Mathematical logic

2544:, but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having a separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis. 10212: 6956: 3514: 214: 320:, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in 7987: 6968: 2175: 3621:"Die Ausführung dieses Vorhabens hat eine wesentliche Verzögerung dadurch erfahren, daß in einem Stadium, in dem die Darstellung schon ihrem Abschuß nahe war, durch das Erscheinen der Arbeiten von Herbrand und von Gödel eine veränderte Situation im Gebiet der Beweistheorie entstand, welche die Berücksichtigung neuer Einsichten zur Aufgabe machte. Dabei ist der Umfang des Buches angewachsen, so daß eine Teilung in zwei Bände angezeigt erschien." 1007:– had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state the incompleteness theorems in generality that could only be implied in the original paper. 7997: 3480:, for there are statements that, according to Brouwer, could not be claimed to be true while their negations also could not be claimed true. Brouwer's philosophy was influential, and the cause of bitter disputes among prominent mathematicians. Kleene and Kreisel would later study formalized versions of intuitionistic logic (Brouwer rejected formalization, and presented his work in unformalized natural language). With the advent of the 8007: 6992: 6980: 3627:"Carrying out this plan has experienced an essential delay because, at the stage at which the exposition was already near to its conclusion, there occurred an altered situation in the area of proof theory due to the appearance of works by Herbrand and Gödel, which necessitated the consideration of new insights. Thus the scope of this book has grown, so that a division into two volumes seemed advisable." 8433: 3500: 917:, establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time. 753:, which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish a second exposition of his result, directly addressing criticisms of his proof. This paper led to the general acceptance of the axiom of choice in the mathematics community. 3426:, was seriously affected by Gödel's incompleteness theorems, which show that the consistency of formal theories of arithmetic cannot be established using methods formalizable in those theories. Gentzen showed that it is possible to produce a proof of the consistency of arithmetic in a finitary system augmented with axioms of 2299:(1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to 3389:

Mathematicians began to search for axiom systems that could be used to formalize large parts of mathematics. In addition to removing ambiguity from previously naive terms such as function, it was hoped that this axiomatization would allow for consistency proofs. In the 19th century, the main method

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Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness

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first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent

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proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of

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properties. Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions

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showed that the continuum hypothesis cannot be proven from the axioms of Zermelo–Fraenkel set theory. This independence result did not completely settle Hilbert's question, however, as it is possible that new axioms for set theory could resolve the hypothesis. Recent work along these lines has been

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in 1826, mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert

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With the development of formal logic, Hilbert asked whether it would be possible to prove that an axiom system is consistent by analyzing the structure of possible proofs in the system, and showing through this analysis that it is impossible to prove a contradiction. This idea led to the study of

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Because proofs are entirely finitary, whereas truth in a structure is not, it is common for work in constructive mathematics to emphasize provability. The relationship between provability in classical (or nonconstructive) systems and provability in intuitionistic (or constructive, respectively)

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include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing.

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states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some

281:(also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish 2614:, which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is 2567:

One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or

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famously stated "God made the integers; all else is the work of man," endorsing a return to the study of finite, concrete objects in mathematics. Although Kronecker's argument was carried forward by constructivists in the 20th century, the mathematical community as a whole rejected them.

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is also included as part of mathematical logic. Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp.

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is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive.

2417:. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of 2940:

Classical recursion theory focuses on the computability of functions from the natural numbers to the natural numbers. The fundamental results establish a robust, canonical class of computable functions with numerous independent, equivalent characterizations using

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In the early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency.

416:', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. Before this emergence, logic was studied with 3073:

is the study of formal proofs in various logical deduction systems. These proofs are represented as formal mathematical objects, facilitating their analysis by mathematical techniques. Several deduction systems are commonly considered, including

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Mathematical logic emerged in the mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics. Mathematical logic, also called 'logistic', 'symbolic logic', the

2917:, which divide the uncomputable functions into sets that have the same level of uncomputability. Recursion theory also includes the study of generalized computability and definability. Recursion theory grew from the work of 3004:

if there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independently by Church and Turing in 1936, showed that the

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showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size. This theorem, known as the

2871:, states that if a first-order theory in a countable language is categorical in some uncountable cardinality, i.e. all models of this cardinality are isomorphic, then it is categorical in all uncountable cardinalities. 978:, a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as the words 2718:. The axiom of choice, first stated by Zermelo, was proved independent of ZF by Fraenkel, but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set 5630: 909: 2676:, which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The 2960:

Generalized recursion theory extends the ideas of recursion theory to computations that are no longer necessarily finite. It includes the study of computability in higher types as well as areas such as

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as one of his 23 problems in 1900. Gödel showed that the continuum hypothesis cannot be disproven from the axioms of Zermelo–Fraenkel set theory (with or without the axiom of choice), by developing the

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in first-order logic. It shows that if a particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The

3445:. At the most accommodating end, proofs in ZF set theory that do not use the axiom of choice are called constructive by many mathematicians. More limited versions of constructivism limit themselves to 2195: 551:

in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century.

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set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of

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Lecture given at the International Congress of Mathematicians, 3 September 1928. Published in English translation as "The Grounding of Elementary Number Theory", in Mancosu 1998, pp. 266–273.

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to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as

937:, which became key tools in proof theory. Gödel gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types. 559:

Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry.

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Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of

2455: 2415: 1538: 1965: 1480: 1393: 1605: 1266: 2842:; classical model theory seeks to determine the properties of models in a particular elementary class, or determine whether certain classes of structures form elementary classes. 1924: 1564: 999:, because early formalizations by Gödel and Kleene relied on recursive definitions of functions. When these definitions were shown equivalent to Turing's formalization involving 1240: 1666: 2857:. He also noted that his methods were equally applicable to algebraically closed fields of arbitrary characteristic. A modern subfield developing from this is concerned with 1747: 1512: 1352: 1300: 1180: 395:, but category theory is not ordinarily considered a subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians including 2610:

was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the

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is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as a foundational theory for mathematics.

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of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of

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and prove the consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the

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asked for an algorithm to determine whether a multivariate polynomial equation with integer coefficients has a solution in the integers. Partial progress was made by

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with particular properties so strong that the existence of such cardinals cannot be proved in ZFC. The existence of the smallest large cardinal typically studied, an

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by Pasch. The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and the

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of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a

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So certainly Hilbert was aware of the importance of Gödel's work by 1934. The second volume in 1939 included a form of Gentzen's consistency proof for arithmetic.

1830: 484:. In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including 1720: 1125: 540:

began to promote it near the turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts.

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states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that

2202: 578:), using a variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time 3472:. This philosophy, poorly understood at first, stated that in order for a mathematical statement to be true to a mathematician, that person must be able to 730:, posed in 1928. This problem asked for a procedure that would decide, given a formalized mathematical statement, whether the statement is true or false. 8591: 6022:(1920). "Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen". 5597:(1930). "Die Vollständigkeit der Axiome des logischen Funktionen-kalküls" [The completeness of the axioms of the calculus of logical functions]. 3476:

the statement, to not only believe its truth but understand the reason for its truth. A consequence of this definition of truth was the rejection of the

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later built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885.

5145: 3131: 2303:. As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. 3093:, in the context of mathematical logic, includes the study of systems in non-classical logic such as intuitionistic logic, as well as the study of 536:, published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until 3457:). A common idea is that a concrete means of computing the values of the function must be known before the function itself can be said to exist. 432:. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. 5958: 9266: 2688: 6416: 6142: 6096: 4883: 634:. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the 3115:) classical logic into intuitionistic logic, allowing some properties about intuitionistic proofs to be transferred back to classical proofs. 5986: 724:

has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's

241: 9349: 8490: 6246: 3552: 896:, which establishes a correspondence between syntax and semantics in first-order logic. Gödel used the completeness theorem to prove the 839:

showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of

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Computer science also contributes to mathematics by developing techniques for the automatic checking or even finding of proofs, such as

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can be used to show that definable sets in particular theories cannot be too complicated. Tarski established quantifier elimination for

7030: 2886:, says that this is true even independently of the continuum hypothesis. Many special cases of this conjecture have been established. 3106: 9663: 3688:"Computability Theory and Foundations of Mathematics / February, 17th – 20th, 2014 / Tokyo Institute of Technology, Tokyo, Japan" 3257: 7740: 7712: 6390:

School of Mathematics, University of Manchester, Prof. Jeff Paris’s Mathematical Logic (course material and unpublished papers)

5379: 3542: 3362:'s axioms for geometry, which had been taught for centuries as an example of the axiomatic method, were incomplete. The use of 673: 8457: 2583: 676:

that the reals and the natural numbers have different cardinalities. Over the next twenty years, Cantor developed a theory of

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in a series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the

17: 8609: 2463: 2326: 9676: 8999: 7616: 6779: 2188: 979: 177: 2219:. These systems, though they differ in many details, share the common property of considering only expressions in a fixed 914: 373: 8089: 7770: 7049: 6337: 3268:, while researchers in mathematical logic often focus on computability as a theoretical concept and on noncomputability. 946: 399:

have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use

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argued in favor of the study of the infinite, saying "No one shall expel us from the Paradise that Cantor has created."

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In the 19th century, mathematicians became aware of logical gaps and inconsistencies in their field. It was shown that

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was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of

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is closely related to the study of computability in mathematical logic. There is a difference of emphasis, however.

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is employed in set theory, model theory, and recursion theory, as well as in the study of intuitionistic mathematics.

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is that a complete theory with less than continuum many nonisomorphic countable models can have only countably many.

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In the mid-19th century, flaws in Euclid's axioms for geometry became known. In addition to the independence of the

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which may be consistent with the logical system). For example, in every logical system capable of expressing the

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theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not

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Reprinted in English translation as "The notion of 'definite' and the independence of the axiom of choice" in

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refers to the possible existence of winning strategies for certain two-player games (the games are said to be

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presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as

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Excerpt reprinted in English translation as "The principles of arithmetic, presented by a new method"in

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Fraenkel proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with

513: 512:, extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of 282: 3453:, and sets of natural numbers (which can be used to represent real numbers, facilitating the study of 2827:

is a structure that gives a concrete interpretation of the theory. Model theory is closely related to

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A first course in logic: an introduction to model theory, proof theory, computability, and complexity

3607: 3572: 3469: 3450: 3441:. The study of constructive mathematics includes many different programs with various definitions of 3315: 3251: 3101:, who showed it is possible to develop a large part of real analysis using only predicative methods. 2780:, already implies the consistency of ZFC. Despite the fact that large cardinals have extremely high 2737: 1725: 668:

developed the fundamental concepts of infinite set theory. His early results developed the theory of

485: 192: 182: 172: 8648: 5629:(1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" [ 2575: 1491: 1331: 1279: 1156: 1026:

studied formal versions of intuitionistic mathematics, particularly in the context of proof theory.

992:, and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics. 791:

Zermelo provided the first set of axioms for set theory. These axioms, together with the additional

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Reprinted in English translation as "The principles of mathematics and the problems of sets" in

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Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl

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Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse

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of proving the consistency of a set of axioms was to provide a model for it. Thus, for example,

2949:, and other systems. More advanced results concern the structure of the Turing degrees and the 2030: 1305: 10155: 9882: 9860: 9827: 9720: 9566: 9551: 9524: 9475: 9359: 9294: 9119: 9085: 9080: 8954: 8785: 8762: 7967: 7798: 7679: 7446: 7436: 7431: 6912: 6809: 6789: 6784: 6713: 6438: 6037: 5185: 4899: 4822:. Cambridge Tracts in Theoretical Computer Science (2nd ed.). Cambridge University Press. 3582: 3422:

to refer to the methods he would allow but not precisely defining them. This project, known as

3391: 3367: 3265: 3179: 3142:"Mathematical logic has been successfully applied not only to mathematics and its foundations ( 2973: 2966: 2846: 2835:, although the methods of model theory focus more on logical considerations than those fields. 2749: 2281: 2095: 1970: 1193: 1019: 817: 583: 521: 187: 162: 65: 39: 4432: 4397:. (suitable as a first course for independent study) (1st ed.). Oxford University Press. 2792:). The existence of these strategies implies structural properties of the real line and other 2235:

and because of their desirable proof-theoretic properties. Stronger classical logics such as

1130: 1081: 1055: 472:, found wide application and acceptance in Western science and mathematics for millennia. The 10085: 9938: 9730: 9448: 9184: 9090: 8949: 8934: 8815: 8790: 8413: 7937: 7907: 7897: 7793: 7707: 7583: 7523: 7490: 7480: 7370: 7335: 7325: 7262: 7131: 7106: 7101: 7066: 6939: 6869: 6746: 6670: 6609: 6594: 6589: 6566: 6448: 6200: 4541: 3562: 3454: 3427: 3308: 3242:)." "Applications have also been made to theology (F. Drewnowski, J. Salamucha, I. Thomas)." 3219: 2777: 1835: 1783: 1424: 1186: 926: 840: 812: 742: 713: 381: 301: 197: 111: 1752: 1679: 1018:. Kleene introduced the concepts of relative computability, foreshadowed by Turing, and the 696:, but was unable to produce a proof for this result, leaving it as an open problem in 1895. 10058: 10020: 9897: 9701: 9541: 9465: 9443: 9271: 9229: 9128: 9095: 8959: 8747: 8658: 8332: 8178: 7697: 7669: 7641: 7636: 7465: 7441: 7393: 7378: 7360: 7350: 7345: 7307: 7257: 7252: 7169: 7115: 6919: 6799: 6794: 6718: 6619: 6234:

Reprinted in English translation as "A new proof of the possibility of a well-ordering" in

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Lewis Carroll: SYMBOLIC LOGIC Part I Elementary. pub. Macmillan 1896. Available online at:

3423: 3418:. Moreover, Hilbert proposed that the analysis should be entirely concrete, using the term 3296: 3280: 3231: 3187: 3167: 3009:

is algorithmically unsolvable. Turing proved this by establishing the unsolvability of the

3006: 2883: 2875: 2715: 2700: 2607: 2557: 2355: 2248: 2153: 1272: 996: 893: 792: 785: 777: 765: 726: 717: 309: 157: 116: 85: 5903: 5819: 5800:. Die Grundlehren der mathematischen Wissenschaften. Vol. 40. Berlin, New York City: 4358: 855: 638:, which sought to axiomatize analysis using properties of the natural numbers. The modern 615:. This would prove to be a major area of research in the first half of the 20th century. 8: 10187: 10078: 10063: 10043: 10000: 9887: 9837: 9763: 9708: 9645: 9438: 9433: 9381: 9149: 9138: 8810: 8710: 8638: 8629: 8625: 8560: 8555: 8287: 8270: 8250: 8213: 8162: 8157: 8099: 8036: 7962: 7887: 7803: 7788: 7553: 7340: 7297: 7292: 7189: 7179: 7151: 6934: 6844: 6766: 6665: 6599: 6556: 6546: 6526: 5946: 5873: 5701: 4768: 4756: 4503: 3655:

surveys the rise of first-order logic over other formal logics in the early 20th century.

3587: 2981: 2930: 2910: 2595: 2579: 2541: 2315: 2310: 2289: 2285: 2277: 2244: 2224: 2163: 1809: 1776: 1048: 1011: 1004: 901: 897: 882: 844: 761: 685: 643: 604: 599: 321: 4845: 3370:, came into question in analysis, as pathological examples such as Weierstrass' nowhere- 3013:, a result with far-ranging implications in both recursion theory and computer science. 1702: 1107: 968:

Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym

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Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in

517: 377: 10216: 9985: 9948: 9933: 9926: 9909: 9713: 9695: 9561: 9487: 9470: 9423: 9236: 9145: 8979: 8964: 8924: 8876: 8861: 8849: 8805: 8780: 8550: 8223: 8109: 7927: 7826: 7702: 7659: 7568: 7510: 7495: 7485: 7277: 7076: 6960: 6879: 6819: 6751: 6741: 6680: 6655: 6531: 6483: 6266: 6225: 6174: 6136: 6058: 6001:

Richard, Jules (1905). "Les principes des mathématiques et le problème des ensembles".

5935: 5884:

Robert Bonola, ed. (1955). "Geometric Investigations on the Theory of Parallel Lines".

5861: 5776: 5683: 5666: 5650: 5614: 5561: 5535:

The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number

5404: 5245: 5211: 5203: 5164: 5105: 5087: 5066: 5058: 5014: 4966: 4877: 4865: 3519: 3481: 3434: 3341: 3333: 3235: 3199: 2832: 2677: 2371: 2236: 1037: 804: 677: 595: 588: 505: 463: 217: 9169: 3764: 2350:, a stronger limitation than the one established by the Löwenheim–Skolem theorem. The 10211: 10151: 9958: 9768: 9758: 9650: 9531: 9366: 9342: 9123: 9107: 9012: 8989: 8866: 8835: 8800: 8695: 8530: 8437: 8408: 8403: 8393: 8327: 8255: 8140: 7947: 7877: 7856: 7818: 7626: 7593: 7573: 7272: 7184: 7058: 6955: 6675: 6660: 6604: 6551: 6377: 6372: 6343: 6270: 6229: 6217: 6178: 6118: 5939: 5927: 5889: 5805: 5780: 5717: 5654: 5618: 5565: 5424: 5408: 5348: 5295: 5249: 5223: 5136: 5122: 4983: 4919: 4915: 4851: 4823: 4801: 4778: 4743: 4717: 4691: 4661: 4638: 4612: 4577: 4527: 4489: 4485: 4477: 4446: 4417: 4398: 4379: 4344: 4321: 4291: 3513: 3505: 3407: 3378: 3319: 3304: 3272: 3211: 3123: 3111: 3079: 3051: 3016:

There are many known examples of undecidable problems from ordinary mathematics. The

2972:

Contemporary research in recursion theory includes the study of applications such as

2879: 2868: 2850: 2828: 2820: 2673: 2650: 2367: 2260: 2228: 2100: 867: 396: 392: 376:

marks not only a milestone in recursion theory and proof theory, but has also led to

213: 6382: 6353: 6302: 5296:"Sur la décomposition des ensembles de points en parties respectivement congruentes" 5261: 5109: 5070: 4335:

Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972).

2323:, and they are a key reason for the prominence of first-order logic in mathematics. 1022:. Kleene later generalized recursion theory to higher-order functionals. Kleene and 324:) rather than trying to find theories in which all of mathematics can be developed. 10165: 10160: 10053: 10010: 9832: 9793: 9788: 9773: 9599: 9556: 9453: 9251: 9201: 8775: 8737: 8342: 8068: 8063: 7780: 7664: 7631: 7426: 7355: 7244: 7230: 7225: 7174: 7161: 7086: 7039: 6889: 6864: 6736: 6584: 6521: 6258: 6209: 6166: 6113: 6105: 6083: 5919: 5851: 5815: 5768: 5694: 5678: 5642: 5606: 5553: 5485: 5452: 5439: 5396: 5340: 5312: 5237: 5215: 5195: 5154: 5097: 5050: 5042: 5006: 4958: 4604: 4519: 4354: 3261: 3207: 3183: 3147: 3083: 2997: 2993: 2918: 2901: 2895: 2858: 2839: 2773: 2634: 2619: 2240: 2120: 1997: 969: 796: 773: 757: 627: 608: 579: 537: 441: 350: 336:

in 1977 makes a rough division of contemporary mathematical logic into four areas:

278: 6183:

Reprinted in English translation as "Proof that every set can be well-ordered" in

5963:

From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s

5513:

Concept Script, a formal language of pure thought modelled upon that of arithmetic

5101: 4092: 2743:

The continuum hypothesis, first proposed as a conjecture by Cantor, was listed by

2295:

Early results from formal logic established limitations of first-order logic. The

10146: 10136: 10090: 10073: 10028: 9990: 9892: 9812: 9619: 9546: 9519: 9507: 9413: 9327: 9301: 9256: 9224: 9025: 8827: 8770: 8720: 8685: 8643: 8188: 8130: 7851: 7745: 7717: 7611: 7563: 7548: 7533: 7388: 7383: 7330: 7220: 7194: 7146: 7091: 6829: 6756: 6685: 6478: 6347: 5823: 5631:

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

5541: 5506: 4815: 4309: 3537: 3446: 3337: 3300: 3223: 3215: 3159: 3094: 3010: 2950: 2946: 2711: 2696: 2646: 2630: 2220: 2215:

At its core, mathematical logic deals with mathematical concepts expressed using

2105: 930: 910:

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

871: 750: 647: 532: 481: 455: 413: 388: 317: 139: 32: 10241: 10131: 10110: 10068: 10048: 9943: 9798: 9396: 9386: 9376: 9371: 9305: 9179: 9055: 8944: 8939: 8917: 8518: 8442: 8135: 8114: 8029: 7957: 7861: 7760: 7606: 7578: 6907: 6834: 6541: 6357: 6109: 6019: 5982: 5835: 5257: 4705: 4375: 3532: 3284: 3195: 3175: 3171: 3039: 2992:

An important subfield of recursion theory studies algorithmic unsolvability; a

2942: 2934: 2765: 2758: 2269: 2216: 2115: 1358: 1015: 1000: 859: 769: 623: 571: 567: 451: 447: 167: 5344: 4608: 4523: 2784:, their existence has many ramifications for the structure of the real line. 10230: 10105: 9783: 9290: 9075: 9065: 9035: 9020: 8690: 8292: 8233: 7846: 7141: 6695: 6627: 6579: 6242: 6221: 6191: 6150: 6071: 5931: 5789: 5752: 5731: 5522: 5501: 5414: 5400: 5291: 5287: 4654:

An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof

4511: 4438: 4305: 3383: 3363: 3222:). Its applications to the history of logic have proven extremely fruitful ( 3191: 3163: 3151: 3143: 3047: 3021: 2922: 2914: 2854: 2744: 2732: 2728: 2684:(ZF), which is now the most widely used foundational theory for mathematics. 2637:

to study the semantics of formal logics. A fundamental example is the use of

2370:, which allow for formulas to provide an infinite amount of information, and 2130: 1023: 958: 941: 824:, which Russell and Whitehead developed in an effort to avoid the paradoxes. 738: 709: 659: 655: 619: 527: 305: 92: 5662: 5626: 5594: 5578: 3029: 889: 520:

made substantial work on algebraization of logic, independently from Boole.

313: 10005: 9852: 9753: 9745: 9625: 9573: 9482: 9418: 9401: 9332: 9191: 9050: 8752: 8535: 8282: 8104: 7942: 7601: 6637: 6632: 6536: 6091: 6038:"Computability Theory and Applications: The Art of Classical Computability" 5970: 5793: 5667:"Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes" 5530: 5488:(1922). "Der Begriff 'definit' und die Unabhängigkeit des Auswahlsaxioms". 5380:"Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" 5317: 3614:), Bernays wrote the following, which is reminiscent of the famous note by 3485: 3465: 3415: 3403: 3292: 3276: 3227: 3155: 3119: 3098: 3069: 3063: 2811: 2805: 2793: 2722:

that contains exactly one element from each set in the collection. The set

2339: 2320: 1930: 962: 746: 693: 665: 575: 501: 356: 345: 270: 266: 6394: 6329: 5078:

Hamkins, Joel David; Löwe, Benedikt (2007). "The modal logic of forcing".

3340:

is precisely the set of languages expressible by sentences of existential

2309:

established the equivalence between semantic and syntactic definitions of

258: 10115: 9995: 9174: 9164: 9111: 8795: 8715: 8700: 8580: 8525: 8317: 8312: 8265: 7932: 7558: 7470: 6839: 6503: 6426: 6129:

Das Kontinuum. Kritische Untersuchungen über die Grund lagen der Analysis

5418: 4731: 4675: 3430:, and the techniques he developed to do so were seminal in proof theory. 3203: 3025: 2926: 2781: 2769: 2642: 2600: 2578:

implies that the only extension of first-order logic satisfying both the

2569: 2540:

Higher-order logics allow for quantification not only of elements of the

2342:, the Gödel sentence holds for the natural numbers but cannot be proved. 2300: 2110: 1570: 921: 821: 669: 262: 130: 56: 6366: 6320: 5509:, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens 3410:, satisfies the axioms of plane geometry except the parallel postulate. 749:

had been unable to obtain. To achieve the proof, Zermelo introduced the

662:

of rational numbers, a definition still employed in contemporary texts.

446:

Theories of logic were developed in many cultures in history, including

9045: 8900: 8871: 8677: 8260: 8228: 8193: 7952: 7882: 7475: 7215: 7071: 6824: 6703: 6498: 6262: 6213: 6170: 5923: 5865: 5772: 5646: 5610: 5557: 5241: 5168: 5062: 5054: 5018: 4970: 4907: 4891: 4797: 2853:, a result which also shows the theory of the field of real numbers is 2753: 2668: 2662: 2615: 2148: 836: 832: 477: 429: 340: 297: 274: 135: 125: 5207: 4258: 4256: 3488:, intuitionism became easier to reconcile with classical mathematics. 3433:

A second thread in the history of foundations of mathematics involves

530:

presented an independent development of logic with quantifiers in his

10197: 10100: 9153: 9070: 9030: 8994: 8930: 8742: 8732: 8705: 8468: 8322: 8183: 8094: 7457: 7418: 6024:

Videnskapsselskapet Skrifter, I. Matematisk-naturvidenskabelig Klasse

5473:

From Kant to Hilbert: A Source Book in the Foundations of Mathematics

5332: 5092: 4683: 4283: 3577: 2374:, which include a portion of set theory directly in their semantics. 2231:

are the most widely studied today, because of their applicability to

1888: 875: 612: 574:

published a set of axioms for arithmetic that came to bear his name (

425: 6389: 5856: 5839: 5159: 5140: 5046: 5010: 4962: 2752:

of set theory in which the continuum hypothesis must hold. In 1963,

582:

showed that the natural numbers are uniquely characterized by their

10182: 9980: 9428: 9133: 8727: 8243: 7518: 7008: 6728: 6647: 6574: 5706:

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

5199: 4687: 4253: 3527: 1672: 1010:

Numerous results in recursion theory were obtained in the 1940s by

995:

The study of computability came to be known as recursion theory or

940:

The first textbook on symbolic logic for the layman was written by

689: 473: 417: 293: 107: 97: 6308: 4837: 4773: 3709: 2740:, is one of many counterintuitive results of the axiom of choice. 2366:

Many logics besides first-order logic are studied. These include

925:

arithmetic using a finitistic system together with a principle of

9778: 8570: 8307: 8238: 6513: 5944:

Translated as "On possibilities in the calculus of relatives" in

5739: 2687:

Other formalizations of set theory have been proposed, including

721: 716:

for the next century. The first two of these were to resolve the

468: 102: 3218:, E. Stamm), and even to metaphysics (J. Salamucha, H. Scholz, 2329:

establish additional limits on first-order axiomatizations. The

8145: 4657: 4554: 4465: 4371: 3377:

Cantor's study of arbitrary infinite sets also drew criticism.

3359: 2618:; this is not true in classical theories of arithmetic such as 2273: 651: 630:

began to construct functions that stretched intuition, such as

6327:

Detlovs, Vilnis, and Podnieks, Karlis (University of Latvia),

4847:

Logical consequences. Theory and applications: An introduction

391:

uses many formal axiomatic methods, and includes the study of

312:

to prove the consistency of foundational theories. Results of

31:

For Quine's theory sometimes called "Mathematical Logic", see

9322: 8668: 8513: 8337: 8052: 5544:(1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". 5448:

English translation as: "Consistency and irrational numbers".

4334: 3547: 3050:. The algorithmic unsolvability of the problem was proved by 2699:(NF). Of these, ZF, NBG, and MK are similar in describing a 400: 289: 4997:

Felscher, Walter (2000). "Bolzano, Cauchy, Epsilon, Delta".

3174:), but also to physics (R. Carnap, A. Dittrich, B. Russell, 3057: 8297: 5280: 3600: 2838:

The set of all models of a particular theory is called an

2764:

Contemporary research in set theory includes the study of

5878:

Geometrishe Untersuchungen zur Theorie der Parellellinien

3118:

Recent developments in proof theory include the study of

8021: 5176:

Soare, Robert I. (1996). "Computability and recursion".

4738:. Springer Monographs in Mathematics. Berlin, New York: 4169: 3967: 3928: 3105:

systems is of particular interest. Results such as the

4814: 4682:. Studies in Logic and the Foundations of Mathematics. 4157: 3984: 3982: 3957: 3955: 3817: 3697: 2815:

studies the models of various formal theories. Here a

492:, but their labors remained isolated and little known. 5226:(1976). "Provability Interpretations of Modal Logic". 3906: 3904: 2987: 2819:

is a set of formulas in a particular formal logic and

2530:{\displaystyle (x=0)\lor (x=1)\lor (x=2)\lor \cdots .} 6076:

A decision method for elementary algebra and geometry

5331:. Synthese Library, Vol. 1. Translated by Otto Bird. 4022: 3733: 3721: 2466: 2423: 2383: 2059: 2033: 2007: 1973: 1940: 1898: 1864: 1838: 1812: 1786: 1755: 1728: 1705: 1682: 1639: 1613: 1580: 1546: 1520: 1494: 1453: 1427: 1401: 1368: 1334: 1308: 1282: 1248: 1222: 1196: 1159: 1133: 1110: 1084: 1058: 803:(ZF). Zermelo's axioms incorporated the principle of 622:, including theories of convergence of functions and 618:

The 19th century saw great advances in the theory of

6247:"Untersuchungen über die Grundlagen der Mengenlehre" 6196:"Neuer Beweis für die Möglichkeit einer Wohlordnung" 4430: 4304: 4051: 4049: 3999: 3997: 3979: 3952: 3916: 3865: 3853: 3793: 3495: 3245: 3024:

in 1955 and independently by W. Boone in 1959. The

2707:

is closely related to generalized recursion theory.

6065:(in Lithuanian). Papie: Per Franciscum Gyrardengum. 5945: 5589:]. doctoral dissertation. University Of Vienna. 5574:, M. E. Szabo, ed., North-Holland, Amsterdam, 1969. 4765:

New York: Van Nostrand. (Ishi Press: 2009 reprint).

4540: 4145: 4097: 4061: 3940: 3901: 3889: 3829: 3805: 3406:on the sphere. The resulting structure, a model of 2984:, as well as new results in pure recursion theory. 953: 900:, demonstrating the finitary nature of first-order 4395:Formal Number Theory and Computability: A Workbook 4205: 4193: 4181: 4121: 4109: 4034: 3877: 3745: 3668: 3264:often focus on concrete programming languages and 3097:systems. An early proponent of predicativism was 2680:, due to Zermelo, was extended slightly to become 2529: 2449: 2409: 2071: 2045: 2019: 1985: 1959: 1918: 1876: 1850: 1824: 1798: 1764: 1741: 1714: 1691: 1660: 1625: 1599: 1558: 1532: 1506: 1474: 1439: 1413: 1387: 1346: 1320: 1294: 1260: 1234: 1208: 1174: 1145: 1119: 1096: 1070: 843:, which is now an important tool for establishing 6155:"Beweis, daß jede Menge wohlgeordnet werden kann" 5844:Transactions of the American Mathematical Society 5146:Transactions of the American Mathematical Society 5080:Transactions of the American Mathematical Society 4936:. Reidel, Dordrecht, 1971 (revised edition 1979). 4557:: Mathematisches Institut der Universität München 4229: 4217: 4133: 4073: 4046: 3994: 3841: 3639:A detailed study of this terminology is given by 2645:in classical propositional logic, and the use of 1003:, it became clear that a new concept – the 688:that no set can have the same cardinality as its 10228: 4431:Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994). 4241: 3781: 3646: 6003:Revue Générale des Sciences Pures et Appliquées 5388:Journal für die Reine und Angewandte Mathematik 4838:Research papers, monographs, texts, and surveys 4093:https://archive.org/details/symboliclogic00carr 2589: 6235: 6184: 6097:Proceedings of the London Mathematical Society 6094:(1939). "Systems of Logic Based on Ordinals". 6012: 5994: 5951:A Source Book in Mathematical Logic, 1879–1931 5788: 5700: 5570:Reprinted in English translation in Gentzen's 5516: 5495: 5370: 3611: 3606:In the foreword to the 1934 first edition of " 8484: 8037: 7048:Note: This template roughly follows the 2012 7024: 6410: 5988:Arithmetices principia, nova methodo exposita 5883: 4934:The Logical Structure of Mathematical Physics 3347: 3332:. The first significant result in this area, 2196: 881:. This counterintuitive fact became known as 304:. In the early 20th century it was shaped by 235: 6303:Polyvalued logic and Quantity Relation Logic 5689:Reprinted in English translation in Gödel's 5362: 5286: 5269:Notices of the American Mathematical Society 5028:"The Road to Modern Logic-An Interpretation" 4977: 4508:A Concise Introduction to Mathematical Logic 4416:(2nd ed.). Cambridge University Press. 4175: 3934: 3765:"Bertić, Vatroslav | Hrvatska enciklopedija" 2929:in the 1930s, which was greatly extended by 2761:, although its importance is not yet clear. 2710:Two famous statements in set theory are the 6424: 5872: 5697:et al., eds. Oxford University Press, 1993. 5077: 4163: 3823: 3553:List of computability and complexity topics 2284:are characterized by the limitation of all 929:. Gentzen's result introduced the ideas of 733: 632:nowhere-differentiable continuous functions 8676: 8491: 8477: 8044: 8030: 7031: 7017: 6417: 6403: 6141:: CS1 maint: location missing publisher ( 6057: 5511:. Halle a. S.: Louis Nebert. Translation: 4882:: CS1 maint: location missing publisher ( 4788: 4502: 3727: 2377:The most well studied infinitary logic is 2203: 2189: 692:. Cantor believed that every set could be 650:in 1817, but remained relatively unknown. 242: 228: 6309:forall x: an introduction to formal logic 6117: 6036:Soare, Robert Irving (22 December 2011). 5902: 5855: 5757:"Probleme der Grundlegung der Mathematik" 5682: 5583:Über die Vollständigkeit des Logikkalküls 5326: 5316: 5189: 5158: 5091: 5025: 4594: 4476: 4262: 4028: 3973: 3715: 3703: 3652: 3058:Proof theory and constructive mathematics 3020:was proved algorithmically unsolvable by 2361: 1029: 772:cannot form a set. Very soon thereafter, 500:In the middle of the nineteenth century, 6241: 6190: 5953:. Harvard Univ. Press. pp. 228–251. 5748:) republished 1980, Open Court, Chicago. 5484: 5451: 5438: 5281:Classical papers, texts, and collections 4996: 4411: 4365: 3988: 3961: 3922: 3871: 3859: 3799: 3258:computability theory in computer science 3206:, U. Klug, P. Oppenheim), to economics ( 3032:in 1962, is another well-known example. 587:of addition and multiplication from the 554: 6149: 6000: 5957: 5751: 5730: 5540: 5413: 5222: 4912:Set theory and the continuum hypothesis 4896:Set Theory and the Continuum Hypothesis 4864: 4843: 4674: 4651: 4584:. Covers logics in close relation with 4277: 4151: 4103: 4067: 3946: 3910: 3835: 3739: 3674: 3291:between proofs and programs relates to 2450:{\displaystyle L_{\omega _{1},\omega }} 2410:{\displaystyle L_{\omega _{1},\omega }} 1533:{\displaystyle A\not \Leftrightarrow B} 654:in 1821 defined continuity in terms of 363:(considered as parts of a single area). 14: 10229: 8498: 7741:Knowledge representation and reasoning 6321:A Problem Course in Mathematical Logic 6090: 6070: 6018: 5840:"Recursive Predicates and Quantifiers" 5834: 5377: 5256: 5135: 4704: 4632: 4392: 4271: 4211: 4199: 4187: 4127: 4115: 4040: 3883: 3543:Knowledge representation and reasoning 3398:to mean a point on a fixed sphere and 3109:show that it is possible to embed (or 1960:{\displaystyle A{\underline {\lor }}B} 1475:{\displaystyle {\overline {A\cdot B}}} 1388:{\displaystyle A{\overline {\land }}B} 764:was the first to state a paradox: the 549:Vorlesungen über die Algebra der Logik 327: 8472: 8025: 7766:Philosophy of artificial intelligence 7012: 6398: 6324:, a free textbook by Stefan Bilaniuk. 6035: 5981: 5969: 5908:"Über Möglichkeiten im Relativkalkül" 5661: 5635:Monatshefte für Mathematik und Physik 5625: 5599:Monatshefte für Mathematik und Physik 5593: 5577: 5468:. Beman, W. W., ed. and trans. Dover. 5175: 4939: 4906: 4890: 4235: 4223: 4139: 4079: 4055: 4016: 4003: 3847: 3787: 3640: 3394:can be proved consistent by defining 3374:continuous function were discovered. 3234:, J. Salamucha, K. Duerr, Z. Jordan, 1600:{\displaystyle A{\overline {\lor }}B} 1261:{\displaystyle A\leftrightharpoons B} 367:Additionally, sometimes the field of 7092:Energy consumption (Green computing) 7038: 6979: 6126: 5882:Reprinted in English translation as 5587:Completeness of the logical calculus 5529:. Breslau: W. Koebner. Translation: 5327:Bochenski, Jozef Maria, ed. (1959). 5116: 4730: 4459: 4368:A mathematical introduction to logic 4247: 3895: 3811: 3751: 2689:von Neumann–Bernays–Gödel set theory 2276:involves only finite expressions as 2254: 8219:Analytic and synthetic propositions 8090:Formal semantics (natural language) 7771:Distributed artificial intelligence 7050:ACM Computing Classification System 6991: 6338:Stanford Encyclopedia of Philosophy 6330:Introduction to Mathematical Logic. 5457:Was sind und was sollen die Zahlen? 3618:when informed of Russell's paradox. 3238:, J. M. Bochenski, S. T. Schayer, 2988:Algorithmically unsolvable problems 2889: 1919:{\displaystyle A\ {\text{XNOR}}\ B} 1559:{\displaystyle A\nleftrightarrow B} 24: 7283:Integrated development environment 5684:10.1111/j.1746-8361.1958.tb01464.x 5475:, 2 vols, Ewald, William B., ed., 5262:"The Continuum Hypothesis, Part I" 4635:Fundamentals of mathematical logic 4482:Introduction to Mathematical Logic 4280:Introduction to Mathematical Logic 3273:semantics of programming languages 3107:Gödel–Gentzen negative translation 2625: 1683: 1235:{\displaystyle A\Leftrightarrow B} 1166: 1163: 1137: 25: 10258: 7751:Automated planning and scheduling 7288:Software configuration management 6278: 5975:Vorlesungen über neuere Geometrie 5444:Stetigkeit und irrationale Zahlen 5365:A question on transfinite numbers 4999:The American Mathematical Monthly 4950:The American Mathematical Monthly 4818:; Schwichtenberg, Helmut (2000). 4777:John Wiley. Dover reprint, 2002. 4626: 4339:. London, Oxford, New York City: 3568:List of mathematical logic topics 3246:Connections with computer science 2584:downward Löwenheim–Skolem theorem 2336:non-standard models of arithmetic 1661:{\displaystyle {\overline {A+B}}} 850: 768:shows that the collection of all 10210: 8431: 8005: 7995: 7986: 7985: 6990: 6978: 6967: 6966: 6954: 4850:. London: College Publications. 4762:Introduction to Metamathematics. 3512: 3498: 2174: 2173: 954:Beginnings of the other branches 947:Alice's Adventures in Wonderland 684:, and used this method to prove 466:(or term logic) as found in the 435: 212: 7996: 7399:Computational complexity theory 6875:Computational complexity theory 5466:Essays on the Theory of Numbers 4980:Computability and Unsolvability 4085: 4009: 3633: 3137: 3076:Hilbert-style deduction systems 2799: 2327:Gödel's incompleteness theorems 1742:{\displaystyle {\overline {A}}} 743:every set could be well-ordered 699: 495: 462:. Greek methods, particularly 369:computational complexity theory 7190:Network performance evaluation 6378:Set Theory & Further Logic 5492:(in German). pp. 253–257. 5329:A Precis of Mathematical Logic 4736:Set Theory: Millennium Edition 4680:Handbook of Mathematical Logic 4660:: Kluwer Academic Publishers. 3757: 3680: 2515: 2503: 2497: 2485: 2479: 2467: 2063: 2011: 1617: 1507:{\displaystyle A\not \equiv B} 1405: 1347:{\displaystyle A\rightarrow B} 1338: 1295:{\displaystyle A\Rightarrow B} 1286: 1252: 1226: 1175:{\displaystyle A\&\&B} 915:Gödel's incompleteness theorem 380:in modal logic. The method of 374:Gödel's incompleteness theorem 334:Handbook of Mathematical Logic 13: 1: 10171:History of mathematical logic 7554:Multimedia information system 7539:Geographic information system 7529:Enterprise information system 7125:Computer systems organization 6367:London Philosophy Study Guide 6063:Calculationes Suiseth Anglici 5423:. Kessinger Legacy Reprints. 5363:Burali-Forti, Cesare (1897). 5229:Israel Journal of Mathematics 5102:10.1090/s0002-9947-07-04297-3 3662: 3366:, and the very definition of 3326:Descriptive complexity theory 3307:are now studied as idealized 3299:. Formal calculi such as the 2874:A trivial consequence of the 2865:Morley's categoricity theorem 2656: 2562:primitive recursive functions 2352:second incompleteness theorem 2243:are also studied, along with 2072:{\displaystyle A\leftarrow B} 2020:{\displaystyle A\Leftarrow B} 1626:{\displaystyle A\downarrow B} 920:Gödel's theorem shows that a 810:In 1910, the first volume of 10096:Primitive recursive function 7913:Computational social science 7501:Theoretical computer science 7321:Software development process 7097:Electronic design automation 7082:Very Large Scale Integration 5798:Grundlagen der Mathematik. I 5515:, by S. Bauer-Mengelberg in 4976:Reprinted as an appendix in 3558:List of first-order theories 3462:Luitzen Egbertus Jan Brouwer 2909:, studies the properties of 2590:Nonclassical and modal logic 2331:first incompleteness theorem 2307:Gödel's completeness theorem 1877:{\displaystyle A\parallel B} 1734: 1653: 1589: 1467: 1377: 807:to avoid Russell's paradox. 603:developed a complete set of 591:and mathematical induction. 566:refers to the theory of the 7: 7736:Natural language processing 7524:Information storage systems 6291:Encyclopedia of Mathematics 5746:The Foundations of Geometry 4337:What is mathematical logic? 4288:World Scientific Publishing 3491: 3460:In the early 20th century, 3289:Curry–Howard correspondence 3214:), to practical questions ( 3186:, P. Fevrier), to biology ( 2955:recursively enumerable sets 2682:Zermelo–Fraenkel set theory 2547:Another type of logics are 1414:{\displaystyle A\uparrow B} 801:Zermelo–Fraenkel set theory 636:arithmetization of analysis 480:, began the development of 10: 10263: 9160:Schröder–Bernstein theorem 8887:Monadic predicate calculus 8546:Foundations of mathematics 7652:Human–computer interaction 7622:Intrusion detection system 7534:Social information systems 7519:Database management system 6925:Films about mathematicians 5965:. Oxford University Press. 5461:Two English translations: 5178:Bulletin of Symbolic Logic 5035:Bulletin of Symbolic Logic 4714:Cambridge University Press 4652:Andrews, Peter B. (2002). 4366:Enderton, Herbert (2001). 4318:Cambridge University Press 4176:Banach & Tarski (1924) 3612:Hilbert & Bernays 1934 3478:law of the excluded middle 3451:number-theoretic functions 3354:Foundations of mathematics 3351: 3348:Foundations of mathematics 3249: 3061: 3002:algorithmically unsolvable 2893: 2803: 2660: 2612:law of the excluded middle 2593: 2258: 2233:foundations of mathematics 2046:{\displaystyle A\subset B} 1321:{\displaystyle A\supset B} 640:(ε, δ)-definition of limit 514:foundations of mathematics 439: 406: 387:The mathematical field of 283:foundations of mathematics 152:Relationship with sciences 29: 10247:Philosophy of mathematics 10206: 10193:Philosophy of mathematics 10142:Automated theorem proving 10124: 10019: 9851: 9744: 9596: 9313: 9289: 9267:Von Neumann–Bernays–Gödel 9212: 9106: 9010: 8908: 8899: 8826: 8761: 8667: 8589: 8506: 8426: 8386: 8358: 8351: 8303:Necessity and sufficiency 8206: 8171: 8123: 8077: 8059: 8051: 7981: 7918:Computational engineering 7893:Computational mathematics 7870: 7817: 7779: 7726: 7688: 7650: 7592: 7509: 7455: 7417: 7369: 7306: 7239: 7203: 7160: 7124: 7057: 7046: 6948: 6898: 6855: 6765: 6727: 6694: 6646: 6618: 6565: 6512: 6494:Philosophy of mathematics 6469: 6434: 6383:Philosophy of Mathematics 6119:21.11116/0000-0001-91CE-3 6045:Department of Mathematics 5345:10.1007/978-94-017-0592-9 4844:Augusto, Luis M. (2017). 4633:Hinman, Peter G. (2005). 4609:10.1007/978-1-4471-4558-5 4524:10.1007/978-1-4419-1221-3 4164:Hamkins & Löwe (2007) 3608:Grundlagen der Mathematik 3573:List of set theory topics 3470:philosophy of mathematics 3316:automated theorem proving 3252:Logic in computer science 2678:first such axiomatization 1986:{\displaystyle A\oplus B} 1209:{\displaystyle A\equiv B} 907:In 1931, Gödel published 646:was already developed by 626:. Mathematicians such as 265:. Major subareas include 7928:Computational healthcare 7923:Differentiable computing 7842:Graphics processing unit 7268:Domain-specific language 7137:Computational complexity 6930:Recreational mathematics 6354:First-order Model Theory 6110:10.1112/plms/s2-45.1.161 5736:Grundlagen der Geometrie 5714:Harvard University Press 5401:10.1515/crll.1874.77.258 5119:A History of Mathematics 5117:Katz, Victor J. (1998). 5026:Ferreirós, José (2001). 4599:. Universitext. Berlin: 4595:van Dalen, Dirk (2013). 4484:(4th ed.). London: 4414:Logic for Mathematicians 4278:Walicki, Michał (2011). 3593: 3439:constructive mathematics 3336:(1974) established that 3330:computational complexity 3128:proof-theoretic ordinals 3091:constructive mathematics 2963:hyperarithmetical theory 2705:Kripke–Platek set theory 2297:Löwenheim–Skolem theorem 1146:{\displaystyle A\&B} 1097:{\displaystyle A\cdot B} 1071:{\displaystyle A\land B} 975:Éléments de mathématique 961:developed the basics of 935:proof-theoretic ordinals 888:In his doctoral thesis, 864:Löwenheim–Skolem theorem 734:Set theory and paradoxes 361:constructive mathematics 9843:Self-verifying theories 9664:Tarski's axiomatization 8615:Tarski's undefinability 8610:incompleteness theorems 7903:Computational chemistry 7837:Photograph manipulation 7728:Artificial intelligence 7544:Decision support system 6815:Mathematical statistics 6805:Mathematical psychology 6775:Engineering mathematics 6709:Algebraic number theory 6047:. University of Chicago 5477:Oxford University Press 5304:Fundamenta Mathematicae 5141:"Categoricity in Power" 4945:Hilbert's tenth problem 4574:Oxford University Press 4470:D. C. Heath and Company 4412:Hamilton, A.G. (1988). 4341:Oxford University Press 4314:Computability and Logic 3036:Hilbert's tenth problem 3018:word problem for groups 2978:computable model theory 2967:α-recursion theory 2772:. Large cardinals are 2693:Morse–Kelley set theory 2348:elementarily equivalent 2126:Functional completeness 1851:{\displaystyle A\mid B} 1799:{\displaystyle A\lor B} 1440:{\displaystyle A\mid B} 1030:Formal logical systems 712:posed a famous list of 27:Subfield of mathematics 10217:Mathematics portal 9828:Proof of impossibility 9476:propositional variable 8786:Propositional calculus 7968:Educational technology 7799:Reinforcement learning 7549:Process control system 7447:Computational geometry 7437:Algorithmic efficiency 7432:Analysis of algorithms 7087:Systems on Chip (SoCs) 6961:Mathematics portal 6810:Mathematical sociology 6790:Mathematical economics 6785:Mathematical chemistry 6714:Analytic number theory 6595:Differential equations 6127:Weyl, Hermann (1918). 5886:Non-Euclidean Geometry 5744:English 1902 edition ( 5378:Cantor, Georg (1874). 5318:10.4064/fm-6-1-244-277 4918:: Dover Publications. 4710:A shorter model theory 4542:Schwichtenberg, Helmut 3629: 3623: 3583:Propositional calculus 3392:non-Euclidean geometry 3266:feasible computability 3202:), to law and morals ( 3086:developed by Gentzen. 3028:problem, developed by 2974:algorithmic randomness 2847:quantifier elimination 2750:constructible universe 2586:is first-order logic. 2560:, like one writes for 2531: 2451: 2411: 2362:Other classical logics 2270:formal system of logic 2217:formal logical systems 2096:Propositional calculus 2073: 2047: 2021: 1987: 1961: 1920: 1878: 1852: 1826: 1800: 1766: 1765:{\displaystyle \sim A} 1743: 1716: 1693: 1692:{\displaystyle \neg A} 1662: 1627: 1601: 1560: 1534: 1508: 1476: 1441: 1415: 1389: 1348: 1322: 1296: 1262: 1236: 1210: 1176: 1147: 1121: 1098: 1072: 1020:arithmetical hierarchy 818:Alfred North Whitehead 522:Charles Sanders Peirce 40:Logic (disambiguation) 10086:Kolmogorov complexity 10039:Computably enumerable 9939:Model complete theory 9731:Principia Mathematica 8791:Propositional formula 8620:Banach–Tarski paradox 8438:Philosophy portal 7938:Electronic publishing 7908:Computational biology 7898:Computational physics 7794:Unsupervised learning 7708:Distributed computing 7584:Information retrieval 7491:Mathematical analysis 7481:Mathematical software 7371:Theory of computation 7336:Software construction 7326:Requirements analysis 7204:Software organization 7132:Computer architecture 7102:Hardware acceleration 7067:Printed circuit board 6940:Mathematics education 6870:Theory of computation 6590:Hypercomplex analysis 6312:, a free textbook by 6251:Mathematische Annalen 6201:Mathematische Annalen 6159:Mathematische Annalen 6131:(in German). Leipzig. 5912:Mathematische Annalen 5761:Mathematische Annalen 5546:Mathematische Annalen 4978:Martin Davis (1985). 4816:Troelstra, Anne Sjerp 4790:Shoenfield, Joseph R. 4460:Katz, Robert (1964). 4393:Fisher, Alec (1982). 3625: 3619: 3563:List of logic symbols 3455:mathematical analysis 3428:transfinite induction 3309:programming languages 2778:inaccessible cardinal 2738:Banach–Tarski paradox 2558:inductive definitions 2532: 2452: 2412: 2154:Programming languages 2074: 2048: 2022: 1988: 1962: 1921: 1879: 1853: 1827: 1801: 1767: 1744: 1717: 1694: 1663: 1628: 1602: 1561: 1535: 1509: 1477: 1442: 1416: 1390: 1349: 1323: 1297: 1263: 1237: 1211: 1177: 1148: 1122: 1099: 1073: 927:transfinite induction 826:Principia Mathematica 813:Principia Mathematica 555:Foundational theories 440:Further information: 18:Mathematical logician 10034:Church–Turing thesis 10021:Computability theory 9230:continuum hypothesis 8748:Square of opposition 8606:Gödel's completeness 7698:Concurrent computing 7670:Ubiquitous computing 7642:Application security 7637:Information security 7466:Discrete mathematics 7442:Randomized algorithm 7394:Computability theory 7379:Model of computation 7351:Software maintenance 7346:Software engineering 7308:Software development 7258:Programming language 7253:Programming paradigm 7170:Network architecture 6920:Informal mathematics 6800:Mathematical physics 6795:Mathematical finance 6780:Mathematical biology 6719:Diophantine geometry 6286:"Mathematical logic" 5874:Lobachevsky, Nikolai 5836:Kleene, Stephen Cole 5702:van Heijenoort, Jean 5537:, 2nd ed. Blackwell. 5486:Fraenkel, Abraham A. 4769:Kleene, Stephen Cole 4757:Kleene, Stephen Cole 4586:computability theory 4504:Rautenberg, Wolfgang 3769:www.enciklopedija.hr 3297:intuitionistic logic 3281:program verification 3007:Entscheidungsproblem 2911:computable functions 2907:computability theory 2884:Robert Lawson Vaught 2876:continuum hypothesis 2859:o-minimal structures 2716:continuum hypothesis 2701:cumulative hierarchy 2633:uses the methods of 2608:Intuitionistic logic 2464: 2421: 2381: 2278:well-formed formulas 2249:intuitionistic logic 2245:Non-classical logics 2057: 2031: 2005: 1971: 1938: 1896: 1862: 1836: 1810: 1784: 1753: 1726: 1703: 1680: 1637: 1611: 1578: 1544: 1518: 1492: 1451: 1425: 1399: 1366: 1332: 1306: 1280: 1246: 1220: 1194: 1157: 1131: 1108: 1082: 1056: 997:computability theory 894:completeness theorem 845:independence results 793:axiom of replacement 766:Burali-Forti paradox 727:Entscheidungsproblem 718:continuum hypothesis 644:continuous functions 117:Discrete mathematics 38:For other uses, see 10188:Mathematical object 10079:P versus NP problem 10044:Computable function 9838:Reverse mathematics 9764:Logical consequence 9641:primitive recursive 9636:elementary function 9409:Free/bound variable 9262:Tarski–Grothendieck 8781:Logical connectives 8711:Logical equivalence 8561:Logical consequence 8100:Philosophy of logic 7973:Document management 7963:Operations research 7888:Enterprise software 7804:Multi-task learning 7789:Supervised learning 7511:Information systems 7341:Software deployment 7298:Software repository 7152:Real-time computing 6935:Mathematics and art 6845:Operations research 6600:Functional analysis 6238:, pp. 183–198. 6236:van Heijenoort 1976 6187:, pp. 139–141. 6185:van Heijenoort 1976 6059:Swineshead, Richard 6015:, pp. 142–144. 6013:van Heijenoort 1976 5995:van Heijenoort 1976 5947:Jean van Heijenoort 5517:van Heijenoort 1976 5498:, pp. 284–289. 5496:van Heijenoort 1976 5371:van Heijenoort 1976 4866:Boehner, Philotheus 4774:Mathematical Logic. 4637:. A K Peters, Ltd. 4597:Logic and Structure 4272:Undergraduate texts 3935:Burali-Forti (1897) 3588:Well-formed formula 3435:nonclassical logics 3262:Computer scientists 2982:reverse mathematics 2880:Vaught's conjecture 2596:Non-classical logic 2580:compactness theorem 2576:Lindström's theorem 2542:domain of discourse 2372:higher-order logics 2358:cannot be reached. 2316:compactness theorem 2311:logical consequence 2290:domain of discourse 2225:propositional logic 2164:Philosophy of logic 1825:{\displaystyle A+B} 1038:Logical connectives 1012:Stephen Cole Kleene 1005:computable function 902:logical consequence 898:compactness theorem 870:cannot control the 762:Cesare Burali-Forti 678:transfinite numbers 605:axioms for geometry 600:Nikolai Lobachevsky 562:In logic, the term 543:From 1890 to 1905, 328:Subfields and scope 322:reverse mathematics 51:Part of a series on 10237:Mathematical logic 9986:Transfer principle 9949:Semantics of logic 9934:Categorical theory 9910:Non-standard model 9424:Logical connective 8551:Information theory 8500:Mathematical logic 8399:Rules of inference 8368:Mathematical logic 8110:Semantics of logic 7756:Search methodology 7703:Parallel computing 7660:Interaction design 7569:Computing platform 7496:Numerical analysis 7486:Information theory 7278:Software framework 7241:Software notations 7180:Network components 7077:Integrated circuit 6880:Numerical analysis 6489:Mathematical logic 6484:Information theory 6373:Mathematical Logic 6263:10.1007/BF01449999 6214:10.1007/BF01450054 6171:10.1007/BF01445300 5924:10.1007/BF01458217 5904:Löwenheim, Leopold 5773:10.1007/BF01782335 5647:10.1007/BF01700692 5611:10.1007/BF01696781 5558:10.1007/BF01565428 5373:, pp. 104–111 5242:10.1007/BF02757006 5224:Solovay, Robert M. 5121:. Addison–Wesley. 4820:Basic Proof Theory 4794:Mathematical Logic 4547:Mathematical Logic 4486:Chapman & Hall 4478:Mendelson, Elliott 4462:Axiomatic Analysis 4434:Mathematical Logic 3824:Lobachevsky (1840) 3520:Mathematics portal 3482:BHK interpretation 3342:second-order logic 3328:relates logics to 3194:), to psychology ( 2851:real-closed fields 2833:algebraic geometry 2651:cylindric algebras 2527: 2447: 2407: 2237:second-order logic 2223:. The systems of 2159:Mathematical logic 2069: 2043: 2017: 1983: 1957: 1952: 1916: 1874: 1848: 1822: 1796: 1762: 1739: 1715:{\displaystyle -A} 1712: 1689: 1658: 1623: 1597: 1556: 1530: 1504: 1472: 1437: 1411: 1385: 1344: 1318: 1292: 1258: 1232: 1206: 1172: 1143: 1120:{\displaystyle AB} 1117: 1094: 1068: 866:, which says that 805:limitation of size 741:gave a proof that 596:parallel postulate 589:successor function 506:Augustus De Morgan 464:Aristotelian logic 255:Mathematical logic 218:Mathematics Portal 10224: 10223: 10156:Abstract category 9959:Theories of truth 9769:Rule of inference 9759:Natural deduction 9740: 9739: 9285: 9284: 8990:Cartesian product 8895: 8894: 8801:Many-valued logic 8776:Boolean functions 8659:Russell's paradox 8634:diagonal argument 8531:First-order logic 8466: 8465: 8422: 8421: 8256:Deductive closure 8202: 8201: 8141:Critical thinking 8019: 8018: 7948:Electronic voting 7878:Quantum Computing 7871:Applied computing 7857:Image compression 7627:Hardware security 7617:Security services 7574:Digital marketing 7361:Open-source model 7273:Modeling language 7185:Network scheduler 7006: 7005: 6605:Harmonic analysis 6333:(hyper-textbook). 5997:, pp. 83–97. 5453:Dedekind, Richard 5440:Dedekind, Richard 4902:: W. A. Benjamin. 4857:978-1-84890-236-7 4829:978-0-521-77911-1 4667:978-1-4020-0763-7 4618:978-1-4471-4557-8 4590:complexity theory 4495:978-0-412-80830-2 4423:978-0-521-36865-0 4404:978-0-19-853188-3 4385:978-0-12-238452-3 4308:; Burgess, John; 4265:, Sec. 0.3, p. 2. 3728:Swineshead (1498) 3718:, Sec. 0.1, p. 1. 3506:Philosophy portal 3424:Hilbert's program 3408:elliptic geometry 3379:Leopold Kronecker 3320:logic programming 3305:combinatory logic 3126:and the study of 3124:Ulrich Kohlenbach 3080:natural deduction 3052:Yuri Matiyasevich 2869:Michael D. Morley 2829:universal algebra 2551:fixed-point logic 2368:infinitary logics 2356:Hilbert's program 2266:First-order logic 2261:First-order logic 2255:First-order logic 2229:first-order logic 2213: 2212: 2082: 2081: 1945: 1912: 1908: 1904: 1737: 1656: 1592: 1470: 1380: 868:first-order logic 856:Leopold Löwenheim 799:, are now called 786:Richard's paradox 778:Russell's paradox 682:diagonal argument 598:, established by 397:Saunders Mac Lane 393:categorical logic 252: 251: 207: 206: 16:(Redirected from 10254: 10215: 10214: 10166:History of logic 10161:Category of sets 10054:Decision problem 9833:Ordinal analysis 9774:Sequent calculus 9672:Boolean algebras 9612: 9611: 9586: 9557:logical/constant 9311: 9310: 9297: 9220:Zermelo–Fraenkel 8971:Set operations: 8906: 8905: 8843: 8674: 8673: 8654:Löwenheim–Skolem 8541:Formal semantics 8493: 8486: 8479: 8470: 8469: 8436: 8435: 8434: 8356: 8355: 8121: 8120: 8085:Computer science 8046: 8039: 8032: 8023: 8022: 8009: 8008: 7999: 7998: 7989: 7988: 7809:Cross-validation 7781:Machine learning 7665:Social computing 7632:Network security 7427:Algorithm design 7356:Programming team 7316:Control variable 7293:Software library 7231:Software quality 7226:Operating system 7175:Network protocol 7040:Computer science 7033: 7026: 7019: 7010: 7009: 6994: 6993: 6982: 6981: 6970: 6969: 6959: 6958: 6890:Computer algebra 6865:Computer science 6585:Complex analysis 6419: 6412: 6405: 6396: 6395: 6315: 6299: 6274: 6233: 6182: 6146: 6140: 6132: 6123: 6121: 6087: 6084:RAND Corporation 6066: 6056: 6054: 6052: 6042: 6031: 6010: 5992: 5991:(in Lithuanian). 5978: 5966: 5954: 5943: 5899: 5881: 5869: 5859: 5831: 5784: 5743: 5727: 5708:(3rd ed.). 5695:Solomon Feferman 5688: 5686: 5677:(3–4): 280–287. 5658: 5622: 5590: 5569: 5542:Gentzen, Gerhard 5493: 5460: 5447: 5434: 5412: 5384: 5368: 5358: 5322: 5320: 5300: 5276: 5266: 5253: 5236:(3–4): 287–304. 5219: 5193: 5172: 5162: 5132: 5113: 5095: 5086:(4): 1793–1818. 5074: 5032: 5022: 4993: 4974: 4947:is unsolvable". 4929: 4903: 4887: 4881: 4873: 4861: 4833: 4811: 4796:(2nd ed.). 4753: 4727: 4701: 4671: 4656:(2nd ed.). 4648: 4622: 4565: 4563: 4562: 4552: 4537: 4510:(3rd ed.). 4499: 4473: 4456: 4437:(2nd ed.). 4427: 4408: 4389: 4370:(2nd ed.). 4362: 4331: 4316:(4th ed.). 4310:Jeffrey, Richard 4301: 4266: 4263:Bochenski (1959) 4260: 4251: 4245: 4239: 4233: 4227: 4221: 4215: 4209: 4203: 4197: 4191: 4185: 4179: 4173: 4167: 4161: 4155: 4149: 4143: 4137: 4131: 4125: 4119: 4113: 4107: 4101: 4095: 4089: 4083: 4077: 4071: 4065: 4059: 4053: 4044: 4038: 4032: 4029:Löwenheim (1915) 4026: 4020: 4013: 4007: 4001: 3992: 3986: 3977: 3974:Ferreirós (2001) 3971: 3965: 3959: 3950: 3944: 3938: 3932: 3926: 3920: 3914: 3908: 3899: 3893: 3887: 3881: 3875: 3869: 3863: 3857: 3851: 3845: 3839: 3833: 3827: 3821: 3815: 3809: 3803: 3797: 3791: 3785: 3779: 3778: 3776: 3775: 3761: 3755: 3749: 3743: 3737: 3731: 3725: 3719: 3716:Bochenski (1959) 3713: 3707: 3704:Ferreirós (2001) 3701: 3695: 3694: 3692: 3684: 3678: 3672: 3656: 3650: 3644: 3637: 3631: 3604: 3522: 3517: 3516: 3508: 3503: 3502: 3501: 3283:(in particular, 3084:sequent calculus 2998:function problem 2994:decision problem 2902:Recursion theory 2896:Recursion theory 2890:Recursion theory 2840:elementary class 2774:cardinal numbers 2672:is the study of 2647:Heyting algebras 2639:Boolean algebras 2635:abstract algebra 2620:Peano arithmetic 2553: 2552: 2536: 2534: 2533: 2528: 2456: 2454: 2453: 2448: 2446: 2445: 2438: 2437: 2416: 2414: 2413: 2408: 2406: 2405: 2398: 2397: 2268:is a particular 2241:infinitary logic 2205: 2198: 2191: 2177: 2176: 2121:Boolean function 2087:Related concepts 2078: 2076: 2075: 2070: 2052: 2050: 2049: 2044: 2026: 2024: 2023: 2018: 1992: 1990: 1989: 1984: 1966: 1964: 1963: 1958: 1953: 1925: 1923: 1922: 1917: 1910: 1909: 1906: 1902: 1883: 1881: 1880: 1875: 1857: 1855: 1854: 1849: 1831: 1829: 1828: 1823: 1805: 1803: 1802: 1797: 1771: 1769: 1768: 1763: 1748: 1746: 1745: 1740: 1738: 1730: 1721: 1719: 1718: 1713: 1698: 1696: 1695: 1690: 1667: 1665: 1664: 1659: 1657: 1652: 1641: 1632: 1630: 1629: 1624: 1606: 1604: 1603: 1598: 1593: 1585: 1565: 1563: 1562: 1557: 1539: 1537: 1536: 1531: 1513: 1511: 1510: 1505: 1481: 1479: 1478: 1473: 1471: 1466: 1455: 1446: 1444: 1443: 1438: 1420: 1418: 1417: 1412: 1394: 1392: 1391: 1386: 1381: 1373: 1353: 1351: 1350: 1345: 1327: 1325: 1324: 1319: 1301: 1299: 1298: 1293: 1267: 1265: 1264: 1259: 1241: 1239: 1238: 1233: 1215: 1213: 1212: 1207: 1181: 1179: 1178: 1173: 1152: 1150: 1149: 1144: 1126: 1124: 1123: 1118: 1103: 1101: 1100: 1095: 1077: 1075: 1074: 1069: 1045: 1044: 1034: 1033: 970:Nicolas Bourbaki 883:Skolem's paradox 835:. Later work by 797:Abraham Fraenkel 774:Bertrand Russell 758:naive set theory 686:Cantor's theorem 628:Karl Weierstrass 580:Richard Dedekind 538:Bertrand Russell 518:Vatroslav Bertić 442:History of logic 414:algebra of logic 351:recursion theory 279:recursion theory 257:is the study of 244: 237: 230: 216: 80: 79: 48: 47: 43: 36: 21: 10262: 10261: 10257: 10256: 10255: 10253: 10252: 10251: 10227: 10226: 10225: 10220: 10209: 10202: 10147:Category theory 10137:Algebraic logic 10120: 10091:Lambda calculus 10029:Church encoding 10015: 9991:Truth predicate 9847: 9813:Complete theory 9736: 9605: 9601: 9597: 9592: 9584: 9304: and 9300: 9295: 9281: 9257:New Foundations 9225:axiom of choice 9208: 9170:Gödel numbering 9110: and 9102: 9006: 8891: 8841: 8822: 8771:Boolean algebra 8757: 8721:Equiconsistency 8686:Classical logic 8663: 8644:Halting problem 8632: and 8608: and 8596: and 8595: 8590:Theorems ( 8585: 8502: 8497: 8467: 8462: 8432: 8430: 8418: 8382: 8373:Boolean algebra 8347: 8198: 8189:Metamathematics 8167: 8119: 8073: 8055: 8050: 8020: 8015: 8006: 7977: 7958:Word processing 7866: 7852:Virtual reality 7813: 7775: 7746:Computer vision 7722: 7718:Multiprocessing 7684: 7646: 7612:Security hacker 7588: 7564:Digital library 7505: 7456:Mathematics of 7451: 7413: 7389:Automata theory 7384:Formal language 7365: 7331:Software design 7302: 7235: 7221:Virtual machine 7199: 7195:Network service 7156: 7147:Embedded system 7120: 7053: 7042: 7037: 7007: 7002: 6953: 6944: 6894: 6851: 6830:Systems science 6761: 6757:Homotopy theory 6723: 6690: 6642: 6614: 6561: 6508: 6479:Category theory 6465: 6430: 6423: 6348:Stewart Shapiro 6344:Classical Logic 6313: 6284: 6281: 6134: 6133: 6092:Turing, Alan M. 6080:Santa Monica CA 6050: 6048: 6040: 6020:Skolem, Thoralf 5983:Peano, Giuseppe 5896: 5857:10.2307/1990131 5812: 5724: 5704:, ed. (1976) . 5691:Collected Works 5572:Collected works 5507:Begriffsschrift 5431: 5395:(77): 258–262. 5382: 5355: 5298: 5283: 5264: 5258:Woodin, W. Hugh 5160:10.2307/1994188 5137:Morley, Michael 5129: 5047:10.2307/2687794 5030: 5011:10.2307/2695743 4990: 4975: 4963:10.2307/2318447 4926: 4875: 4874: 4858: 4840: 4830: 4808: 4750: 4724: 4706:Hodges, Wilfrid 4698: 4668: 4645: 4629: 4619: 4560: 4558: 4550: 4534: 4496: 4453: 4424: 4405: 4386: 4351: 4328: 4298: 4274: 4269: 4261: 4254: 4246: 4242: 4234: 4230: 4222: 4218: 4210: 4206: 4198: 4194: 4186: 4182: 4174: 4170: 4162: 4158: 4150: 4146: 4138: 4134: 4126: 4122: 4114: 4110: 4102: 4098: 4090: 4086: 4078: 4074: 4066: 4062: 4054: 4047: 4039: 4035: 4027: 4023: 4014: 4010: 4002: 3995: 3989:Fraenkel (1922) 3987: 3980: 3972: 3968: 3962:Zermelo (1908b) 3960: 3953: 3945: 3941: 3933: 3929: 3923:Zermelo (1908a) 3921: 3917: 3909: 3902: 3894: 3890: 3882: 3878: 3872:Dedekind (1872) 3870: 3866: 3860:Felscher (2000) 3858: 3854: 3846: 3842: 3834: 3830: 3822: 3818: 3810: 3806: 3800:Dedekind (1888) 3798: 3794: 3786: 3782: 3773: 3771: 3763: 3762: 3758: 3750: 3746: 3738: 3734: 3726: 3722: 3714: 3710: 3702: 3698: 3690: 3686: 3685: 3681: 3673: 3669: 3665: 3660: 3659: 3651: 3647: 3638: 3634: 3605: 3601: 3596: 3538:Universal logic 3518: 3511: 3504: 3499: 3497: 3494: 3447:natural numbers 3356: 3350: 3334:Fagin's theorem 3301:lambda calculus 3254: 3248: 3220:J. M. Bochenski 3180:A. N. Whitehead 3140: 3132:Michael Rathjen 3066: 3060: 3011:halting problem 2990: 2947:λ calculus 2943:Turing machines 2898: 2892: 2808: 2802: 2766:large cardinals 2712:axiom of choice 2697:New Foundations 2665: 2659: 2631:Algebraic logic 2628: 2626:Algebraic logic 2598: 2592: 2550: 2549: 2465: 2462: 2461: 2433: 2429: 2428: 2424: 2422: 2419: 2418: 2393: 2389: 2388: 2384: 2382: 2379: 2378: 2364: 2263: 2257: 2221:formal language 2209: 2168: 2135: 2106:Boolean algebra 2101:Predicate logic 2058: 2055: 2054: 2032: 2029: 2028: 2006: 2003: 2002: 1972: 1969: 1968: 1944: 1939: 1936: 1935: 1905: 1897: 1894: 1893: 1863: 1860: 1859: 1837: 1834: 1833: 1811: 1808: 1807: 1785: 1782: 1781: 1754: 1751: 1750: 1729: 1727: 1724: 1723: 1704: 1701: 1700: 1681: 1678: 1677: 1642: 1640: 1638: 1635: 1634: 1612: 1609: 1608: 1584: 1579: 1576: 1575: 1545: 1542: 1541: 1519: 1516: 1515: 1493: 1490: 1489: 1456: 1454: 1452: 1449: 1448: 1426: 1423: 1422: 1400: 1397: 1396: 1372: 1367: 1364: 1363: 1333: 1330: 1329: 1307: 1304: 1303: 1281: 1278: 1277: 1247: 1244: 1243: 1221: 1218: 1217: 1195: 1192: 1191: 1158: 1155: 1154: 1132: 1129: 1128: 1109: 1106: 1105: 1083: 1080: 1079: 1057: 1054: 1053: 1032: 1001:Turing machines 956: 931:cut elimination 853: 847:in set theory. 816:by Russell and 770:ordinal numbers 751:axiom of choice 736: 702: 568:natural numbers 557: 533:Begriffsschrift 498: 482:predicate logic 444: 438: 409: 389:category theory 330: 318:Gerhard Gentzen 292:frameworks for 248: 203: 202: 153: 145: 144: 140:Decision theory 88: 44: 37: 33:New Foundations 30: 28: 23: 22: 15: 12: 11: 5: 10260: 10250: 10249: 10244: 10239: 10222: 10221: 10207: 10204: 10203: 10201: 10200: 10195: 10190: 10185: 10180: 10179: 10178: 10168: 10163: 10158: 10149: 10144: 10139: 10134: 10132:Abstract logic 10128: 10126: 10122: 10121: 10119: 10118: 10113: 10111:Turing machine 10108: 10103: 10098: 10093: 10088: 10083: 10082: 10081: 10076: 10071: 10066: 10061: 10051: 10049:Computable set 10046: 10041: 10036: 10031: 10025: 10023: 10017: 10016: 10014: 10013: 10008: 10003: 9998: 9993: 9988: 9983: 9978: 9977: 9976: 9971: 9966: 9956: 9951: 9946: 9944:Satisfiability 9941: 9936: 9931: 9930: 9929: 9919: 9918: 9917: 9907: 9906: 9905: 9900: 9895: 9890: 9885: 9875: 9874: 9873: 9868: 9861:Interpretation 9857: 9855: 9849: 9848: 9846: 9845: 9840: 9835: 9830: 9825: 9815: 9810: 9809: 9808: 9807: 9806: 9796: 9791: 9781: 9776: 9771: 9766: 9761: 9756: 9750: 9748: 9742: 9741: 9738: 9737: 9735: 9734: 9726: 9725: 9724: 9723: 9718: 9717: 9716: 9711: 9706: 9686: 9685: 9684: 9682:minimal axioms 9679: 9668: 9667: 9666: 9655: 9654: 9653: 9648: 9643: 9638: 9633: 9628: 9615: 9613: 9594: 9593: 9591: 9590: 9589: 9588: 9576: 9571: 9570: 9569: 9564: 9559: 9554: 9544: 9539: 9534: 9529: 9528: 9527: 9522: 9512: 9511: 9510: 9505: 9500: 9495: 9485: 9480: 9479: 9478: 9473: 9468: 9458: 9457: 9456: 9451: 9446: 9441: 9436: 9431: 9421: 9416: 9411: 9406: 9405: 9404: 9399: 9394: 9389: 9379: 9374: 9372:Formation rule 9369: 9364: 9363: 9362: 9357: 9347: 9346: 9345: 9335: 9330: 9325: 9320: 9314: 9308: 9291:Formal systems 9287: 9286: 9283: 9282: 9280: 9279: 9274: 9269: 9264: 9259: 9254: 9249: 9244: 9239: 9234: 9233: 9232: 9227: 9216: 9214: 9210: 9209: 9207: 9206: 9205: 9204: 9194: 9189: 9188: 9187: 9180:Large cardinal 9177: 9172: 9167: 9162: 9157: 9143: 9142: 9141: 9136: 9131: 9116: 9114: 9104: 9103: 9101: 9100: 9099: 9098: 9093: 9088: 9078: 9073: 9068: 9063: 9058: 9053: 9048: 9043: 9038: 9033: 9028: 9023: 9017: 9015: 9008: 9007: 9005: 9004: 9003: 9002: 8997: 8992: 8987: 8982: 8977: 8969: 8968: 8967: 8962: 8952: 8947: 8945:Extensionality 8942: 8940:Ordinal number 8937: 8927: 8922: 8921: 8920: 8909: 8903: 8897: 8896: 8893: 8892: 8890: 8889: 8884: 8879: 8874: 8869: 8864: 8859: 8858: 8857: 8847: 8846: 8845: 8832: 8830: 8824: 8823: 8821: 8820: 8819: 8818: 8813: 8808: 8798: 8793: 8788: 8783: 8778: 8773: 8767: 8765: 8759: 8758: 8756: 8755: 8750: 8745: 8740: 8735: 8730: 8725: 8724: 8723: 8713: 8708: 8703: 8698: 8693: 8688: 8682: 8680: 8671: 8665: 8664: 8662: 8661: 8656: 8651: 8646: 8641: 8636: 8624:Cantor's 8622: 8617: 8612: 8602: 8600: 8587: 8586: 8584: 8583: 8578: 8573: 8568: 8563: 8558: 8553: 8548: 8543: 8538: 8533: 8528: 8523: 8522: 8521: 8510: 8508: 8504: 8503: 8496: 8495: 8488: 8481: 8473: 8464: 8463: 8461: 8460: 8455: 8445: 8440: 8427: 8424: 8423: 8420: 8419: 8417: 8416: 8411: 8406: 8401: 8396: 8390: 8388: 8384: 8383: 8381: 8380: 8375: 8370: 8364: 8362: 8353: 8349: 8348: 8346: 8345: 8340: 8335: 8330: 8325: 8320: 8315: 8310: 8305: 8300: 8295: 8290: 8285: 8280: 8279: 8278: 8268: 8263: 8258: 8253: 8248: 8247: 8246: 8241: 8231: 8226: 8221: 8216: 8210: 8208: 8204: 8203: 8200: 8199: 8197: 8196: 8191: 8186: 8181: 8175: 8173: 8169: 8168: 8166: 8165: 8160: 8155: 8150: 8149: 8148: 8143: 8133: 8127: 8125: 8118: 8117: 8112: 8107: 8102: 8097: 8092: 8087: 8081: 8079: 8075: 8074: 8072: 8071: 8066: 8060: 8057: 8056: 8049: 8048: 8041: 8034: 8026: 8017: 8016: 8014: 8013: 8003: 7993: 7982: 7979: 7978: 7976: 7975: 7970: 7965: 7960: 7955: 7950: 7945: 7940: 7935: 7930: 7925: 7920: 7915: 7910: 7905: 7900: 7895: 7890: 7885: 7880: 7874: 7872: 7868: 7867: 7865: 7864: 7862:Solid modeling 7859: 7854: 7849: 7844: 7839: 7834: 7829: 7823: 7821: 7815: 7814: 7812: 7811: 7806: 7801: 7796: 7791: 7785: 7783: 7777: 7776: 7774: 7773: 7768: 7763: 7761:Control method 7758: 7753: 7748: 7743: 7738: 7732: 7730: 7724: 7723: 7721: 7720: 7715: 7713:Multithreading 7710: 7705: 7700: 7694: 7692: 7686: 7685: 7683: 7682: 7677: 7672: 7667: 7662: 7656: 7654: 7648: 7647: 7645: 7644: 7639: 7634: 7629: 7624: 7619: 7614: 7609: 7607:Formal methods 7604: 7598: 7596: 7590: 7589: 7587: 7586: 7581: 7579:World Wide Web 7576: 7571: 7566: 7561: 7556: 7551: 7546: 7541: 7536: 7531: 7526: 7521: 7515: 7513: 7507: 7506: 7504: 7503: 7498: 7493: 7488: 7483: 7478: 7473: 7468: 7462: 7460: 7453: 7452: 7450: 7449: 7444: 7439: 7434: 7429: 7423: 7421: 7415: 7414: 7412: 7411: 7406: 7401: 7396: 7391: 7386: 7381: 7375: 7373: 7367: 7366: 7364: 7363: 7358: 7353: 7348: 7343: 7338: 7333: 7328: 7323: 7318: 7312: 7310: 7304: 7303: 7301: 7300: 7295: 7290: 7285: 7280: 7275: 7270: 7265: 7260: 7255: 7249: 7247: 7237: 7236: 7234: 7233: 7228: 7223: 7218: 7213: 7207: 7205: 7201: 7200: 7198: 7197: 7192: 7187: 7182: 7177: 7172: 7166: 7164: 7158: 7157: 7155: 7154: 7149: 7144: 7139: 7134: 7128: 7126: 7122: 7121: 7119: 7118: 7109: 7104: 7099: 7094: 7089: 7084: 7079: 7074: 7069: 7063: 7061: 7055: 7054: 7047: 7044: 7043: 7036: 7035: 7028: 7021: 7013: 7004: 7003: 7001: 7000: 6988: 6976: 6964: 6949: 6946: 6945: 6943: 6942: 6937: 6932: 6927: 6922: 6917: 6916: 6915: 6908:Mathematicians 6904: 6902: 6900:Related topics 6896: 6895: 6893: 6892: 6887: 6882: 6877: 6872: 6867: 6861: 6859: 6853: 6852: 6850: 6849: 6848: 6847: 6842: 6837: 6835:Control theory 6827: 6822: 6817: 6812: 6807: 6802: 6797: 6792: 6787: 6782: 6777: 6771: 6769: 6763: 6762: 6760: 6759: 6754: 6749: 6744: 6739: 6733: 6731: 6725: 6724: 6722: 6721: 6716: 6711: 6706: 6700: 6698: 6692: 6691: 6689: 6688: 6683: 6678: 6673: 6668: 6663: 6658: 6652: 6650: 6644: 6643: 6641: 6640: 6635: 6630: 6624: 6622: 6616: 6615: 6613: 6612: 6610:Measure theory 6607: 6602: 6597: 6592: 6587: 6582: 6577: 6571: 6569: 6563: 6562: 6560: 6559: 6554: 6549: 6544: 6539: 6534: 6529: 6524: 6518: 6516: 6510: 6509: 6507: 6506: 6501: 6496: 6491: 6486: 6481: 6475: 6473: 6467: 6466: 6464: 6463: 6458: 6453: 6452: 6451: 6446: 6435: 6432: 6431: 6422: 6421: 6414: 6407: 6399: 6393: 6392: 6387: 6386: 6385: 6380: 6375: 6363: 6362: 6361: 6358:Wilfrid Hodges 6351: 6334: 6325: 6317: 6305: 6300: 6280: 6279:External links 6277: 6276: 6275: 6257:(2): 261–281. 6243:Zermelo, Ernst 6239: 6192:Zermelo, Ernst 6188: 6165:(4): 514–516. 6151:Zermelo, Ernst 6147: 6124: 6104:(2): 161–228. 6088: 6072:Tarski, Alfred 6033: 6032: 6016: 5998: 5979: 5967: 5961:, ed. (1998). 5959:Mancosu, Paolo 5955: 5918:(4): 447–470. 5900: 5894: 5870: 5832: 5810: 5790:Hilbert, David 5786: 5753:Hilbert, David 5749: 5732:Hilbert, David 5728: 5722: 5698: 5659: 5641:(1): 173–198. 5623: 5591: 5575: 5538: 5523:Frege, Gottlob 5520: 5502:Frege, Gottlob 5499: 5482: 5481: 5480: 5469: 5449: 5429: 5420:Symbolic Logic 5415:Carroll, Lewis 5375: 5374: 5353: 5324: 5323: 5292:Tarski, Alfred 5288:Banach, Stefan 5282: 5279: 5278: 5277: 5254: 5220: 5200:10.2307/420992 5191:10.1.1.35.5803 5184:(3): 284–321. 5173: 5153:(2): 514–538. 5133: 5127: 5114: 5075: 5041:(4): 441–484. 5023: 5005:(9): 844–862. 4994: 4988: 4957:(3): 233–269. 4937: 4930: 4924: 4908:Cohen, Paul J. 4904: 4892:Cohen, Paul J. 4888: 4870:Medieval Logic 4862: 4856: 4839: 4836: 4835: 4834: 4828: 4812: 4806: 4786: 4766: 4754: 4748: 4728: 4722: 4702: 4696: 4678:, ed. (1989). 4672: 4666: 4649: 4643: 4628: 4627:Graduate texts 4625: 4624: 4623: 4617: 4592: 4568:Shawn Hedman, 4566: 4538: 4532: 4500: 4494: 4474: 4457: 4451: 4428: 4422: 4409: 4403: 4390: 4384: 4376:Academic Press 4363: 4349: 4332: 4326: 4306:Boolos, George 4302: 4296: 4273: 4270: 4268: 4267: 4252: 4240: 4228: 4216: 4204: 4192: 4180: 4168: 4156: 4152:Solovay (1976) 4144: 4132: 4120: 4108: 4104:Carroll (1896) 4096: 4084: 4072: 4068:Gentzen (1936) 4060: 4045: 4033: 4021: 4008: 3993: 3978: 3976:, p. 445. 3966: 3951: 3947:Richard (1905) 3939: 3927: 3915: 3911:Zermelo (1904) 3900: 3898:, p. 807. 3888: 3876: 3864: 3852: 3840: 3836:Hilbert (1899) 3828: 3816: 3814:, p. 774. 3804: 3792: 3780: 3756: 3754:, p. 686. 3744: 3742:, p. xiv. 3740:Boehner (1950) 3732: 3720: 3708: 3706:, p. 443. 3696: 3679: 3675:Barwise (1989) 3666: 3664: 3661: 3658: 3657: 3653:Ferreirós 2001 3645: 3632: 3598: 3597: 3595: 3592: 3591: 3590: 3585: 3580: 3575: 3570: 3565: 3560: 3555: 3550: 3545: 3540: 3535: 3533:Informal logic 3530: 3524: 3523: 3509: 3493: 3490: 3372:differentiable 3364:infinitesimals 3352:Main article: 3349: 3346: 3285:model checking 3275:is related to 3271:The theory of 3250:Main article: 3247: 3244: 3224:J. Lukasiewicz 3216:E. C. Berkeley 3212:O. Morgenstern 3184:H. Reichenbach 3139: 3136: 3062:Main article: 3059: 3056: 3040:Julia Robinson 2989: 2986: 2937:in the 1940s. 2915:Turing degrees 2905:, also called 2894:Main article: 2891: 2888: 2882:, named after 2845:The method of 2804:Main article: 2801: 2798: 2759:W. Hugh Woodin 2661:Main article: 2658: 2655: 2627: 2624: 2594:Main article: 2591: 2588: 2538: 2537: 2526: 2523: 2520: 2517: 2514: 2511: 2508: 2505: 2502: 2499: 2496: 2493: 2490: 2487: 2484: 2481: 2478: 2475: 2472: 2469: 2444: 2441: 2436: 2432: 2427: 2404: 2401: 2396: 2392: 2387: 2363: 2360: 2259:Main article: 2256: 2253: 2211: 2210: 2208: 2207: 2200: 2193: 2185: 2182: 2181: 2170: 2169: 2167: 2166: 2161: 2156: 2151: 2145: 2142: 2141: 2137: 2136: 2134: 2133: 2128: 2123: 2118: 2116:Truth function 2113: 2108: 2103: 2098: 2092: 2089: 2088: 2084: 2083: 2080: 2079: 2068: 2065: 2062: 2042: 2039: 2036: 2016: 2013: 2010: 2000: 1994: 1993: 1982: 1979: 1976: 1956: 1951: 1948: 1943: 1933: 1927: 1926: 1915: 1901: 1891: 1885: 1884: 1873: 1870: 1867: 1847: 1844: 1841: 1821: 1818: 1815: 1795: 1792: 1789: 1779: 1773: 1772: 1761: 1758: 1736: 1733: 1711: 1708: 1688: 1685: 1675: 1669: 1668: 1655: 1651: 1648: 1645: 1622: 1619: 1616: 1596: 1591: 1588: 1583: 1573: 1567: 1566: 1555: 1552: 1549: 1529: 1526: 1523: 1503: 1500: 1497: 1487: 1483: 1482: 1469: 1465: 1462: 1459: 1436: 1433: 1430: 1410: 1407: 1404: 1384: 1379: 1376: 1371: 1361: 1355: 1354: 1343: 1340: 1337: 1317: 1314: 1311: 1291: 1288: 1285: 1275: 1269: 1268: 1257: 1254: 1251: 1231: 1228: 1225: 1205: 1202: 1199: 1189: 1183: 1182: 1171: 1168: 1165: 1162: 1142: 1139: 1136: 1116: 1113: 1093: 1090: 1087: 1067: 1064: 1061: 1051: 1041: 1040: 1031: 1028: 1016:Emil Leon Post 955: 952: 860:Thoralf Skolem 852: 851:Symbolic logic 849: 735: 732: 701: 698: 656:infinitesimals 624:Fourier series 607:, building on 572:Giuseppe Peano 556: 553: 545:Ernst Schröder 510:George Peacock 497: 494: 437: 434: 424:, through the 408: 405: 365: 364: 354: 348: 343: 329: 326: 250: 249: 247: 246: 239: 232: 224: 221: 220: 209: 208: 205: 204: 201: 200: 195: 190: 185: 180: 175: 170: 165: 160: 154: 151: 150: 147: 146: 143: 142: 133: 128: 119: 114: 105: 100: 95: 89: 84: 83: 76: 75: 74: 73: 68: 60: 59: 53: 52: 26: 9: 6: 4: 3: 2: 10259: 10248: 10245: 10243: 10240: 10238: 10235: 10234: 10232: 10219: 10218: 10213: 10205: 10199: 10196: 10194: 10191: 10189: 10186: 10184: 10181: 10177: 10174: 10173: 10172: 10169: 10167: 10164: 10162: 10159: 10157: 10153: 10150: 10148: 10145: 10143: 10140: 10138: 10135: 10133: 10130: 10129: 10127: 10123: 10117: 10114: 10112: 10109: 10107: 10106:Recursive set 10104: 10102: 10099: 10097: 10094: 10092: 10089: 10087: 10084: 10080: 10077: 10075: 10072: 10070: 10067: 10065: 10062: 10060: 10057: 10056: 10055: 10052: 10050: 10047: 10045: 10042: 10040: 10037: 10035: 10032: 10030: 10027: 10026: 10024: 10022: 10018: 10012: 10009: 10007: 10004: 10002: 9999: 9997: 9994: 9992: 9989: 9987: 9984: 9982: 9979: 9975: 9972: 9970: 9967: 9965: 9962: 9961: 9960: 9957: 9955: 9952: 9950: 9947: 9945: 9942: 9940: 9937: 9935: 9932: 9928: 9925: 9924: 9923: 9920: 9916: 9915:of arithmetic 9913: 9912: 9911: 9908: 9904: 9901: 9899: 9896: 9894: 9891: 9889: 9886: 9884: 9881: 9880: 9879: 9876: 9872: 9869: 9867: 9864: 9863: 9862: 9859: 9858: 9856: 9854: 9850: 9844: 9841: 9839: 9836: 9834: 9831: 9829: 9826: 9823: 9822:from ZFC 9819: 9816: 9814: 9811: 9805: 9802: 9801: 9800: 9797: 9795: 9792: 9790: 9787: 9786: 9785: 9782: 9780: 9777: 9775: 9772: 9770: 9767: 9765: 9762: 9760: 9757: 9755: 9752: 9751: 9749: 9747: 9743: 9733: 9732: 9728: 9727: 9722: 9721:non-Euclidean 9719: 9715: 9712: 9710: 9707: 9705: 9704: 9700: 9699: 9697: 9694: 9693: 9691: 9687: 9683: 9680: 9678: 9675: 9674: 9673: 9669: 9665: 9662: 9661: 9660: 9656: 9652: 9649: 9647: 9644: 9642: 9639: 9637: 9634: 9632: 9629: 9627: 9624: 9623: 9621: 9617: 9616: 9614: 9609: 9603: 9598:Example 9595: 9587: 9582: 9581: 9580: 9577: 9575: 9572: 9568: 9565: 9563: 9560: 9558: 9555: 9553: 9550: 9549: 9548: 9545: 9543: 9540: 9538: 9535: 9533: 9530: 9526: 9523: 9521: 9518: 9517: 9516: 9513: 9509: 9506: 9504: 9501: 9499: 9496: 9494: 9491: 9490: 9489: 9486: 9484: 9481: 9477: 9474: 9472: 9469: 9467: 9464: 9463: 9462: 9459: 9455: 9452: 9450: 9447: 9445: 9442: 9440: 9437: 9435: 9432: 9430: 9427: 9426: 9425: 9422: 9420: 9417: 9415: 9412: 9410: 9407: 9403: 9400: 9398: 9395: 9393: 9390: 9388: 9385: 9384: 9383: 9380: 9378: 9375: 9373: 9370: 9368: 9365: 9361: 9358: 9356: 9355:by definition 9353: 9352: 9351: 9348: 9344: 9341: 9340: 9339: 9336: 9334: 9331: 9329: 9326: 9324: 9321: 9319: 9316: 9315: 9312: 9309: 9307: 9303: 9298: 9292: 9288: 9278: 9275: 9273: 9270: 9268: 9265: 9263: 9260: 9258: 9255: 9253: 9250: 9248: 9245: 9243: 9242:Kripke–Platek 9240: 9238: 9235: 9231: 9228: 9226: 9223: 9222: 9221: 9218: 9217: 9215: 9211: 9203: 9200: 9199: 9198: 9195: 9193: 9190: 9186: 9183: 9182: 9181: 9178: 9176: 9173: 9171: 9168: 9166: 9163: 9161: 9158: 9155: 9151: 9147: 9144: 9140: 9137: 9135: 9132: 9130: 9127: 9126: 9125: 9121: 9118: 9117: 9115: 9113: 9109: 9105: 9097: 9094: 9092: 9089: 9087: 9086:constructible 9084: 9083: 9082: 9079: 9077: 9074: 9072: 9069: 9067: 9064: 9062: 9059: 9057: 9054: 9052: 9049: 9047: 9044: 9042: 9039: 9037: 9034: 9032: 9029: 9027: 9024: 9022: 9019: 9018: 9016: 9014: 9009: 9001: 8998: 8996: 8993: 8991: 8988: 8986: 8983: 8981: 8978: 8976: 8973: 8972: 8970: 8966: 8963: 8961: 8958: 8957: 8956: 8953: 8951: 8948: 8946: 8943: 8941: 8938: 8936: 8932: 8928: 8926: 8923: 8919: 8916: 8915: 8914: 8911: 8910: 8907: 8904: 8902: 8898: 8888: 8885: 8883: 8880: 8878: 8875: 8873: 8870: 8868: 8865: 8863: 8860: 8856: 8853: 8852: 8851: 8848: 8844: 8839: 8838: 8837: 8834: 8833: 8831: 8829: 8825: 8817: 8814: 8812: 8809: 8807: 8804: 8803: 8802: 8799: 8797: 8794: 8792: 8789: 8787: 8784: 8782: 8779: 8777: 8774: 8772: 8769: 8768: 8766: 8764: 8763:Propositional 8760: 8754: 8751: 8749: 8746: 8744: 8741: 8739: 8736: 8734: 8731: 8729: 8726: 8722: 8719: 8718: 8717: 8714: 8712: 8709: 8707: 8704: 8702: 8699: 8697: 8694: 8692: 8691:Logical truth 8689: 8687: 8684: 8683: 8681: 8679: 8675: 8672: 8670: 8666: 8660: 8657: 8655: 8652: 8650: 8647: 8645: 8642: 8640: 8637: 8635: 8631: 8627: 8623: 8621: 8618: 8616: 8613: 8611: 8607: 8604: 8603: 8601: 8599: 8593: 8588: 8582: 8579: 8577: 8574: 8572: 8569: 8567: 8564: 8562: 8559: 8557: 8554: 8552: 8549: 8547: 8544: 8542: 8539: 8537: 8534: 8532: 8529: 8527: 8524: 8520: 8517: 8516: 8515: 8512: 8511: 8509: 8505: 8501: 8494: 8489: 8487: 8482: 8480: 8475: 8474: 8471: 8459: 8456: 8453: 8449: 8446: 8444: 8441: 8439: 8429: 8428: 8425: 8415: 8414:Logic symbols 8412: 8410: 8407: 8405: 8402: 8400: 8397: 8395: 8392: 8391: 8389: 8385: 8379: 8376: 8374: 8371: 8369: 8366: 8365: 8363: 8361: 8357: 8354: 8350: 8344: 8341: 8339: 8336: 8334: 8331: 8329: 8326: 8324: 8321: 8319: 8316: 8314: 8311: 8309: 8306: 8304: 8301: 8299: 8296: 8294: 8293:Logical truth 8291: 8289: 8286: 8284: 8281: 8277: 8274: 8273: 8272: 8269: 8267: 8264: 8262: 8259: 8257: 8254: 8252: 8249: 8245: 8242: 8240: 8237: 8236: 8235: 8234:Contradiction 8232: 8230: 8227: 8225: 8222: 8220: 8217: 8215: 8212: 8211: 8209: 8205: 8195: 8192: 8190: 8187: 8185: 8182: 8180: 8179:Argumentation 8177: 8176: 8174: 8170: 8164: 8163:Philosophical 8161: 8159: 8158:Non-classical 8156: 8154: 8151: 8147: 8144: 8142: 8139: 8138: 8137: 8134: 8132: 8129: 8128: 8126: 8122: 8116: 8113: 8111: 8108: 8106: 8103: 8101: 8098: 8096: 8093: 8091: 8088: 8086: 8083: 8082: 8080: 8076: 8070: 8067: 8065: 8062: 8061: 8058: 8054: 8047: 8042: 8040: 8035: 8033: 8028: 8027: 8024: 8012: 8004: 8002: 7994: 7992: 7984: 7983: 7980: 7974: 7971: 7969: 7966: 7964: 7961: 7959: 7956: 7954: 7951: 7949: 7946: 7944: 7941: 7939: 7936: 7934: 7931: 7929: 7926: 7924: 7921: 7919: 7916: 7914: 7911: 7909: 7906: 7904: 7901: 7899: 7896: 7894: 7891: 7889: 7886: 7884: 7881: 7879: 7876: 7875: 7873: 7869: 7863: 7860: 7858: 7855: 7853: 7850: 7848: 7847:Mixed reality 7845: 7843: 7840: 7838: 7835: 7833: 7830: 7828: 7825: 7824: 7822: 7820: 7816: 7810: 7807: 7805: 7802: 7800: 7797: 7795: 7792: 7790: 7787: 7786: 7784: 7782: 7778: 7772: 7769: 7767: 7764: 7762: 7759: 7757: 7754: 7752: 7749: 7747: 7744: 7742: 7739: 7737: 7734: 7733: 7731: 7729: 7725: 7719: 7716: 7714: 7711: 7709: 7706: 7704: 7701: 7699: 7696: 7695: 7693: 7691: 7687: 7681: 7680:Accessibility 7678: 7676: 7675:Visualization 7673: 7671: 7668: 7666: 7663: 7661: 7658: 7657: 7655: 7653: 7649: 7643: 7640: 7638: 7635: 7633: 7630: 7628: 7625: 7623: 7620: 7618: 7615: 7613: 7610: 7608: 7605: 7603: 7600: 7599: 7597: 7595: 7591: 7585: 7582: 7580: 7577: 7575: 7572: 7570: 7567: 7565: 7562: 7560: 7557: 7555: 7552: 7550: 7547: 7545: 7542: 7540: 7537: 7535: 7532: 7530: 7527: 7525: 7522: 7520: 7517: 7516: 7514: 7512: 7508: 7502: 7499: 7497: 7494: 7492: 7489: 7487: 7484: 7482: 7479: 7477: 7474: 7472: 7469: 7467: 7464: 7463: 7461: 7459: 7454: 7448: 7445: 7443: 7440: 7438: 7435: 7433: 7430: 7428: 7425: 7424: 7422: 7420: 7416: 7410: 7407: 7405: 7402: 7400: 7397: 7395: 7392: 7390: 7387: 7385: 7382: 7380: 7377: 7376: 7374: 7372: 7368: 7362: 7359: 7357: 7354: 7352: 7349: 7347: 7344: 7342: 7339: 7337: 7334: 7332: 7329: 7327: 7324: 7322: 7319: 7317: 7314: 7313: 7311: 7309: 7305: 7299: 7296: 7294: 7291: 7289: 7286: 7284: 7281: 7279: 7276: 7274: 7271: 7269: 7266: 7264: 7261: 7259: 7256: 7254: 7251: 7250: 7248: 7246: 7242: 7238: 7232: 7229: 7227: 7224: 7222: 7219: 7217: 7214: 7212: 7209: 7208: 7206: 7202: 7196: 7193: 7191: 7188: 7186: 7183: 7181: 7178: 7176: 7173: 7171: 7168: 7167: 7165: 7163: 7159: 7153: 7150: 7148: 7145: 7143: 7142:Dependability 7140: 7138: 7135: 7133: 7130: 7129: 7127: 7123: 7117: 7113: 7110: 7108: 7105: 7103: 7100: 7098: 7095: 7093: 7090: 7088: 7085: 7083: 7080: 7078: 7075: 7073: 7070: 7068: 7065: 7064: 7062: 7060: 7056: 7051: 7045: 7041: 7034: 7029: 7027: 7022: 7020: 7015: 7014: 7011: 6999: 6998: 6989: 6987: 6986: 6977: 6975: 6974: 6965: 6963: 6962: 6957: 6951: 6950: 6947: 6941: 6938: 6936: 6933: 6931: 6928: 6926: 6923: 6921: 6918: 6914: 6911: 6910: 6909: 6906: 6905: 6903: 6901: 6897: 6891: 6888: 6886: 6883: 6881: 6878: 6876: 6873: 6871: 6868: 6866: 6863: 6862: 6860: 6858: 6857:Computational 6854: 6846: 6843: 6841: 6838: 6836: 6833: 6832: 6831: 6828: 6826: 6823: 6821: 6818: 6816: 6813: 6811: 6808: 6806: 6803: 6801: 6798: 6796: 6793: 6791: 6788: 6786: 6783: 6781: 6778: 6776: 6773: 6772: 6770: 6768: 6764: 6758: 6755: 6753: 6750: 6748: 6745: 6743: 6740: 6738: 6735: 6734: 6732: 6730: 6726: 6720: 6717: 6715: 6712: 6710: 6707: 6705: 6702: 6701: 6699: 6697: 6696:Number theory 6693: 6687: 6684: 6682: 6679: 6677: 6674: 6672: 6669: 6667: 6664: 6662: 6659: 6657: 6654: 6653: 6651: 6649: 6645: 6639: 6636: 6634: 6631: 6629: 6628:Combinatorics 6626: 6625: 6623: 6621: 6617: 6611: 6608: 6606: 6603: 6601: 6598: 6596: 6593: 6591: 6588: 6586: 6583: 6581: 6580:Real analysis 6578: 6576: 6573: 6572: 6570: 6568: 6564: 6558: 6555: 6553: 6550: 6548: 6545: 6543: 6540: 6538: 6535: 6533: 6530: 6528: 6525: 6523: 6520: 6519: 6517: 6515: 6511: 6505: 6502: 6500: 6497: 6495: 6492: 6490: 6487: 6485: 6482: 6480: 6477: 6476: 6474: 6472: 6468: 6462: 6459: 6457: 6454: 6450: 6447: 6445: 6442: 6441: 6440: 6437: 6436: 6433: 6428: 6420: 6415: 6413: 6408: 6406: 6401: 6400: 6397: 6391: 6388: 6384: 6381: 6379: 6376: 6374: 6371: 6370: 6368: 6364: 6359: 6355: 6352: 6349: 6345: 6342: 6341: 6339: 6335: 6332: 6331: 6326: 6323: 6322: 6318: 6311: 6310: 6306: 6304: 6301: 6297: 6293: 6292: 6287: 6283: 6282: 6272: 6268: 6264: 6260: 6256: 6252: 6248: 6244: 6240: 6237: 6231: 6227: 6223: 6219: 6215: 6211: 6207: 6204:(in German). 6203: 6202: 6197: 6193: 6189: 6186: 6180: 6176: 6172: 6168: 6164: 6161:(in German). 6160: 6156: 6152: 6148: 6144: 6138: 6130: 6125: 6120: 6115: 6111: 6107: 6103: 6099: 6098: 6093: 6089: 6085: 6081: 6077: 6073: 6069: 6068: 6067: 6064: 6060: 6046: 6039: 6029: 6026:(in German). 6025: 6021: 6017: 6014: 6008: 6005:(in French). 6004: 5999: 5996: 5990: 5989: 5984: 5980: 5976: 5972: 5971:Pasch, Moritz 5968: 5964: 5960: 5956: 5952: 5948: 5941: 5937: 5933: 5929: 5925: 5921: 5917: 5914:(in German). 5913: 5909: 5905: 5901: 5897: 5895:0-486-60027-0 5891: 5887: 5879: 5875: 5871: 5867: 5863: 5858: 5853: 5849: 5845: 5841: 5837: 5833: 5829: 5825: 5821: 5817: 5813: 5811:9783540041344 5807: 5803: 5799: 5795: 5794:Bernays, Paul 5791: 5787: 5782: 5778: 5774: 5770: 5766: 5762: 5758: 5754: 5750: 5747: 5741: 5738:(in German). 5737: 5733: 5729: 5725: 5723:9780674324497 5719: 5715: 5711: 5707: 5703: 5699: 5696: 5692: 5685: 5680: 5676: 5673:(in German). 5672: 5668: 5664: 5660: 5656: 5652: 5648: 5644: 5640: 5637:(in German). 5636: 5632: 5628: 5624: 5620: 5616: 5612: 5608: 5604: 5601:(in German). 5600: 5596: 5592: 5588: 5584: 5580: 5576: 5573: 5567: 5563: 5559: 5555: 5551: 5547: 5543: 5539: 5536: 5532: 5528: 5524: 5521: 5518: 5514: 5510: 5508: 5503: 5500: 5497: 5491: 5487: 5483: 5478: 5474: 5470: 5467: 5464:1963 (1901). 5463: 5462: 5458: 5454: 5450: 5445: 5441: 5437: 5436: 5435: 5432: 5430:9781163444955 5426: 5422: 5421: 5416: 5410: 5406: 5402: 5398: 5394: 5390: 5389: 5381: 5372: 5369:Reprinted in 5366: 5361: 5360: 5359: 5356: 5354:9789048183296 5350: 5346: 5342: 5338: 5334: 5330: 5319: 5314: 5310: 5307:(in French). 5306: 5305: 5297: 5293: 5289: 5285: 5284: 5274: 5270: 5263: 5259: 5255: 5251: 5247: 5243: 5239: 5235: 5231: 5230: 5225: 5221: 5217: 5213: 5209: 5205: 5201: 5197: 5192: 5187: 5183: 5179: 5174: 5170: 5166: 5161: 5156: 5152: 5148: 5147: 5142: 5138: 5134: 5130: 5128:9780321016188 5124: 5120: 5115: 5111: 5107: 5103: 5099: 5094: 5089: 5085: 5081: 5076: 5072: 5068: 5064: 5060: 5056: 5052: 5048: 5044: 5040: 5036: 5029: 5024: 5020: 5016: 5012: 5008: 5004: 5000: 4995: 4991: 4989:9780486614717 4985: 4981: 4972: 4968: 4964: 4960: 4956: 4952: 4951: 4946: 4942: 4941:Davis, Martin 4938: 4935: 4931: 4927: 4925:9780486469218 4921: 4917: 4913: 4909: 4905: 4901: 4900:Menlo Park CA 4897: 4893: 4889: 4885: 4879: 4872:. Manchester. 4871: 4867: 4863: 4859: 4853: 4849: 4848: 4842: 4841: 4831: 4825: 4821: 4817: 4813: 4809: 4807:9781568811352 4803: 4799: 4795: 4791: 4787: 4784: 4783:0-486-42533-9 4780: 4776: 4775: 4770: 4767: 4764: 4763: 4758: 4755: 4751: 4749:9783540440857 4745: 4741: 4737: 4733: 4729: 4725: 4723:9780521587136 4719: 4715: 4711: 4707: 4703: 4699: 4697:9780444863881 4693: 4689: 4685: 4681: 4677: 4673: 4669: 4663: 4659: 4655: 4650: 4646: 4644:1-56881-262-0 4640: 4636: 4631: 4630: 4620: 4614: 4610: 4606: 4602: 4598: 4593: 4591: 4587: 4583: 4582:0-19-852981-3 4579: 4575: 4571: 4567: 4556: 4549: 4548: 4544:(2003–2004). 4543: 4539: 4535: 4533:9781441912206 4529: 4525: 4521: 4517: 4513: 4512:New York City 4509: 4505: 4501: 4497: 4491: 4487: 4483: 4479: 4475: 4471: 4467: 4463: 4458: 4454: 4452:9780387942582 4448: 4444: 4440: 4439:New York City 4436: 4435: 4429: 4425: 4419: 4415: 4410: 4406: 4400: 4396: 4391: 4387: 4381: 4377: 4373: 4369: 4364: 4360: 4356: 4352: 4350:9780198880875 4346: 4342: 4338: 4333: 4329: 4327:9780521007580 4323: 4319: 4315: 4311: 4307: 4303: 4299: 4297:9789814343879 4293: 4289: 4285: 4281: 4276: 4275: 4264: 4259: 4257: 4249: 4244: 4237: 4232: 4225: 4220: 4213: 4212:Morley (1965) 4208: 4201: 4200:Tarski (1948) 4196: 4189: 4188:Woodin (2001) 4184: 4177: 4172: 4165: 4160: 4153: 4148: 4141: 4136: 4129: 4128:Turing (1939) 4124: 4117: 4116:Kleene (1943) 4112: 4105: 4100: 4094: 4088: 4081: 4076: 4069: 4064: 4057: 4052: 4050: 4042: 4041:Skolem (1920) 4037: 4030: 4025: 4018: 4012: 4005: 4000: 3998: 3990: 3985: 3983: 3975: 3970: 3963: 3958: 3956: 3948: 3943: 3936: 3931: 3924: 3919: 3912: 3907: 3905: 3897: 3892: 3885: 3884:Cantor (1874) 3880: 3873: 3868: 3861: 3856: 3849: 3844: 3837: 3832: 3825: 3820: 3813: 3808: 3801: 3796: 3789: 3784: 3770: 3766: 3760: 3753: 3748: 3741: 3736: 3729: 3724: 3717: 3712: 3705: 3700: 3689: 3683: 3676: 3671: 3667: 3654: 3649: 3642: 3636: 3628: 3624:Translation: 3622: 3617: 3613: 3609: 3603: 3599: 3589: 3586: 3584: 3581: 3579: 3576: 3574: 3571: 3569: 3566: 3564: 3561: 3559: 3556: 3554: 3551: 3549: 3546: 3544: 3541: 3539: 3536: 3534: 3531: 3529: 3526: 3525: 3521: 3515: 3510: 3507: 3496: 3489: 3487: 3486:Kripke models 3483: 3479: 3475: 3471: 3468:as a part of 3467: 3463: 3458: 3456: 3452: 3448: 3444: 3440: 3436: 3431: 3429: 3425: 3421: 3417: 3411: 3409: 3405: 3401: 3397: 3393: 3387: 3385: 3384:David Hilbert 3380: 3375: 3373: 3369: 3365: 3361: 3355: 3345: 3343: 3339: 3335: 3331: 3327: 3323: 3321: 3317: 3312: 3310: 3306: 3302: 3298: 3295:, especially 3294: 3290: 3286: 3282: 3278: 3274: 3269: 3267: 3263: 3259: 3256:The study of 3253: 3243: 3241: 3237: 3233: 3230:, A. Becker, 3229: 3226:, H. Scholz, 3225: 3221: 3217: 3213: 3209: 3205: 3201: 3197: 3193: 3189: 3188:J. H. Woodger 3185: 3181: 3177: 3176:C. E. Shannon 3173: 3169: 3168:S. Lesniewski 3165: 3161: 3157: 3153: 3149: 3145: 3135: 3133: 3129: 3125: 3121: 3116: 3114: 3113: 3108: 3102: 3100: 3096: 3092: 3089:The study of 3087: 3085: 3081: 3078:, systems of 3077: 3072: 3071: 3065: 3055: 3053: 3049: 3048:Hilary Putnam 3045: 3041: 3037: 3033: 3031: 3027: 3023: 3022:Pyotr Novikov 3019: 3014: 3012: 3008: 3003: 2999: 2995: 2985: 2983: 2979: 2975: 2970: 2968: 2964: 2958: 2956: 2952: 2948: 2944: 2938: 2936: 2932: 2928: 2924: 2923:Alonzo Church 2920: 2916: 2912: 2908: 2904: 2903: 2897: 2887: 2885: 2881: 2877: 2872: 2870: 2866: 2862: 2860: 2856: 2852: 2848: 2843: 2841: 2836: 2834: 2830: 2826: 2822: 2818: 2814: 2813: 2807: 2797: 2795: 2794:Polish spaces 2791: 2787: 2783: 2779: 2775: 2771: 2767: 2762: 2760: 2757:conducted by 2755: 2751: 2746: 2745:David Hilbert 2741: 2739: 2734: 2733:Alfred Tarski 2730: 2729:Stefan Banach 2725: 2721: 2717: 2713: 2708: 2706: 2702: 2698: 2694: 2690: 2685: 2683: 2679: 2675: 2671: 2670: 2664: 2654: 2652: 2648: 2644: 2641:to represent 2640: 2636: 2632: 2623: 2621: 2617: 2613: 2609: 2605: 2602: 2597: 2587: 2585: 2581: 2577: 2573: 2571: 2565: 2563: 2559: 2555: 2545: 2543: 2524: 2521: 2518: 2512: 2509: 2506: 2500: 2494: 2491: 2488: 2482: 2476: 2473: 2470: 2460: 2459: 2458: 2442: 2439: 2434: 2430: 2425: 2402: 2399: 2394: 2390: 2385: 2375: 2373: 2369: 2359: 2357: 2353: 2349: 2343: 2341: 2337: 2332: 2328: 2324: 2322: 2317: 2312: 2308: 2304: 2302: 2298: 2293: 2291: 2287: 2283: 2279: 2275: 2271: 2267: 2262: 2252: 2250: 2246: 2242: 2238: 2234: 2230: 2226: 2222: 2218: 2206: 2201: 2199: 2194: 2192: 2187: 2186: 2184: 2183: 2180: 2172: 2171: 2165: 2162: 2160: 2157: 2155: 2152: 2150: 2149:Digital logic 2147: 2146: 2144: 2143: 2139: 2138: 2132: 2131:Scope (logic) 2129: 2127: 2124: 2122: 2119: 2117: 2114: 2112: 2109: 2107: 2104: 2102: 2099: 2097: 2094: 2093: 2091: 2090: 2086: 2085: 2066: 2060: 2040: 2037: 2034: 2014: 2008: 2001: 1999: 1996: 1995: 1980: 1977: 1974: 1954: 1949: 1946: 1941: 1934: 1932: 1929: 1928: 1913: 1899: 1892: 1890: 1887: 1886: 1871: 1868: 1865: 1845: 1842: 1839: 1819: 1816: 1813: 1793: 1790: 1787: 1780: 1778: 1775: 1774: 1759: 1756: 1731: 1709: 1706: 1686: 1676: 1674: 1671: 1670: 1649: 1646: 1643: 1620: 1614: 1594: 1586: 1581: 1574: 1572: 1569: 1568: 1553: 1550: 1547: 1527: 1524: 1521: 1501: 1498: 1495: 1488: 1486:nonequivalent 1485: 1484: 1463: 1460: 1457: 1434: 1431: 1428: 1408: 1402: 1382: 1374: 1369: 1362: 1360: 1357: 1356: 1341: 1335: 1315: 1312: 1309: 1289: 1283: 1276: 1274: 1271: 1270: 1255: 1249: 1229: 1223: 1203: 1200: 1197: 1190: 1188: 1185: 1184: 1169: 1160: 1140: 1134: 1114: 1111: 1091: 1088: 1085: 1065: 1062: 1059: 1052: 1050: 1047: 1046: 1043: 1042: 1039: 1036: 1035: 1027: 1025: 1024:Georg Kreisel 1021: 1017: 1013: 1008: 1006: 1002: 998: 993: 991: 990: 986: 982: 977: 976: 971: 966: 964: 960: 959:Alfred Tarski 951: 949: 948: 943: 942:Lewis Carroll 938: 936: 932: 928: 923: 918: 916: 912: 911: 905: 903: 899: 895: 891: 886: 884: 880: 877: 873: 872:cardinalities 869: 865: 862:obtained the 861: 857: 848: 846: 842: 838: 834: 829: 827: 823: 819: 815: 814: 808: 806: 802: 798: 794: 789: 787: 783: 782:Jules Richard 780:in 1901, and 779: 775: 771: 767: 763: 759: 754: 752: 748: 744: 740: 739:Ernst Zermelo 731: 729: 728: 723: 719: 715: 711: 706: 697: 695: 691: 687: 683: 679: 675: 671: 667: 663: 661: 660:Dedekind cuts 657: 653: 649: 645: 641: 637: 633: 629: 625: 621: 620:real analysis 616: 614: 610: 609:previous work 606: 601: 597: 592: 590: 585: 581: 577: 573: 569: 565: 560: 552: 550: 546: 541: 539: 535: 534: 529: 528:Gottlob Frege 525: 523: 519: 515: 511: 507: 503: 493: 491: 487: 483: 479: 476:, especially 475: 471: 470: 465: 461: 460:Islamic world 457: 453: 449: 443: 436:Early history 433: 431: 427: 423: 422:calculationes 419: 415: 404: 402: 398: 394: 390: 385: 383: 379: 378:Löb's theorem 375: 370: 362: 358: 355: 352: 349: 347: 344: 342: 339: 338: 337: 335: 325: 323: 319: 315: 311: 307: 306:David Hilbert 303: 299: 295: 291: 286: 284: 280: 276: 272: 268: 264: 260: 256: 245: 240: 238: 233: 231: 226: 225: 223: 222: 219: 215: 211: 210: 199: 196: 194: 191: 189: 186: 184: 181: 179: 176: 174: 171: 169: 166: 164: 161: 159: 156: 155: 149: 148: 141: 137: 134: 132: 129: 127: 123: 120: 118: 115: 113: 109: 106: 104: 101: 99: 96: 94: 93:Number theory 91: 90: 87: 82: 81: 78: 77: 72: 69: 67: 64: 63: 62: 61: 58: 55: 54: 50: 49: 46: 41: 34: 19: 10208: 10006:Ultraproduct 9853:Model theory 9818:Independence 9754:Formal proof 9746:Proof theory 9729: 9702: 9659:real numbers 9631:second-order 9542:Substitution 9419:Metalanguage 9360:conservative 9333:Axiom schema 9277:Constructive 9247:Morse–Kelley 9213:Set theories 9192:Aleph number 9185:inaccessible 9091:Grothendieck 8975:intersection 8862:Higher-order 8850:Second-order 8796:Truth tables 8753:Venn diagram 8536:Formal proof 8499: 8333:Substitution 8153:Mathematical 8152: 8078:Major fields 7943:Cyberwarfare 7602:Cryptography 6995: 6983: 6971: 6952: 6885:Optimization 6747:Differential 6671:Differential 6638:Order theory 6633:Graph theory 6537:Group theory 6488: 6328: 6319: 6314:P. D. Magnus 6307: 6289: 6254: 6250: 6205: 6199: 6162: 6158: 6128: 6101: 6095: 6075: 6062: 6049:. Retrieved 6044: 6034: 6027: 6023: 6006: 6002: 5987: 5974: 5962: 5950: 5915: 5911: 5885: 5880:(in German). 5877: 5850:(1): 41–73. 5847: 5843: 5797: 5764: 5760: 5745: 5735: 5710:Cambridge MA 5705: 5690: 5674: 5670: 5638: 5634: 5602: 5598: 5586: 5582: 5571: 5549: 5545: 5534: 5531:J. L. Austin 5526: 5512: 5505: 5489: 5472: 5465: 5456: 5446:(in German). 5443: 5419: 5392: 5386: 5376: 5364: 5328: 5325: 5308: 5302: 5272: 5268: 5233: 5227: 5181: 5177: 5150: 5144: 5118: 5093:math/0509616 5083: 5079: 5038: 5034: 5002: 4998: 4979: 4954: 4948: 4933: 4932:J.D. Sneed, 4911: 4895: 4869: 4846: 4819: 4793: 4772: 4760: 4735: 4732:Jech, Thomas 4709: 4679: 4676:Barwise, Jon 4653: 4634: 4596: 4569: 4559:. Retrieved 4546: 4507: 4481: 4461: 4433: 4413: 4394: 4367: 4336: 4313: 4279: 4243: 4236:Davis (1973) 4231: 4224:Soare (2011) 4219: 4207: 4195: 4183: 4171: 4159: 4147: 4140:Gödel (1931) 4135: 4123: 4111: 4099: 4087: 4080:Gödel (1958) 4075: 4063: 4056:Gödel (1929) 4036: 4024: 4011: 4004:Cohen (1966) 3969: 3942: 3930: 3918: 3891: 3879: 3867: 3855: 3848:Pasch (1882) 3843: 3831: 3819: 3807: 3795: 3788:Peano (1889) 3783: 3772:. Retrieved 3768: 3759: 3747: 3735: 3723: 3711: 3699: 3682: 3670: 3648: 3635: 3626: 3620: 3602: 3473: 3466:intuitionism 3459: 3443:constructive 3442: 3432: 3419: 3416:proof theory 3412: 3404:great circle 3399: 3395: 3388: 3376: 3357: 3324: 3313: 3293:proof theory 3277:model theory 3270: 3255: 3200:C. G. Hempel 3141: 3138:Applications 3120:proof mining 3117: 3110: 3103: 3099:Hermann Weyl 3090: 3088: 3070:Proof theory 3068: 3067: 3064:Proof theory 3044:Martin Davis 3034: 3015: 3001: 2991: 2971: 2959: 2939: 2906: 2900: 2899: 2873: 2867:, proved by 2863: 2844: 2837: 2812:Model theory 2810: 2809: 2806:Model theory 2800:Model theory 2789: 2785: 2763: 2742: 2723: 2719: 2709: 2686: 2667: 2666: 2643:truth values 2629: 2606: 2601:Modal logics 2599: 2574: 2566: 2548: 2546: 2539: 2376: 2365: 2351: 2344: 2340:Peano axioms 2330: 2325: 2321:model theory 2305: 2294: 2280:, while its 2265: 2264: 2214: 2158: 2140:Applications 1009: 994: 988: 984: 980: 973: 967: 963:model theory 957: 945: 944:, author of 939: 919: 908: 906: 887: 854: 830: 825: 811: 809: 795:proposed by 790: 755: 747:Georg Cantor 737: 725: 707: 703: 700:20th century 694:well-ordered 666:Georg Cantor 664: 617: 593: 576:Peano axioms 563: 561: 558: 548: 542: 531: 526: 502:George Boole 499: 496:19th century 467: 445: 421: 410: 386: 366: 357:proof theory 346:model theory 333: 331: 287: 271:proof theory 267:model theory 259:formal logic 254: 253: 121: 45: 10116:Type theory 10064:undecidable 9996:Truth value 9883:equivalence 9562:non-logical 9175:Enumeration 9165:Isomorphism 9112:cardinality 9096:Von Neumann 9061:Ultrafilter 9026:Uncountable 8960:equivalence 8877:Quantifiers 8867:Fixed-point 8836:First-order 8716:Consistency 8701:Proposition 8678:Traditional 8649:Lindström's 8639:Compactness 8581:Type theory 8526:Cardinality 8448:WikiProject 8318:Proposition 8313:Probability 8266:Description 8207:Foundations 7953:Video games 7933:Digital art 7690:Concurrency 7559:Data mining 7471:Probability 7211:Interpreter 6997:WikiProject 6840:Game theory 6820:Probability 6557:Homological 6547:Multilinear 6527:Commutative 6504:Type theory 6471:Foundations 6427:mathematics 6208:: 107–128. 5663:Gödel, Kurt 5627:Gödel, Kurt 5605:: 349–360. 5595:Gödel, Kurt 5579:Gödel, Kurt 5552:: 132–213. 5311:: 244–277. 5055:11441/38373 4771:. (1967), 3896:Katz (1998) 3812:Katz (1998) 3752:Katz (1998) 3196:F. B. Fitch 3095:predicative 3026:busy beaver 2927:Alan Turing 2919:Rózsa Péter 2786:Determinacy 2782:cardinality 2770:determinacy 2570:fuzzy logic 2556:that allow 2301:isomorphism 2288:to a fixed 2286:quantifiers 2111:Truth table 972:to publish 950:, in 1896. 922:consistency 892:proved the 822:type theory 784:discovered 776:discovered 745:, a result 714:23 problems 670:cardinality 516:. In 1847, 428:, and with 263:mathematics 183:Linguistics 173:Computation 168:Geosciences 131:Probability 57:Mathematics 10231:Categories 9927:elementary 9620:arithmetic 9488:Quantifier 9466:functional 9338:Expression 9056:Transitive 9000:identities 8985:complement 8918:hereditary 8901:Set theory 8378:Set theory 8276:Linguistic 8271:Entailment 8261:Definition 8229:Consequent 8224:Antecedent 8011:Glossaries 7883:E-commerce 7476:Statistics 7419:Algorithms 7216:Middleware 7072:Peripheral 6825:Statistics 6704:Arithmetic 6666:Arithmetic 6532:Elementary 6499:Set theory 5820:60.0017.02 5742:: Teubner. 5693:, vol II, 5671:Dialectica 5479:: 787–832. 4916:Mineola NY 4798:A K Peters 4561:2016-02-24 4359:0251.02001 4017:Cohen 2008 3774:2023-05-01 3663:References 3641:Soare 1996 3402:to mean a 3240:D. Ingalls 3236:P. Boehner 3208:J. Neumann 3156:P. Bernays 3152:D. Hilbert 3148:B. Russell 3082:, and the 3030:Tibor Radó 2823:, while a 2790:determined 2754:Paul Cohen 2695:(MK), and 2669:Set theory 2663:Set theory 2657:Set theory 2616:computable 1187:equivalent 989:surjection 890:Kurt Gödel 837:Paul Cohen 833:urelements 564:arithmetic 547:published 478:Chrysippus 430:philosophy 341:set theory 314:Kurt Gödel 298:arithmetic 275:set theory 193:Philosophy 136:Statistics 126:Set theory 10198:Supertask 10101:Recursion 10059:decidable 9893:saturated 9871:of models 9794:deductive 9789:axiomatic 9709:Hilbert's 9696:Euclidean 9677:canonical 9600:axiomatic 9532:Signature 9461:Predicate 9350:Extension 9272:Ackermann 9197:Operation 9076:Universal 9066:Recursive 9041:Singleton 9036:Inhabited 9021:Countable 9011:Types of 8995:power set 8965:partition 8882:Predicate 8828:Predicate 8743:Syllogism 8733:Soundness 8706:Inference 8696:Tautology 8598:paradoxes 8409:Fallacies 8404:Paradoxes 8394:Logicians 8328:Statement 8323:Reference 8288:Induction 8251:Deduction 8214:Abduction 8184:Metalogic 8131:Classical 8095:Inference 7832:Rendering 7827:Animation 7458:computing 7409:Semantics 7107:Processor 6752:Geometric 6742:Algebraic 6681:Euclidean 6656:Algebraic 6552:Universal 6296:EMS Press 6271:120085563 6245:(1908b). 6230:119924143 6222:0025-5831 6194:(1908a). 6179:124189935 6137:cite book 6051:23 August 5940:116581304 5932:0025-5831 5888:. Dover. 5781:122870563 5726:. (pbk.). 5655:197663120 5619:123343522 5566:122719892 5471:1996. In 5409:199545885 5333:Dordrecht 5250:121226261 5186:CiteSeerX 4982:. Dover. 4943:(1973). " 4910:(2008) . 4878:cite book 4792:(2001) . 4759:.(1952), 4684:Amsterdam 4284:Singapore 4248:Weyl 1918 4015:See also 3578:Mereology 3204:K. Menger 3192:A. Tarski 3172:T. Skolem 3164:R. Carnap 3160:H. Scholz 3112:translate 3054:in 1970. 2855:decidable 2821:signature 2522:⋯ 2519:∨ 2501:∨ 2483:∨ 2443:ω 2431:ω 2403:ω 2391:ω 2282:semantics 2064:← 2038:⊂ 2012:⇐ 1978:⊕ 1950:_ 1947:∨ 1869:∥ 1843:∣ 1791:∨ 1757:∼ 1735:¯ 1707:− 1684:¬ 1654:¯ 1618:↓ 1590:¯ 1587:∨ 1551:↮ 1468:¯ 1461:⋅ 1432:∣ 1406:↑ 1378:¯ 1375:∧ 1339:→ 1313:⊃ 1287:⇒ 1253:⇋ 1227:⇔ 1201:≡ 1167:& 1164:& 1138:& 1089:⋅ 1063:∧ 985:injection 981:bijection 876:countable 708:In 1900, 613:real line 584:induction 504:and then 426:syllogism 290:axiomatic 198:Education 188:Economics 163:Chemistry 10183:Logicism 10176:timeline 10152:Concrete 10011:Validity 9981:T-schema 9974:Kripke's 9969:Tarski's 9964:semantic 9954:Strength 9903:submodel 9898:spectrum 9866:function 9714:Tarski's 9703:Elements 9690:geometry 9646:Robinson 9567:variable 9552:function 9525:spectrum 9515:Sentence 9471:variable 9414:Language 9367:Relation 9328:Automata 9318:Alphabet 9302:language 9156:-jection 9134:codomain 9120:Function 9081:Universe 9051:Infinite 8955:Relation 8738:Validity 8728:Argument 8626:theorem, 8443:Category 8343:Validity 8244:Antinomy 8172:Theories 8136:Informal 7991:Category 7819:Graphics 7594:Security 7263:Compiler 7162:Networks 7059:Hardware 6973:Category 6729:Topology 6676:Discrete 6661:Analytic 6648:Geometry 6620:Discrete 6575:Calculus 6567:Analysis 6522:Abstract 6461:Glossary 6444:Timeline 6153:(1904). 6074:(1948). 6061:(1498). 5985:(1889). 5973:(1882). 5949:(1967). 5906:(1915). 5876:(1840). 5838:(1943). 5802:Springer 5796:(1934). 5755:(1929). 5734:(1899). 5665:(1958). 5581:(1929). 5533:, 1974. 5525:(1884), 5504:(1879), 5455:(1888). 5442:(1872). 5417:(1896). 5337:Springer 5294:(1924). 5260:(2001). 5139:(1965). 5110:14724471 5071:43258676 4894:(1966). 4868:(1950). 4740:Springer 4734:(2003). 4708:(1997). 4688:Elsevier 4601:Springer 4576:, 2004, 4516:Springer 4506:(2010). 4480:(1997). 4443:Springer 4312:(2002). 3528:Argument 3492:See also 3464:founded 3420:finitary 3368:function 3279:, as is 3232:E. Moody 3228:B. Mates 3144:G. Frege 2913:and the 2714:and the 2582:and the 2457:such as 2247:such as 2179:Category 1998:converse 1525:⇎ 1499:≢ 722:integers 690:powerset 458:and the 418:rhetoric 302:analysis 294:geometry 112:Analysis 108:Calculus 98:Geometry 10125:Related 9922:Diagram 9820: ( 9799:Hilbert 9784:Systems 9779:Theorem 9657:of the 9602:systems 9382:Formula 9377:Grammar 9293: ( 9237:General 8950:Forcing 8935:Element 8855:Monadic 8630:paradox 8571:Theorem 8507:General 8458:changes 8450: ( 8308:Premise 8239:Paradox 8069:History 8064:Outline 8001:Outline 6985:Commons 6767:Applied 6737:General 6514:Algebra 6439:History 6365:In the 6336:In the 6298:, 2001 6030:: 1–36. 5866:1990131 5828:0237246 5767:: 1–9. 5740:Leipzig 5633:]. 5216:5894394 5169:1994188 5063:2687794 5019:2695743 4971:2318447 3287:). 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