: 2073: 2069: 2065: 2061: 2054: 2045: 2044: 2043: 2041: 2035: 2025: 2023: 2019: 2015: 2011: 2009: 2005: 2001: 1997: 1938: 1931: 1924: 1920: 1915: 1911: 1907: 1904:explains how 1903: 1899: 1897: 1893: 1889: 1876: 1872: 1871:Kleene (1943) 1868: 1866: 1862: 1856: 1846: 1844: 1840: 1836: 1835:predicativism 1828: 1824: 1822: 1818: 1814: 1810: 1806: 1802: 1798: 1793: 1784: 1782: 1778: 1771:greater than 1770: 1762: 1758: 1754: 1750: 1748: 1744: 1740: 1739:Shelah (1974) 1736: 1733: 1729: 1725: 1721: 1717: 1713: 1708: 1706: 1701: 1696: 1694: 1689: 1687: 1683: 1679: 1675: 1671: 1667: 1661: 1651: 1649: 1645: 1641: 1633: 1628: 1624: 1611: 1540: 1538: 1534: 1528: 1522: 1513: 1504: 1495: 1486: 1473: 1464: 1455: 1438: 1429: 1420: 1403: 1394: 1385: 1376: 1367: 1358: 1349: 1340: 1329: 1322: 1316: 1309: 1305: 1298: 1294: 1290: 1283: 1275: 1270: 1263: 1256: 1252: 1245: 1233: 1228: 1221: 1204: 1203: 1202: 1194: 1188: 1182: 1175: 1164: 1161: 1148: 1128: 1121: 1117: 1113: 1105: 1093: 1089: 1057: 1055: 1051: 1047: 1025: 1021: 1007: 1003: 1002: 996: 994: 993: 987: 978: 976: 975:Alfred Tarski 972: 968: 959: 938:for a system 932: 928: 924: 920: 910: 908: 904: 892: 888: 884: 880: 875: 873: 869: 859: 855: 846: 842: 838: 833: 827: 823: 818: 812: 808: 803: 799: 785: 781: 770: 768: 764: 760: 742: 737: 724: 720: 711: 708: 707:Gödel numbers 703: 701: 691: 687: 680: 676: 660: 647: 643: 636: 632: 624: 620: 604: 600: 581: 577: 566: 562: 547: 543: 529: 527: 513: 509: 502: 488: 483: 481: 477: 473: 469: 464: 454: 450: 446: 444: 443:contradictory 440: 436: 432: 421: 419: 415: 411: 407: 404: 399: 397: 393: 389: 384: 380: 375: 370: 360: 358: 354: 349: 347: 342: 340: 336: 326: 317: 312: 310: 306: 296: 294: 289: 284: 282: 278: 274: 270: 266: 261: 257: 255: 251: 247: 242: 238: 234: 233:syntactically 224: 221: 216: 214: 210: 206: 203:(also called 202: 192: 188: 186: 182: 178: 174: 170: 166: 156: 154: 150: 146: 145: 140: 136: 132: 127: 124: 122: 118: 114: 110: 106: 101: 99: 95: 91: 87: 83: 79: 75: 71: 65: 48: 44: 40: 33: 28: 19: 8209:Metatheorems 8204:Epistemology 8199:Proof theory 8194:Model theory 8155: 7953:Ultraproduct 7800:Model theory 7765:Independence 7701:Formal proof 7693:Proof theory 7676: 7649: 7606:real numbers 7578:second-order 7489:Substitution 7366:Metalanguage 7307:conservative 7280:Axiom schema 7224:Constructive 7194:Morse–Kelley 7160:Set theories 7139:Aleph number 7132:inaccessible 7038:Grothendieck 6922:intersection 6809:Higher-order 6797:Second-order 6743:Truth tables 6700:Venn diagram 6556: 6483:Formal proof 6395:Independence 6370:Decidability 6365:Completeness 6354: 6315: 6261: 6226: 6203: 6166: 6157: 6145: 6138:"Kurt Gödel" 6127: 6074:math/0508572 6064: 6048: 6037: 6012: 5994: 5968: 5943: 5919: 5915: 5885: 5879: 5848: 5844: 5827: 5817: 5811: 5785: 5763: 5746: 5742: 5704: 5700: 5648: 5631: 5611: 5607: 5589:. Retrieved 5585:the original 5569: 5540: 5523: 5520:Paul Bernays 5498: 5465: 5459: 5444: 5416:A. K. Peters 5411: 5392: 5383: 5377: 5345: 5325: 5303: 5289: 5275: 5254: 5243:. Retrieved 5239:the original 5215: 5200: 5176: 5169:Ernest Nagel 5162: 5152: 5140: 5127: 5124:Stanley Jaki 5093: 5062: 5033: 5005: 4997: 4963: 4922:math/0102189 4912: 4906: 4871: 4867: 4848: 4836: 4832: 4825: 4821: 4794: 4790: 4774:. Retrieved 4750:(1): 41–73. 4747: 4743: 4713: 4709: 4698:the original 4651: 4647: 4633: 4614:. Elsevier. 4611: 4583: 4579: 4566: 4562: 4532: 4506: 4502: 4485: 4452: 4440:Boolos (1998 4437: 4433: 4427: 4397: 4387: 4384:Martin Davis 4365: 4343: 4327: 4319: 4312: 4293: 4270: 4223: 4211: 4199: 4179: 4167: 4155: 4139: 4127: 4115: 4103: 4096:Finsler 1926 4091: 4079: 4067: 4055: 4043: 4027: 4015: 3995: 3976:Franzén 2005 3971: 3964:Shapiro 2002 3959: 3947: 3931: 3904: 3892: 3885:Hellman 1981 3880: 3868: 3856: 3840: 3824: 3817:Franzén 2005 3812: 3792: 3780: 3772: 3767: 3755: 3748:Franzén 2005 3743: 3690: 3674: 3662: 3655:Franzén 2005 3650: 3605:Proof theory 3592: 3565: 3518:Berto (2009) 3504: 3502: 3495: 3493: 3472: 3449: 3444: 3441:Wittgenstein 3427: 3411: 3392: 3390: 3382: 3378: 3368: 3360: 3358: 3353: 3350: 3345: 3343: 3337: 3331: 3329: 3320: 3317:Announcement 3299: 3283: 3271:habilitation 3264: 3255:Régis Debray 3247:anti-realist 3211: 3183: 3177: 3167: 3162: 3157: 3152:strange loop 3151: 3145: 3139: 3134: 3129: 3120: 3113: 3107:, or by the 3094: 3068: 3049:proposed by 3044: 3028: 2988: 2977: 2973: 2922: 2912: 2905: 2890:Paulson 2014 2862:Shankar 1994 2851: 2840: 2809:liar paradox 2799: 2783: 2779: 2768: 2729: 2722: 2687: 2683: 2679: 2668: 2664: 2660: 2653: 2642: 2638: 2634: 2623: 2619: 2615: 2608: 2600:ω-consistent 2597: 2593: 2572: 2568: 2564: 2556: 2549: 2542: 2538: 2534: 2530: 2520: 2516: 2509: 2501: 2497: 2493: 2489: 2470: 2458: 2454: 2442: 2438: 2434: 2423: 2419: 2416: 2401: 2397: 2392: 2388: 2386: 2367: 2363: 2359: 2345: 2341: 2321: 2314: 2310: 2295: 2291: 2284: 2280: 2277: 2228: 2220: 2216: 2215:is called a 2206: 2202: 2199: 2195: 2185: 2181: 2177: 2173: 2170: 2156: 2143: 2130:Gödel number 2127: 2120: 2112: 2108: 2086: 2048: 2037: 2012: 1998: 1936: 1929: 1922: 1918: 1914:Matiyasevich 1900: 1869: 1858: 1825: 1794: 1790: 1777:liar paradox 1751: 1747:group theory 1737: 1731: 1709: 1697: 1690: 1663: 1631: 1612: 1541: 1529: 1520: 1511: 1502: 1493: 1484: 1471: 1462: 1453: 1436: 1427: 1418: 1401: 1392: 1383: 1374: 1365: 1356: 1347: 1338: 1335: 1326: 1321:modus ponens 1314: 1307: 1303: 1296: 1292: 1288: 1281: 1268: 1261: 1254: 1250: 1243: 1226: 1219: 1192: 1180: 1173: 1170: 1149: 1134: 1119: 1115: 1111: 1091: 1090: 1063: 1023: 1019: 1006:ω-consistent 1005: 1001:ω-consistent 999: 997: 990: 988: 984: 967:Gödel number 960: 931:liar paradox 923:liar paradox 916: 907:Franzén 2005 891:Franzén 2005 876: 857: 853: 844: 840: 836: 825: 821: 810: 806: 802:Franzén 2005 783: 779: 776: 767:Franzén 2005 763:MRDP theorem 761:). Via the 712: 704: 689: 685: 678: 674: 666: 645: 641: 634: 630: 622: 618: 602: 598: 579: 575: 564: 560: 545: 541: 530: 511: 507: 504: 486: 485: 467: 466: 451: 447: 427: 408: 400: 376: 366: 350: 343: 324: 313: 309:inconsistent 308: 302: 285: 267:without the 262: 258: 254:semantically 253: 245: 236: 232: 230: 227:Completeness 217: 204: 200: 198: 189: 184: 172: 162: 142: 129:Employing a 128: 125: 102: 69: 68: 46: 37:This is the 31: 8063:Type theory 8011:undecidable 7943:Truth value 7830:equivalence 7509:non-logical 7122:Enumeration 7112:Isomorphism 7059:cardinality 7043:Von Neumann 7008:Ultrafilter 6973:Uncountable 6907:equivalence 6824:Quantifiers 6814:Fixed-point 6783:First-order 6663:Consistency 6648:Proposition 6625:Traditional 6596:Lindström's 6586:Compactness 6528:Type theory 6473:Cardinality 6385:Metatheorem 6343:of geometry 6328:Consistency 6181:Kurt Gödel. 6128:In Our Time 5949:. Picador. 5936:Sokal, Alan 5808:Sidney Hook 5197:Rudy Rucker 4845:Jon Barwise 4776:November 7, 4586:: 676–682. 4324:Dawson 1997 4232:Rodych 2003 4192:Priest 2004 4184:Rodych 2003 4172:Dawson 1996 4144:Dawson 1996 4132:Dawson 1996 4120:Dawson 1996 4072:Dawson 1996 4060:Dawson 1996 4048:Dawson 1996 4020:Dawson 1997 3952:Priest 2006 3940:Priest 2006 3936:Priest 1984 3897:Putnam 1960 3873:Boolos 1998 3861:Boolos 1998 3849:Kleene 1967 3785:Kleene 1943 3760:Shelah 1974 3680:independent 3488:Karl Menger 3481:as well as 3477:, although 3369:Monatshefte 3361:Monatshefte 3338:Ignorabimus 3333:Ignorabimus 3294:von Neumann 3198:dialetheism 3130:knowability 3097:J. R. Lucas 2813:Saul Kripke 2510:By letting 2117:Alan Turing 2049:"statement 1726:), and the 1700:truth value 1693:independent 1640:wellfounded 1602:, and thus 554:, and thus 439:maximal set 381:of a given 299:Consistency 98:mathematics 47:5 July 2024 8178:Categories 7874:elementary 7567:arithmetic 7435:Quantifier 7413:functional 7285:Expression 7003:Transitive 6947:identities 6932:complement 6865:hereditary 6848:Set theory 6140:entry by 5845:Dialectica 5714:2104.14260 5658:cs/0505034 5591:2018-10-24 5319:0198534507 5298:0195046722 5283:0192801414 5245:2005-10-29 5149:J.R. Lucas 5023:0724.03003 4762:Davis 1965 4332:Davis 1965 4236:Berto 2009 4204:Berto 2009 3833:Jones 1980 3829:Davis 2006 3667:Smith 2007 3637:References 3403:Criticisms 3311:Königsberg 3253:criticize 3185:dialetheia 2900:See also: 2835:See also: 2704:, because 2602:, and let 2577:; rather, 2221:class-sign 2200:A formula 1853:See also: 1801:Harrington 1724:set theory 1712:Paul Cohen 1658:See also: 1586:, such as 1189:. Letting 1032:such that 723:Smith 2007 721:relation ( 700:Smith 2007 528:sentence. 461:See also: 445:theorems. 431:Smith 2007 305:consistent 181:set theory 82:Kurt Gödel 8145:Supertask 8048:Recursion 8006:decidable 7840:saturated 7818:of models 7741:deductive 7736:axiomatic 7656:Hilbert's 7643:Euclidean 7624:canonical 7547:axiomatic 7479:Signature 7408:Predicate 7297:Extension 7219:Ackermann 7144:Operation 7023:Universal 7013:Recursive 6988:Singleton 6983:Inhabited 6968:Countable 6958:Types of 6942:power set 6912:partition 6829:Predicate 6775:Predicate 6690:Syllogism 6680:Soundness 6653:Inference 6643:Tautology 6545:paradoxes 6360:Soundness 6296:Metalogic 6233:EMS Press 6101:162131413 5563:(2007) . 5482:0022-362X 5342:Wang, Hao 5323:—, 2013. 5302:—, 1994. 5288:—, 1992. 4939:0039-7857 4685:, 1900, " 4668:0029-4624 4600:121054124 4569:(4): 414. 4216:Wang 1996 4036:Zach 2003 4032:Zach 2007 3642:Citations 3321:The 1930 3286:Ackermann 3243:Platonism 3035:formalism 2874:HOL Light 2393:beweisbar 2387:The name 2230:bona fide 2142:The word 1795:In 1977, 1460:. But if 734:Π 353:paradoxes 117:algorithm 115:(i.e. an 8130:Logicism 8123:timeline 8099:Concrete 7958:Validity 7928:T-schema 7921:Kripke's 7916:Tarski's 7911:semantic 7901:Strength 7850:submodel 7845:spectrum 7813:function 7661:Tarski's 7650:Elements 7637:geometry 7593:Robinson 7514:variable 7499:function 7472:spectrum 7462:Sentence 7418:variable 7361:Language 7314:Relation 7275:Automata 7265:Alphabet 7249:language 7103:-jection 7081:codomain 7067:Function 7028:Universe 6998:Infinite 6902:Relation 6685:Validity 6675:Argument 6573:theorem, 6184:Archived 6047:, 1996, 6045:Hao Wang 6029:(2010). 5993:(2006). 5967:(1967). 5942:(1999). 5874:(1974). 5826:, 2010, 5784:(2006). 5731:13913592 5695:(2014). 5685:15610367 5630:(1967). 5539:(1979). 5443:, 2005, 5434:36104240 5344:(1996). 5274:, 1987. 5161:, 2022. 5139:, 1997. 5061:, 1979. 5032:(2005). 4947:16657040 4908:Synthese 4898:(2003). 4811:52083793 4554:(2006). 4478:Archived 4451:(1998). 3712:⊬ 3524:See also 3505:Nachlass 3468:Nachlass 3245:and the 3059:Bob Hale 3047:logicism 2943:, where 2920:itself. 2886:Isabelle 2773:or "not 2550:and the 2354:exists: 2275:3 = 6". 1527:either. 1127:itself. 929:". 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Knowledge

Gödel's incompleteness theorems

Index