Church–Turing thesis

508:, finally goes on to formally name "Church's Thesis" and "Turing's Thesis", using his theory of recursive realizability. Kleene having switched from presenting his work in the terminology of Church-Kleene lambda definability, to that of Gödel-Kleene recursiveness (partial recursive functions). In this transition, Kleene modified Gödel's general recursive functions to allow for proofs of the unsolvability of problems in the Intuitionism of E. J. Brouwer. In his graduate textbook on logic, "Church's thesis" is introduced and basic mathematical results are demonstrated to be unrealizable. Next, Kleene proceeds to present "Turing's thesis", where results are shown to be uncomputable, using his simplified derivation of a Turing machine based on the work of Emil Post. Both theses are proven equivalent by use of "Theorem XXX". 556:), parallelism, and crystalline automata, led him to propose four "principles (or constraints) ... which it is argued, any machine must satisfy". His most-important fourth, "the principle of causality" is based on the "finite velocity of propagation of effects and signals; contemporary physics rejects the possibility of instantaneous action at a distance". From these principles and some additional constraints—(1a) a lower bound on the linear dimensions of any of the parts, (1b) an upper bound on speed of propagation (the velocity of light), (2) discrete progress of the machine, and (3) deterministic behavior—he produces a theorem that "What can be calculated by a device satisfying principles I–IV is computable." 6055: 257:) addresses the notion of "effective computability" as follows: "Clearly the existence of CC and RC (Church's and Rosser's proofs) presupposes a precise definition of 'effective'. 'Effective method' is here used in the rather special sense of a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps". Thus the adverb-adjective "effective" is used in a sense of "1a: producing a decided, decisive, or desired effect", and "capable of producing a result". 1054: 348:, and they were able to prove that several large classes of functions frequently encountered in number theory were λ-definable. The debate began when Church proposed to Gödel that one should define the "effectively computable" functions as the λ-definable functions. Gödel, however, was not convinced and called the proposal "thoroughly unsatisfactory". Rather, in correspondence with Church (c. 1934–1935), Gödel proposed 497:...the thesis has the character of an hypothesis—a point emphasized by Post and by Church. If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about the application of the mathematical theory developed from the definition. For the acceptance of the hypothesis, there are, as we have suggested, quite compelling grounds. 695:. Because all these different attempts at formalizing the concept of "effective calculability/computability" have yielded equivalent results, it is now generally assumed that the Church–Turing thesis is correct. In fact, Gödel (1936) proposed something stronger than this; he observed that there was something "absolute" about the concept of "reckonable in S 1891:

edited by Adam Olszewski et al. 2006, Peter Smith's criticism of a paper by Muraswski and Wolenski suggests 4 "lines" re the status of the Church–Turing Thesis: (1) empirical hypothesis (2) axiom or theorem, (3) definition, (4) explication. But Smith opines that (4) is indistinguishable from (3), cf.

387:

Many years later in a letter to Davis (c. 1965), Gödel said that "he was, at the time of these lectures, not at all convinced that his concept of recursion comprised all possible recursions". By 1963–1964 Gödel would disavow Herbrand–Gödel recursion and the λ-calculus in favor of the Turing machine

1378:

Church's paper was presented to the American Mathematical Society on 19 April 1935 and published on 15 April 1936. Turing, who had made substantial progress in writing up his own results, was disappointed to learn of Church's proof upon its publication. Turing quickly completed his paper and rushed

974:

states that it is an open empirical question whether there are actual deterministic physical processes that, in the long run, elude simulation by a Turing machine; furthermore, he states that it is an open empirical question whether any such processes are involved in the working of the human brain.

596:

The thesis can be viewed as nothing but an ordinary mathematical definition. Comments by Gödel on the subject suggest this view, e.g. "the correct definition of mechanical computability was established beyond any doubt by Turing". The case for viewing the thesis as nothing more than a definition is

440:

abstract computational model). And in a proof-sketch added as an "Appendix" to his 1936–1937 paper, Turing showed that the classes of functions defined by λ-calculus and Turing machines coincided. Church was quick to recognise how compelling Turing's analysis was. In his review of Turing's paper he

1141:

such as inductive Turing machines disprove the Church–Turing thesis. His argument relies on a definition of algorithm broader than the ordinary one, so that non-computable functions obtained from some inductive Turing machines are called computable. This interpretation of the Church–Turing thesis

720:

Proofs in computability theory often invoke the Church–Turing thesis in an informal way to establish the computability of functions while avoiding the (often very long) details which would be involved in a rigorous, formal proof. To establish that a function is computable by Turing machine, it is

284:

It was stated ... that "a function is effectively calculable if its values can be found by some purely mechanical process". We may take this literally, understanding that by a purely mechanical process one which could be carried out by a machine. The development ... leads to ... an

563:

analyzed Turing's and Gandy's notions of "effective calculability" with the intent of "sharpening the informal notion, formulating its general features axiomatically, and investigating the axiomatic framework". In his 1997 and 2002 work Sieg presents a series of constraints on the behavior of a

836:

In order to make the above example completely rigorous, one would have to carefully construct a Turing machine, or λ-function, or carefully invoke recursion axioms, or at best, cleverly invoke various theorems of computability theory. But because the computability theorist believes that Turing

260:

In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turing-machine or equivalent mechanical device". Turing's "definitions" given in a footnote in his 1938 Ph.D. thesis

1794:

which also gives these definitions for "effective" – the first as the definition for sense "1a" of the word "effective", and the second as part of the "Synonym Discussion of EFFECTIVE" there, (in the introductory part, where it summarizes the similarities between the meanings of the words

953:

Eugene Eberbach and Peter Wegner claim that the Church–Turing thesis is sometimes interpreted too broadly, stating "Though Turing machines express the behavior of algorithms, the broader assertion that algorithms precisely capture what can be computed is invalid". They claim that forms of

584:"(D) (Determinacy) The immediately recognizable (sub-)configuration determines uniquely the next computation step (and id )"; stated another way: "A computor's internal state together with the observed configuration fixes uniquely the next computation step and the next internal state." 312:, which asked whether there was a mechanical procedure for separating mathematical truths from mathematical falsehoods. This quest required that the notion of "algorithm" or "effective calculability" be pinned down, at least well enough for the quest to begin. But from the very outset 1383:

on 28 May 1936, read on 12 November 1936, and published in series 2, volume 42 (1936–1937); it appeared in two sections: in Part 3 (pages 230–240), issued on 30 November 1936 and in Part 4 (pages 241–265), issued on 23 December 1936; Turing added corrections in volume 43 (1937), pp.

1142:

differs from the interpretation commonly accepted in computability theory, discussed above. The argument that super-recursive algorithms are indeed algorithms in the sense of the Church–Turing thesis has not found broad acceptance within the computability research community.

181:

created a theoretical model for machines, now called Turing machines, that could carry out calculations from inputs by manipulating symbols on a tape. Given a suitable encoding of the natural numbers as sequences of symbols, a function on the natural numbers is called

531:

Kleene, finally, uses for the first time the term the "Church-Turing thesis" in a section in which he helps to give clarifications to concepts in Alan Turing's paper "The Word Problem in Semi-Groups with Cancellation", as demanded in a critique from William Boone.

198:. This has led mathematicians and computer scientists to believe that the concept of computability is accurately characterized by these three equivalent processes. Other formal attempts to characterize computability have subsequently strengthened this belief (see 942:. This would not however invalidate the original Church–Turing thesis, since a quantum computer can always be simulated by a Turing machine, but it would invalidate the classical complexity-theoretic Church–Turing thesis for efficiency reasons. Consequently, the 436:: Within just a short time, Turing's 1936–1937 paper "On Computable Numbers, with an Application to the Entscheidungsproblem" appeared. In it he stated another notion of "effective computability" with the introduction of his a-machines (now known as the 895:(1997). The complexity-theoretic Church–Turing thesis, then, posits that all 'reasonable' models of computation yield the same class of problems that can be computed in polynomial time. Assuming the conjecture that probabilistic polynomial time ( 711:. Thus the concept 'computable' is in a certain definite sense 'absolute', while practically all other familiar metamathematical concepts (e.g. provable, definable, etc.) depend quite essentially on the system to which they are defined ... 721:

usually considered sufficient to give an informal English description of how the function can be effectively computed, and then conclude "by the Church–Turing thesis" that the function is Turing computable (equivalently, partial recursive).

356:

His only idea at the time was that it might be possible, in terms of effective calculability as an undefined notion, to state a set of axioms which would embody the generally accepted properties of this notion, and to do something on that

837:

computability correctly captures what can be computed effectively, and because an effective procedure is spelled out in English for deciding the set B, the computability theorist accepts this as proof that the set is indeed recursive.

1130:, a problem known to be unsolvable by Turing machines. Since the busy beaver function cannot be computed by Turing machines, the Church–Turing thesis states that this function cannot be effectively computed by any method. 410:

Actually the work already done by Church and others carries this identification considerably beyond the working hypothesis stage. But to mask this identification under a definition… blinds us to the need of its continual

581:"(L.2) (Locality) A computor can shift attention from one symbolic configuration to another one, but the new observed configurations must be within a bounded distance of the immediately previously observed configuration. 271:

We shall use the expression "computable function" to mean a function calculable by a machine, and let "effectively calculable" refer to the intuitive idea without particular identification with any one of these

526:

Turing's thesis: Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem

1809: 406:'s 1936 paper had appeared and was certified independent of Turing's work. Post strongly disagreed with Church's "identification" of effective computability with the λ-calculus and recursion, stating: 3617:, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in 1544: 209:

notion of an effectively calculable function. Although the thesis has near-universal acceptance, it cannot be formally proven, as the concept of effective calculability is only informally defined.

828:'s is increasing we have to produce at most k+1 elements of the list and compare them with k. If none of them is equal to k, then k not in B. Since this test is effective, B is decidable and, 361:

But Gödel offered no further guidance. Eventually, he would suggest his recursion, modified by Herbrand's suggestion, that Gödel had detailed in his 1934 lectures in Princeton NJ (Kleene and

1710: 372:

produced proofs (1933, 1935) to show that the two calculi are equivalent. Church subsequently modified his methods to include use of Herbrand–Gödel recursion and then proved (1936) that the

368:

Next, it was necessary to identify and prove the equivalence of two notions of effective calculability. Equipped with the λ-calculus and "general" recursion, Kleene with help of Church and

1508: 194:, and Turing proved that these three formally defined classes of computable functions coincide: a function is λ-computable if and only if it is Turing computable, and if and only if it is 1635: 1578: 998:

The universe is not equivalent to a Turing machine (i.e., the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a

1736: 1465: 1601: 448:

formal definition of mechanical computing agent was the correct one. Thus, by 1939, both Church (1934) and Turing (1939) had individually proposed that their "formal systems" should be

1661: 911:'Reasonable' machines can simulate each other within a polynomially bounded overhead in time and a constant-factor overhead in space." The thesis originally appeared in a paper at 492:

Since a precise mathematical definition of the term effectively calculable (effectively decidable) has been wanting, we can take this thesis ... as a definition of it ...

521:

Theorem XXX: The following classes of partial functions are coextensive, i.e. have the same members: (a) the partial recursive functions, (b) the computable functions ...

2513: 688:

model of Kolmogorov and Uspensky (1953, 1958): "... they just wanted to ... convince themselves that there is no way to extend the notion of computable function."

852:

The Church–Turing thesis says nothing about the efficiency with which one model of computation can simulate another. It has been proved for instance that a (multi-tape)

280:. Church also stated that "No computational procedure will be considered as an algorithm unless it can be represented as a Turing Machine". Turing stated it this way: 1997: 1017:, and it is possible to build physical devices to harness this property and calculate non-recursive functions. For example, it is an open question whether all 1818: 859:

A variation of the Church–Turing thesis addresses whether an arbitrary but "reasonable" model of computation can be efficiently simulated. This is called the

212:

Since its inception, variations on the original thesis have arisen, including statements about what can physically be realized by a computer in our universe (

3135: 1040:

Philosophical aspects of the thesis, regarding both physical and biological computers, are also discussed in Odifreddi's 1989 textbook on recursion theory.

3951: 1865:) English Translation: David Hilbert and Wilhelm Ackermann: Principles of Mathematical Logic, AMS Chelsea Publishing, Providence, Rhode Island, USA, 1950. 1037:

There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept.

4434: 2562: 2284:

In a footnote Sieg breaks Post's 1936 (B) into (B.1) and (B.2) and (L) into (L.1) and (L.2) and describes (D) differently. With respect to his proposed

1287:

The essential Turing : seminal writings in computing, logic, philosophy, artificial intelligence, and artificial life, plus the secrets of Enigma

4042: 481:

This heuristic fact ... led Church to state the following thesis. The same thesis is implicit in Turing's description of computing machines.

2285: 560: 441:

made clear that Turing's notion made "the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately".

5109: 3500: 3745: 975:

There are also some important open questions which cover the relationship between the Church–Turing thesis and physics, and the possibility of

1156: 31: 5192: 4333: 601:, where it is also argued that Turing's definition of computability is no less likely to be correct than the epsilon-delta definition of a 205:

On the other hand, the Church–Turing thesis states that the above three formally-defined classes of computable functions coincide with the

2965: 1126:

states can print before halting, when run with no input. Finding an upper bound on the busy beaver function is equivalent to solving the

691:

All these contributions involve proofs that the models are computationally equivalent to the Turing machine; such models are said to be

1033:

have suggested that the human mind might be the result of some kind of quantum-mechanically enhanced, "non-algorithmic" computation.

3453: 3075: 2472: 6245: 5506: 4092: 1161: 988: 295: 1857:

David Hilbert and Wilhelm Ackermann: Grundzüge der theoretischen Logik, Berlin, Germany, Springer, 1st ed. 1928. (6th ed. 1972,

1064: 931: 896: 107:

to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to

5664: 4279: 3885: 3866: 3800: 3778: 3729: 3362: 3222:

Church, Alonzo (March 1937). "Review: A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem".

3095: 2954: 2905: 2854: 2815: 2784: 2744: 2598: 2545: 2388: 2020: 6159: 4452: 2327: 1512: 5519: 4842: 3321: 365:

transcribed the notes). But he did not think that the two ideas could be satisfactorily identified "except heuristically".

17: 572:"(B.1) (Boundedness) There is a fixed bound on the number of symbolic configurations a computor can immediately recognize. 6091: 4033: 4017: 2716: 621:) have been proposed for describing effective calculability/computability. Kleene (1952) adds to the list the functions " 5524: 5514: 5251: 5104: 4457: 4063: 1176: 1133:

Several computational models allow for the computation of (Church-Turing) non-computable functions. These are known as

903:), the word 'probabilistic' is optional in the complexity-theoretic Church–Turing thesis. A similar thesis, called the 724:

Dirk van Dalen gives the following example for the sake of illustrating this informal use of the Church–Turing thesis:

548:

computation (as opposed to human-computation acted out by a Turing machine). Gandy's curiosity about, and analysis of,

263: 40: 6154: 4448: 2026: 6199: 5660: 4295: 3514: 1862: 1665: 1298: 1097: 5002: 5757: 5501: 4326: 2638: 1469: 912: 660:

expanded the model to two or more tapes and greatly simplified the tapes into "up-down counters", which Melzak and

1605: 1548: 5062: 4755: 3380:. Kleene and Rosser (lecture note-takers); Institute for Advanced Study (lecture sponsor). New York: Raven Press. 3174: 1714: 1443: 876: 221: 6184: 4496: 3990: 1582: 415:

Rather, he regarded the notion of "effective calculability" as merely a "working hypothesis" that might lead by

6240: 6018: 5720: 5483: 5478: 5303: 4724: 4408: 3344:(1980). "Church's Thesis and the Principles for Mechanisms". In H. J. Barwise; H. J. Keisler; K. Kunen (eds.). 1025:

are equivalent to deterministic Turing machines. (They are not necessarily efficiently equivalent; see above.)

3928:

Turing, A. M. (1938). "On Computable Numbers, with an Application to the Entscheidungsproblem: A correction".

2569: 934:, it would invalidate the complexity-theoretic Church–Turing thesis. In other words, there would be efficient 6230: 6013: 5796: 5713: 5426: 5357: 5234: 4476: 4193: 4178: 1639: 320:

the notion of "effective calculability" to be (i) an "axiom or axioms" in an axiomatic system, (ii) merely a

213: 2885:. Studies in Logic and the Foundations of Mathematics. Vol. 125. Amsterdam, Netherlands: North Holland. 5938: 5764: 5450: 5084: 4683: 4183: 4120: 3372:

Gödel, Kurt (1965) . "On Undecidable Propositions of Formal Mathematical Systems". In Davis, Martin (ed.).

3269: 3260:

Cooper, S. B.; Odifreddi, P. (2003). "Incomputability in Nature". In S. B. Cooper; S. S. Goncharov (eds.).

880: 845:

The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the

649: 1110:

One can formally define functions that are not computable. A well-known example of such a function is the

915:'84, which was the first paper to show that polynomial-time overhead and constant-space overhead could be 6209: 5816: 5811: 5421: 5160: 5089: 4418: 4319: 4085: 3899: 2635:

On tape versus core: an application of space efficient perfect hash functions to the invariance of space

1952:

Sieg 1997:160 quoting from the 1935 letter written by Church to Kleene, cf. Footnote 3 in Gödel 1934 in

6204: 6142: 5745: 5335: 4729: 4697: 4388: 3276:

The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions

2537: 128: 3282:

Includes original papers by Gödel, Church, Turing, Rosser, Kleene, and Post mentioned in this section.

1075: 883:

can efficiently simulate any realistic model of computation." The word 'efficiently' here means up to

430:'s suggestion to Church in 1934–1935 that the thesis might be expressed as an axiom or set of axioms. 6035: 5984: 5881: 5379: 5340: 4817: 4462: 4142: 4137: 3705: 3558: 3025: 2844: 2805: 2552:

Sections 1.4, "Machines as strings and the universal Turing machine" and 1.7, "Proof of theorem 1.9".

1201: 1196: 1138: 884: 171: 4491: 3676: 3432: 3039: 2492: 174:

if the corresponding function on the Church numerals can be represented by a term of the λ-calculus.

103:. Before the precise definition of computable function, mathematicians often used the informal term 5806: 5345: 5197: 5180: 4903: 4383: 3316: 2984: 1340: 1079: 1026: 939: 853: 553: 352:

the notion of "effective calculability"; indeed, in a 1935 letter to Kleene, Church reported that:

4024: 575:"(B.2) (Boundedness) There is a fixed bound on the number of internal states a computor can be in. 477:: This left the overt expression of a "thesis" to Kleene. In 1943 Kleene proposed his "Thesis I": 6147: 6084: 5708: 5685: 5646: 5532: 5473: 5119: 5039: 4883: 4827: 4440: 2376: 1991:

For a detailed discussion of Gödel's adoption of Turing's machines as models of computation, see

1245: 668:

model. In the late 1960s and early 1970s researchers expanded the counter machine model into the

653: 875:, which is not due to Church or Turing, but rather was realized gradually in the development of 6179: 5998: 5725: 5703: 5670: 5563: 5409: 5394: 5367: 5318: 5202: 5137: 4962: 4928: 4923: 4797: 4628: 4605: 4188: 4078: 3671: 3427: 3311: 3034: 3005: 2979: 2487: 2433:

p. 83, differing in the use of the word 'reckonable' in the translation in Kleene (1952) p. 321

1335: 1022: 947: 729: 104: 80: 2380: 2362: 1229: 5928: 5781: 5573: 5291: 5027: 4933: 4792: 4777: 4658: 4633: 4173: 2878: 1272: 920: 578:"(L.1) (Locality) A computor can change only elements of an observed symbolic configuration. 514:

Every effectively calculable function (effectively decidable predicate) is general recursive

487:

Every effectively calculable function (effectively decidable predicate) is general recursive

6235: 6174: 6169: 6121: 5901: 5863: 5740: 5544: 5384: 5308: 5286: 5114: 5072: 4971: 4938: 4802: 4590: 4501: 4274: 1230:"Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory" 1191: 1181: 423:" rather than by "a definition or an axiom". This idea was "sharply" criticized by Church. 402:

is unsolvable) was delivered orally, but had not yet appeared in print. On the other hand,

399: 373: 301: 132: 4218: 1071: 8: 6115: 6030: 5921: 5906: 5886: 5843: 5730: 5680: 5606: 5551: 5488: 5281: 5276: 5224: 4992: 4981: 4653: 4553: 4481: 4472: 4468: 4403: 4398: 4208: 4163: 4028: 3524: 3300: 2468: 1206: 1171: 984: 645: 602: 416: 377: 341: 191: 183: 152: 131:: the smallest class of functions (with arbitrarily many arguments) that is closed under 76: 3412: 3064: 1285: 6077: 6059: 5828: 5791: 5776: 5769: 5752: 5556: 5538: 5404: 5330: 5313: 5266: 5079: 4988: 4822: 4807: 4767: 4719: 4704: 4692: 4648: 4623: 4393: 4342: 4238: 4203: 3982: 3974: 3918: 3859: 3842: 3834: 3739: 3689: 3602: 3544: 3496: 3484: 3445: 3374: 3274: 3239: 3210: 3202: 3166: 3158: 3122: 2997: 2685: 2665: 2505: 1361: 1353: 967: 935: 229: 148: 5012: 3385:

Gödel, Kurt (1936). "Über die Lāange von Beweisen" [On The Length of Proofs].

2288:

he later adds LC.1, LC.2, GA.1 and GA.2. These are complicated; see Sieg 1998–1999 in

1780: 966:

Philosophers have interpreted the Church–Turing thesis as having implications for the

186:

if some Turing machine computes the corresponding function on encoded natural numbers.

6054: 5994: 5801: 5611: 5601: 5493: 5374: 5209: 5185: 4966: 4950: 4855: 4832: 4709: 4678: 4643: 4538: 4373: 4243: 3881: 3862: 3784: 3774: 3753: 3725: 3638: 3510: 3488: 3358: 3091: 2950: 2911: 2901: 2860: 2850: 2821: 2811: 2780: 2750: 2740: 2594: 2541: 2394: 2384: 2308: 2016: 1897: 1858: 1294: 1018: 1007: 677: 549: 369: 309: 250: 4058: 4008: 3846: 3817:(1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem". 3693: 3214: 3170: 3001: 2707: 2689: 444:

In a few years (1939) Turing would propose, like Church and Kleene before him, that

30:"Church's thesis" redirects here. For the axiom CT in constructive mathematics, see 6137: 6008: 6003: 5896: 5853: 5675: 5636: 5631: 5616: 5442: 5399: 5296: 5094: 5044: 4618: 4580: 4258: 4233: 4228: 4168: 3986: 3966: 3933: 3910: 3826: 3681: 3592: 3567: 3536: 3476: 3464: 3449: 3437: 3231: 3194: 3150: 3114: 3044: 2989: 2677: 2497: 2366: 2008: 1365: 1345: 1247:

Turing, Church, Gödel, Computability, Complexity and Randomization: A Personal View

1241: 1186: 1165: 1151: 1134: 1014: 999: 987:

is physically impossible. This has been termed the strong Church–Turing thesis, or

976: 955: 681: 669: 568:—"a human computing agent who proceeds mechanically". These constraints reduce to: 381: 245: 124: 88: 3856:

Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman

3662:

Kugel, Peter (November 2005). "It's time to think outside the computational box".

3571: 2509: 220:). These variations are not due to Church or Turing, but arise from later work in 6104: 5989: 5979: 5933: 5916: 5871: 5833: 5735: 5655: 5462: 5389: 5362: 5350: 5256: 5170: 5144: 5099: 5067: 4868: 4670: 4613: 4563: 4528: 4486: 4213: 4130: 4115: 3922: 3806: 2772: 2711: 2531: 2370: 2334: 1323: 1127: 992: 954:

computation not captured by the thesis are relevant today, terms which they call

703:

It may also be shown that a function which is computable in one of the systems S

692: 685: 665: 614: 598: 345: 225: 167: 163: 84: 6069: 1878:

in Davis 1965:88. Church uses the words "effective calculability" on page 100ff.

1021:

events are Turing-computable, although it is known that rigorous models such as

6194: 5974: 5953: 5911: 5891: 5786: 5641: 5239: 5229: 5219: 5214: 5148: 5022: 4898: 4787: 4782: 4760: 4361: 4253: 3937: 3914: 3796: 3701: 2732: 1119: 900: 892: 661: 618: 437: 388:

as the definition of "algorithm" or "mechanical procedure" or "formal system".

92: 3048: 2838:

Also the description of "the non-algorithmic nature of mathematical insight",

2681: 2501: 2012: 166:. Within λ-calculus, he defined an encoding of the natural numbers called the 72: 6224: 5948: 5626: 5133: 4918: 4908: 4878: 4863: 4533: 4101: 4012: 3788: 3626: 3399: 3060: 3056: 2942: 2938: 2915: 2864: 2840: 2825: 2801: 2768: 2754: 2703: 2398: 1030: 971: 856:

only suffers a logarithmic slowdown factor in simulating any Turing machine.

733: 657: 313: 305: 159: 144: 112: 108: 96: 3685: 3480: 2993: 2846:

The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics

2807:

The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics

2358: 630: 540:

An attempt to better understand the notion of "effective computability" led

427: 140: 120: 5848: 5695: 5596: 5588: 5468: 5416: 5325: 5261: 5244: 5175: 5034: 4893: 4595: 4378: 3814: 3650: 3291: 2473:"Computationalism, the Church–Turing Thesis, and the Church–Turing Fallacy" 1993: 1225: 1003: 362: 3642: 3441: 3105:

Church, Alonzo (1932). "A set of Postulates for the Foundation of Logic".

820:. B is decidable. For, in order to test k in B we must check if k = m 6189: 6132: 5958: 5838: 5017: 5007: 4954: 4638: 4558: 4543: 4423: 4368: 4300: 4223: 4198: 4158: 3895: 3717: 3506: 3341: 2970: 2934: 1111: 541: 420: 395: 178: 100: 4043:"Transcendental idealism and Post's variant of the Church-Turing thesis" 3900:"On Computable Numbers, with an Application to the Entscheidungsproblem" 4888: 4743: 4714: 4520: 4248: 3978: 3838: 3606: 3548: 3243: 3206: 3162: 3126: 1268: 588:

The matter remains in active discussion within the academic community.

3556:

Kleene, Stephen Cole (1936). "Lambda-Definability and Recursiveness".

1357: 707:, or even in a system of transfinite type, is already computable in S 6164: 6100: 6040: 5943: 4996: 4913: 4873: 4837: 4773: 4585: 4575: 4548: 4311: 3308:

Bulletin of the European Association for Theoretical Computer Science

979:. When applied to physics, the thesis has several possible meanings: 907:, was introduced by Cees F. Slot and Peter van Emde Boas. It states: 849:

states: "All physically computable functions are Turing-computable."

739:

Proof: Let A be infinite RE. We list the elements of A effectively, n

634: 403: 136: 3970: 3830: 3657:. Vol. 1/Fundamental Algorithms (2nd ed.). Addison–Wesley. 3597: 3580: 3540: 3235: 3198: 3154: 3118: 459:

Rosser (1939) formally identified the three notions-as-definitions:

6025: 5823: 5271: 4976: 4570: 3185:

Church, Alonzo (June 1936b). "A Note on the Entscheidungsproblem".

2619: 1349: 673: 3413:"Sequential Abstract State Machines Capture Sequential Algorithms" 2777:

The Blackwell guide to the philosophy of computing and information

376:

is unsolvable: there is no algorithm that can determine whether a

5621: 4413: 3527:(January 1935). "A Theory of Positive Integers in Formal Logic". 3404:

Bulletin of European Association for Theoretical Computer Science

3072:

Bulletin of European Association for Theoretical Computer Science

3023:

Bernstein, E.; Vazirani, U. (1997). "Quantum complexity theory".

452:

of "effective calculability"; neither framed their statements as

300:

One of the important problems for logicians in the 1930s was the

4070: 3854:

Sieg, Wilfried; Sommer, Richard; Talcott, Carolyn, eds. (2002).

2666:"Philosophy of mind is (in part) philosophy of computer science" 983:

The universe is equivalent to a Turing machine; thus, computing

2614:

van Emde Boas, Peter (1990). "Machine Models and Simulations".

3700: 950:

can efficiently simulate any realistic model of computation."

340:

In the course of studying the problem, Church and his student

328:

to be verified by observation of natural events, or (iv) just

285:

identification of computability with effective calculability.

278:

Every effectively calculable function is a computable function

217: 5165: 4511: 4356: 3769:

Olszewski, Adam; Woleński, Jan; Janusz, Robert, eds. (2006).

4037:—a comprehensive philosophical treatment of relevant issues. 3878:

Hypercomputation: Computing Beyond the Church–Turing Barrier

3795: 316:'s attempts began with a debate that continues to this day. 1002:. For example, a universe in which physics involves random 239: 3402:(June 1988). "On Kolmogorov Machines and Related Issues". 1795:"effective", "effectual", "efficient", and "efficacious"). 794:, etc. this yields an effective listing of the subset B={m 652:

greatly simplified the one-tape Turing-machine model (see

2589:

Kaye, Phillip; Laflamme, Raymond; Mosca, Michele (2007).

927: 1803: 1801: 467:

are equivalent, so it does not matter which one is used.

434:

Turing adds another definition, Rosser equates all three

3768: 2737:

Philosophy of Mind: Classical and Contemporary Readings

2443: 2302: 889:

computational complexity-theoretic Church–Turing thesis

3932:. 2. Vol. 43 (published 1937). pp. 544–546. 3467:(1932). "Sur la non-contradiction de l'arithmétique". 3264:. Kluwer Academic/Plenum Publishers. pp. 137–160. 2900:. Monographs in Computer Science. New York: Springer. 1817:(PhD). Princeton University. p. 8. Archived from 1722: 1647: 1587: 1539:{\displaystyle \lambda {\mbox{-}}K{\mbox{-definable}}} 1530: 1520: 1451: 758:

From this list we extract an increasing sublist: put m

3348:. North-Holland Publishing Company. pp. 123–148. 2877: 2849:. Oxford: Oxford University Press. pp. 416–418. 2328:"Did Church and Turing Have a Thesis about Machines?" 1798: 1717: 1668: 1642: 1608: 1585: 1551: 1515: 1472: 1446: 324:

that "identified" two or more propositions, (iii) an

3712:. Upper Saddle River, New Jersey, US: Prentice-Hall. 3499:"Chapter XVII: Church, Turing, Tarski, and Others". 3262:

Computability and Models: Perspectives East and West

3355:

The universal Turing Machine: A Half-Century Survey

3259: 3136:"An Unsolvable Problem of Elementary Number Theory" 3086:Burgin, Mark (2005). "Super-recursive algorithms". 2933: 2810:. Oxford: Oxford University Press. pp. 47–49. 2207: 2205: 162:created a method for defining functions called the 95:. The thesis is named after American mathematician 3373: 3273: 2730:For a good place to encounter original papers see 1898:http://www.logicmatters.net/resources/pdfs/CTT.pdf 1730: 1704: 1655: 1629: 1595: 1572: 1538: 1502: 1459: 6099: 3853: 3585:Transactions of the American Mathematical Society 3357:. New York: Wien Springer–Verlag. pp. 51ff. 3288:Logical Dilemmas: The Life and Work of Kurt Gödel 3022: 2632: 2588: 2289: 2272: 1913:An Unsolvable Problem of Elementary Number Theory 1876:An Unsolvable Problem of Elementary Number Theory 1326:(September 1996). "Computability and Recursion". 1118:and returns the largest number of symbols that a 944:quantum complexity-theoretic Church–Turing thesis 544:(Turing's student and friend) in 1980 to analyze 426:Thus Post in his 1936 paper was also discounting 6222: 2202: 2048: 2046: 1168:can be simulated by a universal computing device 267:, supervised by Church, are virtually the same: 177:Also in 1936, before learning of Church's work, 3469:Journal für die Reine und Angewandte Mathematik 2949:. Amsterdam: North-Holland Publishing Company. 1705:{\displaystyle {\stackrel {Kleene}{\implies }}} 233: 3930:Proceedings of the London Mathematical Society 3907:Proceedings of the London Mathematical Society 3298: 3065:"Algorithms: A Quest for Absolute Definitions" 2966:"The Church-Turing Thesis and its Look-Alikes" 2651: 1381:Proceedings of the London Mathematical Society 1277: 938:that perform tasks that do not have efficient 664:further evolved into what is now known as the 170:. A function on the natural numbers is called 6085: 4327: 4086: 3055: 2633:Slot, C.; van Emde Boas, P. (December 1984). 2613: 2043: 1846:What is effectively calculable is computable. 1503:{\displaystyle {\stackrel {triv}{\implies }}} 961: 672:, a close cousin to the modern notion of the 591: 3502:Gödel, Escher, Bach: an Eternal Golden Braid 1630:{\displaystyle {\stackrel {161}{\implies }}} 1573:{\displaystyle {\stackrel {160}{\implies }}} 332:for the sake of argument (i.e. a "thesis")? 127:, formalized the definition of the class of 4040: 3387:Ergenbnisse Eines Mathematishen Kolloquiums 2843:(1990). "Where lies the physics of mind?". 1765:Merriam Webster's New Collegiate Dictionary 1731:{\displaystyle \lambda {\mbox{-definable}}} 1460:{\displaystyle \lambda {\mbox{-definable}}} 1261: 1157:Church's thesis in constructive mathematics 1043: 715: 228:. The thesis also has implications for the 6092: 6078: 4519: 4334: 4320: 4093: 4079: 3875: 3758:American Mathematical Society Translations 3744:: CS1 maint: location missing publisher ( 2804:(1990). "Algorithms and Turing machines". 2616:Handbook of Theoretical Computer Science A 2529: 2066:Finite Combinatory Process. Formulation I. 1939: 1937: 1677: 1673: 1617: 1613: 1596:{\displaystyle {\mbox{Turing computable}}} 1560: 1556: 1481: 1477: 1379:it to publication; it was received by the 1372: 199: 32:Church's thesis (constructive mathematics) 3675: 3596: 3431: 3353:Gandy, Robin (1994). Herken, Rolf (ed.). 3315: 3299:Eberbach, E.; Wegner, P. (October 2003). 3038: 2983: 2963: 2593:. Oxford University Press. pp. 5–6. 2491: 2467: 1339: 1098:Learn how and when to remove this message 869:complexity-theoretic Church–Turing thesis 613:Other formalisms (besides recursion, the 4067:was devoted to the Church–Turing thesis. 3950:Turing, Alan Mathison (December 1937b). 3858:. Lecture Notes in Logic. Vol. 15. 3463: 3410: 3398: 3255:. Princeton: Princeton University Press. 2767: 2731: 2702: 2663: 2429:Translation of Gödel (1936) by Davis in 2174: 1978:Davis's commentary before Gödel 1934 in 1070:Relevant discussion may be found on the 899:) equals deterministic polynomial time ( 766:, after finitely many steps we find an n 608: 240:Statement in Church's and Turing's words 218:Church–Turing thesis (complexity theory) 216:) and what can be efficiently computed ( 3949: 3894: 3420:ACM Transactions on Computational Logic 3184: 3133: 2839: 2800: 2584: 2582: 2303:Olszewski, Woleński & Janusz (2006) 2301:A collection of papers can be found in 2052: 1992: 1934: 1909: 1656:{\displaystyle \mu {\mbox{-recursive}}} 1437: 1415: 1393: 1318: 1316: 1293:. Oxford: Clarendon Press. p. 44. 1114:function. This function takes an input 14: 6223: 4341: 3927: 3813: 3752: 3632: 3622: 3618: 3581:"Recursive Predicates and Quantifiers" 3578: 3555: 3523: 3495: 3392: 3331: 3285: 3250: 3221: 3104: 3085: 2895: 2455: 2363:"Undecidable Diophantine Propositions" 2325: 2223: 2211: 2196: 2180: 2164: 2125: 1928: 1874:Davis's commentary before Church 1936 1807: 1746: 1426: 1404: 1283: 1082:to additional sources at this section. 392:A hypothesis leading to a natural law? 254: 6073: 4315: 4280:University of California, Los Angeles 4074: 3943: 3909:, 2, vol. 42, pp. 230–265, 3802:Computability in Analysis and Physics 3756:(1960) . "The Theory of Algorithms". 3716: 3710:Elements of the Theory of Computation 3661: 3649: 3621::300) and name it "Church's Thesis" ( 3612: 3384: 3371: 3352: 3340: 3268: 2739:. New York: Oxford University Press. 2444:Olszewski, Woleński & Janusz 2006 2357: 2306: 2259: 2247: 2235: 2184: 2168: 2151: 2147: 2113: 2082: 2069: 1979: 1966: 1953: 1916: 1841: 1750: 1322: 1240: 535: 91:if and only if it is computable by a 27:Thesis on the nature of computability 2591:An introduction to quantum computing 2579: 2533:Complexity Theory: A Modern Approach 2530:Arora, Sanjeev; Barak, Boaz (2009). 2305:. Also a review of this collection: 1313: 1047: 930:is shown to be a strict superset of 887:. This thesis was originally called 786:. We repeat this procedure to find m 4034:Stanford Encyclopedia of Philosophy 4018:Stanford Encyclopedia of Philosophy 2717:Stanford Encyclopedia of Philosophy 2064:cf. Editor's footnote to Post 1936 1782:Merriam-Webster's Online Dictionary 824:for some i. Since the sequence of m 296:History of the Church–Turing thesis 24: 4064:Notre Dame Journal of Formal Logic 3952:"Computability and λ-Definability" 3722:Mathematical Theory of Computation 3505:(Twentieth-anniversary ed.). 1998:"Gödel on Turing on Computability" 1811:Systems of Logic Based on Ordinals 1063:relies largely or entirely upon a 335: 264:Systems of Logic Based on Ordinals 25: 6257: 4296:Alonzo Church (college president) 4100: 4041:Kaznatcheev, Artem (2014-09-11). 4025:"Computation in Physical Systems" 4002: 6053: 3459:from the original on 2003-10-16. 3327:from the original on 2016-03-15. 3290:. Wellesley, Massachusetts, US: 3253:The Calculi of Lambda-Conversion 3081:from the original on 2004-07-27. 2519:from the original on 2008-04-24. 2309:"Church's Thesis after 70 Years" 1848:He calls this "Church's Thesis". 1440:. Proof outline on p. 153: 1052: 1010:, would fall into this category. 398:'s paper (also proving that the 6160:Gödel's incompleteness theorems 3655:The Art of Computer Programming 3635:Introduction to Metamathematics 3529:American Journal of Mathematics 3334:Handbook of Philosophical Logic 3143:American Journal of Mathematics 2889: 2871: 2832: 2793: 2779:. Wiley-Blackwell. p. 15. 2761: 2724: 2696: 2657: 2645: 2626: 2607: 2555: 2523: 2461: 2449: 2436: 2423: 2414: 2405: 2351: 2318: 2295: 2290:Sieg, Sommer & Talcott 2002 2278: 2273:Sieg, Sommer & Talcott 2002 2265: 2253: 2241: 2229: 2217: 2190: 2157: 2140: 2131: 2119: 2106: 2097: 2088: 2075: 2058: 1985: 1972: 1959: 1946: 1922: 1902: 1881: 1868: 1851: 1835: 1771: 1756: 1740: 1431: 1420: 1162:Church–Turing–Deutsch principle 989:Church–Turing–Deutsch principle 919:achieved for a simulation of a 806:,...} of A, with the property m 506:Introduction To Metamathematics 6246:Philosophy of computer science 4061:(Vol. 28, No. 4, 1987) of the 3876:Syropoulos, Apostolos (2008). 3771:Church's Thesis After 70 Years 3134:Church, Alonzo (April 1936a). 3088:Monographs in computer science 2563:"Official Problem Description" 2275::390ff.; also Sieg 1997:154ff. 2005:Church's Thesis After 70 Years 1894:Church's Thesis after 70 Years 1889:Church's Thesis after 70 Years 1674: 1614: 1557: 1478: 1409: 1398: 1387: 1234: 1219: 732:(RE) set contains an infinite 99:and the British mathematician 13: 1: 6014:History of mathematical logic 3819:The Journal of Symbolic Logic 3633:Kleene, Stephen Cole (1952). 3579:Kleene, Stephen Cole (1943). 3572:10.1215/s0012-7094-36-00227-2 3389:(in German) (7). Heft: 23–24. 3336:. Vol. 1 (2nd ed.). 2926: 873:extended Church–Turing thesis 847:physical Church–Turing thesis 840: 276:The thesis can be stated as: 214:physical Church-Turing thesis 6155:Gödel's completeness theorem 5939:Primitive recursive function 4121:Simply typed lambda calculus 3535:(1): 153–173 & 219–244. 3411:Gurevich, Yuri (July 2000). 3286:Dawson, John W. Jr. (1997). 2094:Post 1936 in Davis 1952:291. 1212: 881:probabilistic Turing machine 7: 2307:Smith, Peter (2007-07-11). 1145: 129:general recursive functions 10: 6262: 6143:Foundations of mathematics 5003:Schröder–Bernstein theorem 4730:Monadic predicate calculus 4389:Foundations of mathematics 4009:"The Church–Turing Thesis" 2898:Super-Recursive Algorithms 2883:Classical Recursion Theory 2771:(2004). "Computation". In 2708:"The Church-Turing Thesis" 2652:Eberbach & Wegner 2003 2538:Cambridge University Press 2150::100, also Turing 1939 in 2103:Sieg 1997:171 and 176–177. 1844::123) states it this way: 1328:Bulletin of Symbolic Logic 1164:, which states that every 1139:super-recursive algorithms 1137:. Mark Burgin argues that 962:Philosophical implications 885:polynomial-time reductions 592:The thesis as a definition 293: 289: 243: 29: 6111: 6049: 6036:Philosophy of mathematics 5985:Automated theorem proving 5967: 5862: 5694: 5587: 5439: 5156: 5132: 5110:Von Neumann–Bernays–Gödel 5055: 4949: 4853: 4751: 4742: 4669: 4604: 4510: 4432: 4349: 4288: 4267: 4151: 4108: 4047:Journal of Symbolic Logic 3959:Journal of Symbolic Logic 3799:; Richards, J.I. (1989). 3664:Communications of the ACM 3559:Duke Mathematical Journal 3224:Journal of Symbolic Logic 3187:Journal of Symbolic Logic 3049:10.1137/S0097539796300921 3026:SIAM Journal on Computing 2964:Ben-Amram, A. M. (2005). 2682:10.1007/s11023-011-9236-0 2664:Abramson, Darren (2011). 2502:10.1007/s11229-005-0194-z 2375:. Vol. 3. New York: 2137:Turing 1939 in Davis:160. 2013:10.1515/9783110325461.393 1202:Super-recursive algorithm 1197:Oracle (computer science) 991:, and is a foundation of 344:introduced the notion of 6185:Löwenheim–Skolem theorem 3938:10.1112/plms/s2-43.6.544 3915:10.1112/plms/s2-42.1.230 3301:"Beyond Turing Machines" 3280:. New York: Raven Press. 2187::274. Footnotes omitted. 1044:Non-computable functions 956:super-Turing computation 940:probabilistic algorithms 854:universal Turing machine 716:Informal usage in proofs 502:The Church–Turing Thesis 87:can be calculated by an 57:Church–Turing conjecture 6210:Use–mention distinction 5686:Self-verifying theories 5507:Tarski's axiomatization 4458:Tarski's undefinability 4453:incompleteness theorems 3686:10.1145/1096000.1096001 3481:10.1515/crll.1932.166.1 3251:Church, Alonzo (1941). 2994:10.1145/1086649.1086651 2377:Oxford University Press 2326:Hodges, Andrew (2005). 1267:Correspondence between 1250:. Event occurs at 9:36 1023:quantum Turing machines 985:non-recursive functions 891:by Ethan Bernstein and 728:Example: Each infinite 676:. Other models include 6205:Type–token distinction 6060:Mathematics portal 5671:Proof of impossibility 5319:propositional variable 4629:Propositional calculus 3497:Hofstadter, Douglas R. 3332:Gabbay, D. M. (2001). 1808:Turing, A. M. (1938). 1732: 1706: 1657: 1631: 1597: 1574: 1540: 1504: 1461: 948:quantum Turing machine 834: 730:recursively enumerable 713: 529: 523: 518: 499: 494: 490: 469: 413: 359: 287: 274: 105:effectively calculable 6241:Theory of computation 5929:Kolmogorov complexity 5882:Computably enumerable 5782:Model complete theory 5574:Principia Mathematica 4634:Propositional formula 4463:Banach–Tarski paradox 4174:George Alfred Barnard 4143:Church–Rosser theorem 4138:Frege–Church ontology 3442:10.1145/343369.343384 3107:Annals of Mathematics 2896:Burgin, Mark (2005). 2879:Piergiorgio Odifreddi 1733: 1707: 1658: 1632: 1598: 1575: 1541: 1505: 1462: 1284:Turing, Alan (2004). 923:on a Turing machine. 921:Random Access Machine 863:, also known as the ( 726: 701: 609:Success of the thesis 554:Conway's game of life 524: 519: 510: 504:: Stephen Kleene, in 495: 483: 479: 461: 408: 354: 346:λ-definable functions 282: 269: 6231:Computability theory 6128:Church–Turing thesis 6122:Entscheidungsproblem 5877:Church–Turing thesis 5864:Computability theory 5073:continuum hypothesis 4591:Square of opposition 4449:Gödel's completeness 4275:Princeton University 4126:Church–Turing thesis 3773:. Frankfurt: Ontos. 3525:Kleene, Stephen Cole 3509:. pp. 559–585. 3346:The Kleene Symposium 2947:The Kleene Symposium 2469:Piccinini, Gualtiero 2112:Turing 1936–1937 in 1715: 1666: 1640: 1606: 1583: 1549: 1513: 1470: 1444: 1273:Alonzo Church papers 1192:Model of computation 1177:Computability theory 1076:improve this article 684:. Gurevich adds the 400:Entscheidungsproblem 374:Entscheidungsproblem 326:empirical hypothesis 302:Entscheidungsproblem 77:computable functions 75:about the nature of 53:Turing–Church thesis 49:computability thesis 45:Church–Turing thesis 41:computability theory 18:Church-Turing thesis 6031:Mathematical object 5922:P versus NP problem 5887:Computable function 5681:Reverse mathematics 5607:Logical consequence 5484:primitive recursive 5479:elementary function 5252:Free/bound variable 5105:Tarski–Grothendieck 4624:Logical connectives 4554:Logical equivalence 4404:Logical consequence 4209:Stephen Cole Kleene 4164:C. Anthony Anderson 4029:Gualtiero Piccinini 3706:Papadimitriou, C.H. 2146:cf. Church 1934 in 1965:cf. Church 1936 in 1892:Smith (2007-07-11) 1207:Turing completeness 1172:Computability logic 654:Post–Turing machine 603:continuous function 597:made explicitly by 417:inductive reasoning 378:well formed formula 79:. It states that a 65:Church's conjecture 5829:Transfer principle 5792:Semantics of logic 5777:Categorical theory 5753:Non-standard model 5267:Logical connective 4394:Information theory 4343:Mathematical logic 4239:Hartley Rogers, Jr 4204:John George Kemeny 3860:A. K. Peters, Ltd. 2733:Chalmers, David J. 2670:Minds and Machines 2271:Sieg 1998–1999 in 1908:cf. footnote 3 in 1728: 1726: 1702: 1653: 1651: 1627: 1593: 1591: 1570: 1536: 1534: 1524: 1500: 1457: 1455: 1019:quantum mechanical 1013:The universe is a 968:philosophy of mind 936:quantum algorithms 861:feasibility thesis 830:by Church's thesis 559:In the late 1990s 536:Later developments 230:philosophy of mind 6218: 6217: 6067: 6066: 5999:Abstract category 5802:Theories of truth 5612:Rule of inference 5602:Natural deduction 5583: 5582: 5128: 5127: 4833:Cartesian product 4738: 4737: 4644:Many-valued logic 4619:Boolean functions 4502:Russell's paradox 4477:diagonal argument 4374:First-order logic 4309: 4308: 4244:J. Barkley Rosser 3887:978-0-387-30886-9 3868:978-1-56881-169-7 3780:978-3-938793-09-1 3731:978-0-486-43238-0 3637:. North-Holland. 3625::317) (i.e., the 3465:Herbrand, Jacques 3364:978-3-211-82637-9 3097:978-0-387-95569-8 2956:978-0-444-85345-5 2907:978-0-387-95569-8 2856:978-0-19-851973-7 2817:978-0-19-851973-7 2786:978-0-631-22919-3 2769:Copeland, B. Jack 2746:978-0-19-514581-6 2704:Copeland, B. Jack 2600:978-0-19-857049-3 2547:978-0-521-42426-4 2390:978-0-19-507255-6 2367:Feferman, Solomon 2085::291, footnote 8. 2022:978-3-11-032494-5 1887:In his review of 1725: 1699: 1650: 1624: 1590: 1589:Turing computable 1567: 1533: 1523: 1497: 1454: 1242:Rabin, Michael O. 1108: 1107: 1100: 905:invariance thesis 877:complexity theory 682:Markov algorithms 678:combinatory logic 637:'s (1943, 1946) " 550:cellular automata 370:J. Barkley Rosser 310:Wilhelm Ackermann 222:complexity theory 196:general recursive 184:Turing computable 16:(Redirected from 6253: 6138:Effective method 6116:Cantor's theorem 6094: 6087: 6080: 6071: 6070: 6058: 6057: 6009:History of logic 6004:Category of sets 5897:Decision problem 5676:Ordinal analysis 5617:Sequent calculus 5515:Boolean algebras 5455: 5454: 5429: 5400:logical/constant 5154: 5153: 5140: 5063:Zermelo–Fraenkel 4814:Set operations: 4749: 4748: 4686: 4517: 4516: 4497:Löwenheim–Skolem 4384:Formal semantics 4336: 4329: 4322: 4313: 4312: 4259:Raymond Smullyan 4234:Nicholas Rescher 4229:Michael O. Rabin 4095: 4088: 4081: 4072: 4071: 4054: 4013:B. Jack Copeland 3997: 3995: 3989:. Archived from 3956: 3941: 3925: 3904: 3891: 3872: 3850: 3810: 3792: 3765: 3749: 3743: 3735: 3713: 3697: 3679: 3658: 3646: 3610: 3600: 3575: 3552: 3520: 3492: 3460: 3458: 3435: 3417: 3407: 3390: 3381: 3379: 3368: 3349: 3337: 3328: 3326: 3319: 3305: 3295: 3281: 3279: 3265: 3256: 3247: 3218: 3181: 3179: 3173:. Archived from 3140: 3130: 3101: 3082: 3080: 3069: 3063:(October 2003). 3052: 3042: 3033:(5): 1411–1473. 3019: 3017: 3016: 3010: 3004:. Archived from 2987: 2960: 2920: 2919: 2893: 2887: 2886: 2875: 2869: 2868: 2836: 2830: 2829: 2797: 2791: 2790: 2773:Floridi, Luciano 2765: 2759: 2758: 2728: 2722: 2721: 2712:Zalta, Edward N. 2700: 2694: 2693: 2661: 2655: 2649: 2643: 2642: 2630: 2624: 2623: 2611: 2605: 2604: 2586: 2577: 2576: 2574: 2568:. Archived from 2567: 2559: 2553: 2551: 2527: 2521: 2520: 2518: 2495: 2477: 2471:(January 2007). 2465: 2459: 2453: 2447: 2440: 2434: 2427: 2421: 2418: 2412: 2409: 2403: 2402: 2355: 2349: 2348: 2346: 2345: 2339: 2333:. Archived from 2332: 2322: 2316: 2315: 2313: 2299: 2293: 2282: 2276: 2269: 2263: 2257: 2251: 2245: 2239: 2233: 2227: 2221: 2215: 2209: 2200: 2194: 2188: 2183:, p. 60 in 2178: 2172: 2161: 2155: 2144: 2138: 2135: 2129: 2123: 2117: 2110: 2104: 2101: 2095: 2092: 2086: 2079: 2073: 2062: 2056: 2050: 2041: 2040: 2038: 2037: 2031: 2025:. Archived from 2002: 1989: 1983: 1976: 1970: 1963: 1957: 1950: 1944: 1941: 1932: 1926: 1920: 1906: 1900: 1885: 1879: 1872: 1866: 1855: 1849: 1839: 1833: 1832: 1830: 1829: 1823: 1816: 1805: 1796: 1793: 1791: 1790: 1775: 1769: 1768: 1760: 1754: 1744: 1738: 1737: 1735: 1734: 1729: 1727: 1723: 1711: 1709: 1708: 1703: 1701: 1700: 1698: 1678: 1671: 1662: 1660: 1659: 1654: 1652: 1648: 1636: 1634: 1633: 1628: 1626: 1625: 1623: 1618: 1611: 1602: 1600: 1599: 1594: 1592: 1588: 1579: 1577: 1576: 1571: 1569: 1568: 1566: 1561: 1554: 1545: 1543: 1542: 1537: 1535: 1531: 1525: 1521: 1509: 1507: 1506: 1501: 1499: 1498: 1496: 1482: 1475: 1466: 1464: 1463: 1458: 1456: 1452: 1435: 1429: 1424: 1418: 1413: 1407: 1402: 1396: 1391: 1385: 1376: 1370: 1369: 1343: 1324:Soare, Robert I. 1320: 1311: 1310: 1308: 1307: 1292: 1281: 1275: 1265: 1259: 1258: 1256: 1255: 1238: 1232: 1223: 1187:Hypercomputation 1166:physical process 1152:Abstract machine 1103: 1096: 1092: 1089: 1083: 1056: 1055: 1048: 1008:computable reals 1006:, as opposed to 977:hypercomputation 972:B. Jack Copeland 910: 879:. It states: "A 670:register machine 644:". In the 1950s 472:Kleene proposes 382:beta normal form 319: 251:J. B. Rosser 246:Effective method 125:Jacques Herbrand 89:effective method 21: 6261: 6260: 6256: 6255: 6254: 6252: 6251: 6250: 6221: 6220: 6219: 6214: 6107: 6105:metamathematics 6098: 6068: 6063: 6052: 6045: 5990:Category theory 5980:Algebraic logic 5963: 5934:Lambda calculus 5872:Church encoding 5858: 5834:Truth predicate 5690: 5656:Complete theory 5579: 5448: 5444: 5440: 5435: 5427: 5147: and 5143: 5138: 5124: 5100:New Foundations 5068:axiom of choice 5051: 5013:Gödel numbering 4953: and 4945: 4849: 4734: 4684: 4665: 4614:Boolean algebra 4600: 4564:Equiconsistency 4529:Classical logic 4506: 4487:Halting problem 4475: and 4451: and 4439: and 4438: 4433:Theorems ( 4428: 4345: 4340: 4310: 4305: 4284: 4263: 4214:Simon B. Kochen 4147: 4131:Church encoding 4116:Lambda calculus 4104: 4099: 4005: 4000: 3993: 3971:10.2307/2268280 3954: 3902: 3888: 3869: 3831:10.2307/2269059 3807:Springer Verlag 3781: 3737: 3736: 3732: 3677:10.1.1.137.6939 3614:The Undecidable 3598:10.2307/1990131 3541:10.2307/2372027 3517: 3456: 3433:10.1.1.146.3017 3415: 3376:The Undecidable 3365: 3324: 3310:(81): 279–304. 3303: 3236:10.2307/2268810 3199:10.2307/2269326 3177: 3155:10.2307/2371045 3138: 3119:10.2307/1968337 3098: 3078: 3067: 3040:10.1.1.655.1186 3014: 3012: 3008: 2957: 2945:, eds. (1980). 2929: 2924: 2923: 2908: 2894: 2890: 2876: 2872: 2857: 2837: 2833: 2818: 2798: 2794: 2787: 2766: 2762: 2747: 2729: 2725: 2701: 2697: 2662: 2658: 2650: 2646: 2631: 2627: 2612: 2608: 2601: 2587: 2580: 2572: 2565: 2561: 2560: 2556: 2548: 2528: 2524: 2516: 2493:10.1.1.360.9796 2475: 2466: 2462: 2454: 2450: 2441: 2437: 2431:The Undecidable 2428: 2424: 2420:Gurevich 1988:2 2419: 2415: 2411:Kleene 1952:320 2410: 2406: 2391: 2372:Collected Works 2356: 2352: 2343: 2341: 2337: 2330: 2323: 2319: 2311: 2300: 2296: 2283: 2279: 2270: 2266: 2258: 2254: 2246: 2242: 2234: 2230: 2222: 2218: 2210: 2203: 2195: 2191: 2179: 2175: 2163:italics added, 2162: 2158: 2145: 2141: 2136: 2132: 2124: 2120: 2111: 2107: 2102: 2098: 2093: 2089: 2080: 2076: 2063: 2059: 2051: 2044: 2035: 2033: 2029: 2023: 2000: 1990: 1986: 1977: 1973: 1964: 1960: 1951: 1947: 1942: 1935: 1927: 1923: 1907: 1903: 1886: 1882: 1873: 1869: 1856: 1852: 1840: 1836: 1827: 1825: 1821: 1814: 1806: 1799: 1788: 1786: 1785:(11th ed.) 1778: 1776: 1772: 1767:(9th ed.). 1762: 1761: 1757: 1745: 1741: 1721: 1716: 1713: 1712: 1679: 1672: 1670: 1669: 1667: 1664: 1663: 1646: 1641: 1638: 1637: 1619: 1612: 1610: 1609: 1607: 1604: 1603: 1586: 1584: 1581: 1580: 1562: 1555: 1553: 1552: 1550: 1547: 1546: 1529: 1519: 1514: 1511: 1510: 1483: 1476: 1474: 1473: 1471: 1468: 1467: 1450: 1445: 1442: 1441: 1436: 1432: 1425: 1421: 1414: 1410: 1403: 1399: 1392: 1388: 1377: 1373: 1321: 1314: 1305: 1303: 1301: 1290: 1282: 1278: 1266: 1262: 1253: 1251: 1239: 1235: 1224: 1220: 1215: 1148: 1128:halting problem 1104: 1093: 1087: 1084: 1078:by introducing 1069: 1057: 1053: 1046: 993:digital physics 964: 908: 843: 827: 823: 813: 809: 805: 801: 797: 793: 789: 785: 781: 777: 773: 769: 765: 761: 754: 750: 746: 742: 718: 710: 706: 698: 693:Turing complete 686:pointer machine 666:counter machine 628: 625:in the system S 611: 599:Robert I. Soare 594: 538: 394:: In late 1936 338: 336:Circa 1930–1952 317: 298: 292: 248: 242: 226:digital physics 168:Church numerals 143:, and includes 85:natural numbers 69:Turing's thesis 61:Church's thesis 47:(also known as 35: 28: 23: 22: 15: 12: 11: 5: 6259: 6249: 6248: 6243: 6238: 6233: 6216: 6215: 6213: 6212: 6207: 6202: 6197: 6195:Satisfiability 6192: 6187: 6182: 6180:Interpretation 6177: 6172: 6167: 6162: 6157: 6152: 6151: 6150: 6140: 6135: 6130: 6125: 6118: 6112: 6109: 6108: 6097: 6096: 6089: 6082: 6074: 6065: 6064: 6050: 6047: 6046: 6044: 6043: 6038: 6033: 6028: 6023: 6022: 6021: 6011: 6006: 6001: 5992: 5987: 5982: 5977: 5975:Abstract logic 5971: 5969: 5965: 5964: 5962: 5961: 5956: 5954:Turing machine 5951: 5946: 5941: 5936: 5931: 5926: 5925: 5924: 5919: 5914: 5909: 5904: 5894: 5892:Computable set 5889: 5884: 5879: 5874: 5868: 5866: 5860: 5859: 5857: 5856: 5851: 5846: 5841: 5836: 5831: 5826: 5821: 5820: 5819: 5814: 5809: 5799: 5794: 5789: 5787:Satisfiability 5784: 5779: 5774: 5773: 5772: 5762: 5761: 5760: 5750: 5749: 5748: 5743: 5738: 5733: 5728: 5718: 5717: 5716: 5711: 5704:Interpretation 5700: 5698: 5692: 5691: 5689: 5688: 5683: 5678: 5673: 5668: 5658: 5653: 5652: 5651: 5650: 5649: 5639: 5634: 5624: 5619: 5614: 5609: 5604: 5599: 5593: 5591: 5585: 5584: 5581: 5580: 5578: 5577: 5569: 5568: 5567: 5566: 5561: 5560: 5559: 5554: 5549: 5529: 5528: 5527: 5525:minimal axioms 5522: 5511: 5510: 5509: 5498: 5497: 5496: 5491: 5486: 5481: 5476: 5471: 5458: 5456: 5437: 5436: 5434: 5433: 5432: 5431: 5419: 5414: 5413: 5412: 5407: 5402: 5397: 5387: 5382: 5377: 5372: 5371: 5370: 5365: 5355: 5354: 5353: 5348: 5343: 5338: 5328: 5323: 5322: 5321: 5316: 5311: 5301: 5300: 5299: 5294: 5289: 5284: 5279: 5274: 5264: 5259: 5254: 5249: 5248: 5247: 5242: 5237: 5232: 5222: 5217: 5215:Formation rule 5212: 5207: 5206: 5205: 5200: 5190: 5189: 5188: 5178: 5173: 5168: 5163: 5157: 5151: 5134:Formal systems 5130: 5129: 5126: 5125: 5123: 5122: 5117: 5112: 5107: 5102: 5097: 5092: 5087: 5082: 5077: 5076: 5075: 5070: 5059: 5057: 5053: 5052: 5050: 5049: 5048: 5047: 5037: 5032: 5031: 5030: 5023:Large cardinal 5020: 5015: 5010: 5005: 5000: 4986: 4985: 4984: 4979: 4974: 4959: 4957: 4947: 4946: 4944: 4943: 4942: 4941: 4936: 4931: 4921: 4916: 4911: 4906: 4901: 4896: 4891: 4886: 4881: 4876: 4871: 4866: 4860: 4858: 4851: 4850: 4848: 4847: 4846: 4845: 4840: 4835: 4830: 4825: 4820: 4812: 4811: 4810: 4805: 4795: 4790: 4788:Extensionality 4785: 4783:Ordinal number 4780: 4770: 4765: 4764: 4763: 4752: 4746: 4740: 4739: 4736: 4735: 4733: 4732: 4727: 4722: 4717: 4712: 4707: 4702: 4701: 4700: 4690: 4689: 4688: 4675: 4673: 4667: 4666: 4664: 4663: 4662: 4661: 4656: 4651: 4641: 4636: 4631: 4626: 4621: 4616: 4610: 4608: 4602: 4601: 4599: 4598: 4593: 4588: 4583: 4578: 4573: 4568: 4567: 4566: 4556: 4551: 4546: 4541: 4536: 4531: 4525: 4523: 4514: 4508: 4507: 4505: 4504: 4499: 4494: 4489: 4484: 4479: 4467:Cantor's 4465: 4460: 4455: 4445: 4443: 4430: 4429: 4427: 4426: 4421: 4416: 4411: 4406: 4401: 4396: 4391: 4386: 4381: 4376: 4371: 4366: 4365: 4364: 4353: 4351: 4347: 4346: 4339: 4338: 4331: 4324: 4316: 4307: 4306: 4304: 4303: 4298: 4292: 4290: 4286: 4285: 4283: 4282: 4277: 4271: 4269: 4265: 4264: 4262: 4261: 4256: 4254:Norman Shapiro 4251: 4246: 4241: 4236: 4231: 4226: 4221: 4219:Maurice L'Abbé 4216: 4211: 4206: 4201: 4196: 4191: 4189:William Easton 4186: 4181: 4176: 4171: 4166: 4161: 4155: 4153: 4149: 4148: 4146: 4145: 4140: 4135: 4134: 4133: 4128: 4123: 4112: 4110: 4106: 4105: 4098: 4097: 4090: 4083: 4075: 4069: 4068: 4055: 4038: 4022: 4004: 4003:External links 4001: 3999: 3998: 3996:on 2020-08-09. 3965:(4): 153–163. 3947: 3892: 3886: 3873: 3867: 3851: 3811: 3797:Pour-El, M. B. 3793: 3779: 3766: 3750: 3730: 3714: 3698: 3659: 3647: 3630: 3576: 3566:(2): 340–353. 3553: 3521: 3515: 3493: 3461: 3408: 3400:Gurevich, Yuri 3396: 3382: 3369: 3363: 3350: 3338: 3329: 3317:10.1.1.61.9759 3296: 3283: 3272:, ed. (1965). 3266: 3257: 3248: 3219: 3182: 3180:on 2020-02-27. 3149:(2): 345–363. 3131: 3113:(2): 346–366. 3102: 3096: 3083: 3061:Gurevich, Yuri 3057:Blass, Andreas 3053: 3020: 2985:10.1.1.74.7308 2978:(3): 113–116. 2961: 2955: 2943:Kunen, Kenneth 2930: 2928: 2925: 2922: 2921: 2906: 2888: 2870: 2855: 2841:Penrose, Roger 2831: 2816: 2802:Penrose, Roger 2792: 2785: 2760: 2745: 2735:, ed. (2002). 2723: 2706:(2017-11-10). 2695: 2676:(2): 203–219. 2656: 2654:, p. 287. 2644: 2625: 2606: 2599: 2578: 2575:on 2005-11-24. 2554: 2546: 2522: 2460: 2448: 2435: 2422: 2413: 2404: 2389: 2350: 2317: 2294: 2277: 2264: 2252: 2240: 2228: 2216: 2201: 2189: 2173: 2156: 2139: 2130: 2118: 2105: 2096: 2087: 2074: 2057: 2042: 2021: 2007:. De Gruyter. 1996:(2006-06-15). 1984: 1971: 1958: 1945: 1943:Sieg 1997:160. 1933: 1921: 1901: 1880: 1867: 1850: 1834: 1797: 1770: 1755: 1739: 1720: 1697: 1694: 1691: 1688: 1685: 1682: 1676: 1645: 1622: 1616: 1565: 1559: 1528: 1518: 1495: 1492: 1489: 1486: 1480: 1449: 1430: 1419: 1408: 1397: 1386: 1371: 1350:10.2307/420992 1341:10.1.1.35.5803 1334:(3): 284–321. 1312: 1299: 1276: 1271:and Church in 1260: 1233: 1217: 1216: 1214: 1211: 1210: 1209: 1204: 1199: 1194: 1189: 1184: 1179: 1174: 1169: 1159: 1154: 1147: 1144: 1135:hypercomputers 1120:Turing machine 1106: 1105: 1074:. Please help 1060: 1058: 1051: 1045: 1042: 1035: 1034: 1011: 996: 963: 960: 917:simultaneously 893:Umesh Vazirani 842: 839: 825: 821: 811: 807: 803: 799: 795: 791: 787: 783: 779: 775: 771: 767: 763: 759: 752: 748: 744: 740: 717: 714: 708: 704: 696: 626: 619:Turing machine 610: 607: 593: 590: 586: 585: 582: 579: 576: 573: 537: 534: 438:Turing machine 342:Stephen Kleene 337: 334: 294:Main article: 291: 288: 241: 238: 188: 187: 175: 156: 111:the notion of 93:Turing machine 26: 9: 6: 4: 3: 2: 6258: 6247: 6244: 6242: 6239: 6237: 6234: 6232: 6229: 6228: 6226: 6211: 6208: 6206: 6203: 6201: 6198: 6196: 6193: 6191: 6188: 6186: 6183: 6181: 6178: 6176: 6173: 6171: 6168: 6166: 6163: 6161: 6158: 6156: 6153: 6149: 6146: 6145: 6144: 6141: 6139: 6136: 6134: 6131: 6129: 6126: 6124: 6123: 6119: 6117: 6114: 6113: 6110: 6106: 6102: 6095: 6090: 6088: 6083: 6081: 6076: 6075: 6072: 6062: 6061: 6056: 6048: 6042: 6039: 6037: 6034: 6032: 6029: 6027: 6024: 6020: 6017: 6016: 6015: 6012: 6010: 6007: 6005: 6002: 6000: 5996: 5993: 5991: 5988: 5986: 5983: 5981: 5978: 5976: 5973: 5972: 5970: 5966: 5960: 5957: 5955: 5952: 5950: 5949:Recursive set 5947: 5945: 5942: 5940: 5937: 5935: 5932: 5930: 5927: 5923: 5920: 5918: 5915: 5913: 5910: 5908: 5905: 5903: 5900: 5899: 5898: 5895: 5893: 5890: 5888: 5885: 5883: 5880: 5878: 5875: 5873: 5870: 5869: 5867: 5865: 5861: 5855: 5852: 5850: 5847: 5845: 5842: 5840: 5837: 5835: 5832: 5830: 5827: 5825: 5822: 5818: 5815: 5813: 5810: 5808: 5805: 5804: 5803: 5800: 5798: 5795: 5793: 5790: 5788: 5785: 5783: 5780: 5778: 5775: 5771: 5768: 5767: 5766: 5763: 5759: 5758:of arithmetic 5756: 5755: 5754: 5751: 5747: 5744: 5742: 5739: 5737: 5734: 5732: 5729: 5727: 5724: 5723: 5722: 5719: 5715: 5712: 5710: 5707: 5706: 5705: 5702: 5701: 5699: 5697: 5693: 5687: 5684: 5682: 5679: 5677: 5674: 5672: 5669: 5666: 5665:from ZFC 5662: 5659: 5657: 5654: 5648: 5645: 5644: 5643: 5640: 5638: 5635: 5633: 5630: 5629: 5628: 5625: 5623: 5620: 5618: 5615: 5613: 5610: 5608: 5605: 5603: 5600: 5598: 5595: 5594: 5592: 5590: 5586: 5576: 5575: 5571: 5570: 5565: 5564:non-Euclidean 5562: 5558: 5555: 5553: 5550: 5548: 5547: 5543: 5542: 5540: 5537: 5536: 5534: 5530: 5526: 5523: 5521: 5518: 5517: 5516: 5512: 5508: 5505: 5504: 5503: 5499: 5495: 5492: 5490: 5487: 5485: 5482: 5480: 5477: 5475: 5472: 5470: 5467: 5466: 5464: 5460: 5459: 5457: 5452: 5446: 5441:Example 5438: 5430: 5425: 5424: 5423: 5420: 5418: 5415: 5411: 5408: 5406: 5403: 5401: 5398: 5396: 5393: 5392: 5391: 5388: 5386: 5383: 5381: 5378: 5376: 5373: 5369: 5366: 5364: 5361: 5360: 5359: 5356: 5352: 5349: 5347: 5344: 5342: 5339: 5337: 5334: 5333: 5332: 5329: 5327: 5324: 5320: 5317: 5315: 5312: 5310: 5307: 5306: 5305: 5302: 5298: 5295: 5293: 5290: 5288: 5285: 5283: 5280: 5278: 5275: 5273: 5270: 5269: 5268: 5265: 5263: 5260: 5258: 5255: 5253: 5250: 5246: 5243: 5241: 5238: 5236: 5233: 5231: 5228: 5227: 5226: 5223: 5221: 5218: 5216: 5213: 5211: 5208: 5204: 5201: 5199: 5198:by definition 5196: 5195: 5194: 5191: 5187: 5184: 5183: 5182: 5179: 5177: 5174: 5172: 5169: 5167: 5164: 5162: 5159: 5158: 5155: 5152: 5150: 5146: 5141: 5135: 5131: 5121: 5118: 5116: 5113: 5111: 5108: 5106: 5103: 5101: 5098: 5096: 5093: 5091: 5088: 5086: 5085:Kripke–Platek 5083: 5081: 5078: 5074: 5071: 5069: 5066: 5065: 5064: 5061: 5060: 5058: 5054: 5046: 5043: 5042: 5041: 5038: 5036: 5033: 5029: 5026: 5025: 5024: 5021: 5019: 5016: 5014: 5011: 5009: 5006: 5004: 5001: 4998: 4994: 4990: 4987: 4983: 4980: 4978: 4975: 4973: 4970: 4969: 4968: 4964: 4961: 4960: 4958: 4956: 4952: 4948: 4940: 4937: 4935: 4932: 4930: 4929:constructible 4927: 4926: 4925: 4922: 4920: 4917: 4915: 4912: 4910: 4907: 4905: 4902: 4900: 4897: 4895: 4892: 4890: 4887: 4885: 4882: 4880: 4877: 4875: 4872: 4870: 4867: 4865: 4862: 4861: 4859: 4857: 4852: 4844: 4841: 4839: 4836: 4834: 4831: 4829: 4826: 4824: 4821: 4819: 4816: 4815: 4813: 4809: 4806: 4804: 4801: 4800: 4799: 4796: 4794: 4791: 4789: 4786: 4784: 4781: 4779: 4775: 4771: 4769: 4766: 4762: 4759: 4758: 4757: 4754: 4753: 4750: 4747: 4745: 4741: 4731: 4728: 4726: 4723: 4721: 4718: 4716: 4713: 4711: 4708: 4706: 4703: 4699: 4696: 4695: 4694: 4691: 4687: 4682: 4681: 4680: 4677: 4676: 4674: 4672: 4668: 4660: 4657: 4655: 4652: 4650: 4647: 4646: 4645: 4642: 4640: 4637: 4635: 4632: 4630: 4627: 4625: 4622: 4620: 4617: 4615: 4612: 4611: 4609: 4607: 4606:Propositional 4603: 4597: 4594: 4592: 4589: 4587: 4584: 4582: 4579: 4577: 4574: 4572: 4569: 4565: 4562: 4561: 4560: 4557: 4555: 4552: 4550: 4547: 4545: 4542: 4540: 4537: 4535: 4534:Logical truth 4532: 4530: 4527: 4526: 4524: 4522: 4518: 4515: 4513: 4509: 4503: 4500: 4498: 4495: 4493: 4490: 4488: 4485: 4483: 4480: 4478: 4474: 4470: 4466: 4464: 4461: 4459: 4456: 4454: 4450: 4447: 4446: 4444: 4442: 4436: 4431: 4425: 4422: 4420: 4417: 4415: 4412: 4410: 4407: 4405: 4402: 4400: 4397: 4395: 4392: 4390: 4387: 4385: 4382: 4380: 4377: 4375: 4372: 4370: 4367: 4363: 4360: 4359: 4358: 4355: 4354: 4352: 4348: 4344: 4337: 4332: 4330: 4325: 4323: 4318: 4317: 4314: 4302: 4299: 4297: 4294: 4293: 4291: 4287: 4281: 4278: 4276: 4273: 4272: 4270: 4266: 4260: 4257: 4255: 4252: 4250: 4247: 4245: 4242: 4240: 4237: 4235: 4232: 4230: 4227: 4225: 4222: 4220: 4217: 4215: 4212: 4210: 4207: 4205: 4202: 4200: 4197: 4195: 4194:Alfred Foster 4192: 4190: 4187: 4185: 4182: 4180: 4179:William Boone 4177: 4175: 4172: 4170: 4169:Peter Andrews 4167: 4165: 4162: 4160: 4157: 4156: 4154: 4150: 4144: 4141: 4139: 4136: 4132: 4129: 4127: 4124: 4122: 4119: 4118: 4117: 4114: 4113: 4111: 4109:Notable ideas 4107: 4103: 4102:Alonzo Church 4096: 4091: 4089: 4084: 4082: 4077: 4076: 4073: 4066: 4065: 4060: 4059:special issue 4056: 4053:(3): 103–105. 4052: 4048: 4044: 4039: 4036: 4035: 4030: 4026: 4023: 4020: 4019: 4014: 4010: 4007: 4006: 3992: 3988: 3984: 3980: 3976: 3972: 3968: 3964: 3960: 3953: 3948: 3945: 3939: 3935: 3931: 3924: 3920: 3916: 3912: 3908: 3901: 3897: 3896:Turing, A. M. 3893: 3889: 3883: 3879: 3874: 3870: 3864: 3861: 3857: 3852: 3848: 3844: 3840: 3836: 3832: 3828: 3824: 3820: 3816: 3815:Rosser, J. B. 3812: 3808: 3804: 3803: 3798: 3794: 3790: 3786: 3782: 3776: 3772: 3767: 3763: 3759: 3755: 3754:Markov, A. A. 3751: 3747: 3741: 3733: 3727: 3723: 3719: 3715: 3711: 3707: 3703: 3699: 3695: 3691: 3687: 3683: 3678: 3673: 3670:(11): 32–37. 3669: 3665: 3660: 3656: 3652: 3651:Knuth, Donald 3648: 3644: 3640: 3636: 3631: 3628: 3627:Church thesis 3624: 3620: 3616: 3615: 3611:Reprinted in 3608: 3604: 3599: 3594: 3590: 3586: 3582: 3577: 3573: 3569: 3565: 3561: 3560: 3554: 3550: 3546: 3542: 3538: 3534: 3530: 3526: 3522: 3518: 3516:0-465-02656-7 3512: 3508: 3504: 3503: 3498: 3494: 3490: 3486: 3482: 3478: 3474: 3471:(in French). 3470: 3466: 3462: 3455: 3451: 3447: 3443: 3439: 3434: 3429: 3426:(1): 77–111. 3425: 3421: 3414: 3409: 3405: 3401: 3397: 3394: 3393:Kleene (1952) 3388: 3383: 3378: 3377: 3370: 3366: 3360: 3356: 3351: 3347: 3343: 3339: 3335: 3330: 3323: 3318: 3313: 3309: 3302: 3297: 3293: 3289: 3284: 3278: 3277: 3271: 3270:Davis, Martin 3267: 3263: 3258: 3254: 3249: 3245: 3241: 3237: 3233: 3229: 3225: 3220: 3216: 3212: 3208: 3204: 3200: 3196: 3192: 3188: 3183: 3176: 3172: 3168: 3164: 3160: 3156: 3152: 3148: 3144: 3137: 3132: 3128: 3124: 3120: 3116: 3112: 3108: 3103: 3099: 3093: 3089: 3084: 3077: 3073: 3066: 3062: 3058: 3054: 3050: 3046: 3041: 3036: 3032: 3028: 3027: 3021: 3011:on 2017-07-06 3007: 3003: 2999: 2995: 2991: 2986: 2981: 2977: 2973: 2972: 2967: 2962: 2958: 2952: 2948: 2944: 2940: 2939:Keisler, H.J. 2936: 2932: 2931: 2917: 2913: 2909: 2903: 2899: 2892: 2884: 2880: 2874: 2866: 2862: 2858: 2852: 2848: 2847: 2842: 2835: 2827: 2823: 2819: 2813: 2809: 2808: 2803: 2796: 2788: 2782: 2778: 2774: 2770: 2764: 2756: 2752: 2748: 2742: 2738: 2734: 2727: 2719: 2718: 2713: 2709: 2705: 2699: 2691: 2687: 2683: 2679: 2675: 2671: 2667: 2660: 2653: 2648: 2640: 2636: 2629: 2621: 2617: 2610: 2602: 2596: 2592: 2585: 2583: 2571: 2564: 2558: 2549: 2543: 2539: 2535: 2534: 2526: 2515: 2511: 2507: 2503: 2499: 2494: 2489: 2486:(1): 97–120. 2485: 2481: 2474: 2470: 2464: 2457: 2452: 2445: 2439: 2432: 2426: 2417: 2408: 2400: 2396: 2392: 2386: 2382: 2378: 2374: 2373: 2368: 2364: 2360: 2354: 2340:on 2016-03-04 2336: 2329: 2321: 2310: 2304: 2298: 2291: 2287: 2286:Gandy machine 2281: 2274: 2268: 2261: 2256: 2249: 2244: 2237: 2232: 2225: 2220: 2213: 2208: 2206: 2198: 2193: 2186: 2182: 2177: 2170: 2166: 2160: 2153: 2149: 2143: 2134: 2127: 2122: 2115: 2109: 2100: 2091: 2084: 2081:Post 1936 in 2078: 2071: 2067: 2061: 2054: 2049: 2047: 2032:on 2015-12-17 2028: 2024: 2018: 2014: 2010: 2006: 1999: 1995: 1994:Shagrir, Oron 1988: 1981: 1975: 1968: 1962: 1955: 1949: 1940: 1938: 1930: 1925: 1918: 1914: 1911: 1905: 1899: 1895: 1890: 1884: 1877: 1871: 1864: 1863:3-540-05843-5 1860: 1854: 1847: 1843: 1838: 1824:on 2012-10-23 1820: 1813: 1812: 1804: 1802: 1784: 1783: 1779:"effective". 1774: 1766: 1763:"effective". 1759: 1752: 1748: 1743: 1718: 1695: 1692: 1689: 1686: 1683: 1680: 1643: 1620: 1563: 1526: 1516: 1493: 1490: 1487: 1484: 1447: 1439: 1434: 1428: 1423: 1417: 1412: 1406: 1401: 1395: 1390: 1382: 1375: 1367: 1363: 1359: 1355: 1351: 1347: 1342: 1337: 1333: 1329: 1325: 1319: 1317: 1302: 1300:9780198250791 1296: 1289: 1288: 1280: 1274: 1270: 1264: 1249: 1248: 1244:(June 2012). 1243: 1237: 1231: 1227: 1222: 1218: 1208: 1205: 1203: 1200: 1198: 1195: 1193: 1190: 1188: 1185: 1183: 1180: 1178: 1175: 1173: 1170: 1167: 1163: 1160: 1158: 1155: 1153: 1150: 1149: 1143: 1140: 1136: 1131: 1129: 1125: 1121: 1117: 1113: 1102: 1099: 1091: 1088:November 2017 1081: 1077: 1073: 1067: 1066: 1065:single source 1061:This section 1059: 1050: 1049: 1041: 1038: 1032: 1031:Roger Penrose 1028: 1024: 1020: 1016: 1015:hypercomputer 1012: 1009: 1005: 1001: 1000:hypercomputer 997: 994: 990: 986: 982: 981: 980: 978: 973: 969: 959: 957: 951: 949: 945: 941: 937: 933: 929: 924: 922: 918: 914: 906: 902: 898: 894: 890: 886: 882: 878: 874: 870: 866: 862: 857: 855: 850: 848: 838: 833: 831: 819: 815: 756: 737: 735: 734:recursive set 731: 725: 722: 712: 700: 694: 689: 687: 683: 679: 675: 671: 667: 663: 659: 658:Marvin Minsky 655: 651: 647: 643: 640: 636: 632: 624: 620: 616: 606: 604: 600: 589: 583: 580: 577: 574: 571: 570: 569: 567: 562: 561:Wilfried Sieg 557: 555: 551: 547: 543: 533: 528: 522: 517: 515: 509: 507: 503: 498: 493: 489: 488: 482: 478: 476: 475: 468: 466: 460: 457: 455: 451: 447: 442: 439: 435: 431: 429: 424: 422: 418: 412: 411:verification. 407: 405: 401: 397: 393: 389: 385: 383: 379: 375: 371: 366: 364: 358: 353: 351: 347: 343: 333: 331: 327: 323: 315: 314:Alonzo Church 311: 307: 306:David Hilbert 303: 297: 286: 281: 279: 273: 268: 266: 265: 258: 256: 252: 247: 237: 235: 231: 227: 223: 219: 215: 210: 208: 203: 201: 197: 193: 185: 180: 176: 173: 169: 165: 161: 160:Alonzo Church 157: 154: 150: 146: 142: 138: 134: 130: 126: 122: 118: 117: 116: 114: 113:computability 110: 106: 102: 98: 97:Alonzo Church 94: 90: 86: 82: 78: 74: 70: 66: 62: 58: 54: 50: 46: 42: 37: 33: 19: 6200:Independence 6175:Decidability 6170:Completeness 6127: 6120: 6051: 5876: 5849:Ultraproduct 5696:Model theory 5661:Independence 5597:Formal proof 5589:Proof theory 5572: 5545: 5502:real numbers 5474:second-order 5385:Substitution 5262:Metalanguage 5203:conservative 5176:Axiom schema 5120:Constructive 5090:Morse–Kelley 5056:Set theories 5035:Aleph number 5028:inaccessible 4934:Grothendieck 4818:intersection 4705:Higher-order 4693:Second-order 4639:Truth tables 4596:Venn diagram 4379:Formal proof 4268:Institutions 4184:Martin Davis 4125: 4062: 4050: 4046: 4032: 4016: 3991:the original 3962: 3958: 3929: 3906: 3880:. Springer. 3877: 3855: 3825:(2): 53–60. 3822: 3818: 3801: 3770: 3761: 3757: 3721: 3718:Manna, Zohar 3709: 3667: 3663: 3654: 3634: 3613: 3591:(1): 41–73. 3588: 3584: 3563: 3557: 3532: 3528: 3501: 3472: 3468: 3423: 3419: 3406:(35): 71–82. 3403: 3386: 3375: 3354: 3345: 3342:Gandy, Robin 3333: 3307: 3292:A. K. Peters 3287: 3275: 3261: 3252: 3230:(1): 42–43. 3227: 3223: 3193:(1): 40–41. 3190: 3186: 3175:the original 3146: 3142: 3110: 3106: 3090:. Springer. 3087: 3071: 3030: 3024: 3013:. Retrieved 3006:the original 2975: 2969: 2946: 2935:Barwise, Jon 2897: 2891: 2882: 2873: 2845: 2834: 2806: 2795: 2776: 2763: 2736: 2726: 2715: 2698: 2673: 2669: 2659: 2647: 2634: 2628: 2622:. p. 5. 2615: 2609: 2590: 2570:the original 2557: 2532: 2525: 2483: 2479: 2463: 2451: 2438: 2430: 2425: 2416: 2407: 2371: 2353: 2342:. Retrieved 2335:the original 2320: 2297: 2280: 2267: 2255: 2243: 2231: 2219: 2192: 2176: 2159: 2142: 2133: 2121: 2108: 2099: 2090: 2077: 2065: 2060: 2053:Turing 1937a 2034:. Retrieved 2027:the original 2004: 1987: 1974: 1961: 1948: 1924: 1912: 1910:Church 1936a 1904: 1893: 1888: 1883: 1875: 1870: 1853: 1845: 1837: 1826:. Retrieved 1819:the original 1810: 1787:. Retrieved 1781: 1773: 1764: 1758: 1742: 1438:Turing 1937b 1433: 1422: 1416:Turing 1937a 1411: 1400: 1394:Church 1936a 1389: 1380: 1374: 1331: 1327: 1304:. Retrieved 1286: 1279: 1263: 1252:. Retrieved 1246: 1236: 1226:Robert Soare 1221: 1182:Decidability 1132: 1123: 1115: 1109: 1094: 1085: 1062: 1039: 1036: 1004:real numbers 965: 952: 943: 925: 916: 904: 888: 872: 868: 864: 860: 858: 851: 846: 844: 835: 832:, recursive. 829: 817: 816: 757: 738: 727: 723: 719: 702: 690: 650:Martin Davis 641: 638: 622: 612: 595: 587: 565: 558: 545: 539: 530: 525: 520: 513: 511: 505: 501: 500: 496: 491: 486: 484: 480: 473: 471: 470: 464: 462: 458: 453: 449: 445: 443: 433: 432: 425: 414: 409: 391: 390: 386: 367: 360: 355: 350:axiomatizing 349: 339: 329: 325: 321: 299: 283: 277: 275: 272:definitions. 270: 262: 259: 249: 211: 206: 204: 195: 189: 172:λ-computable 141:minimization 68: 64: 60: 56: 52: 48: 44: 38: 36: 6236:Alan Turing 6190:Metatheorem 6148:of geometry 6133:Consistency 5959:Type theory 5907:undecidable 5839:Truth value 5726:equivalence 5405:non-logical 5018:Enumeration 5008:Isomorphism 4955:cardinality 4939:Von Neumann 4904:Ultrafilter 4869:Uncountable 4803:equivalence 4720:Quantifiers 4710:Fixed-point 4679:First-order 4559:Consistency 4544:Proposition 4521:Traditional 4492:Lindström's 4482:Compactness 4424:Type theory 4369:Cardinality 4301:A. C. Croom 4224:Gary R. Mar 4199:Leon Henkin 4159:Alan Turing 3942:(See also: 3764:(15): 1–14. 3702:Lewis, H.R. 3623:Kleene 1952 3619:Kleene 1952 3507:Basic Books 2971:SIGACT News 2456:Gabbay 2001 2442:Horsten in 2359:Gödel, Kurt 2224:Kleene 1952 2212:Kleene 1952 2197:Kleene 1952 2181:Kleene 1943 2165:Rosser 1939 2126:Church 1937 1929:Dawson 1997 1842:Gandy (1980 1747:Rosser 1939 1427:Kleene 1936 1405:Kleene 1936 1112:Busy Beaver 946:states: "A 770:such that n 552:(including 542:Robin Gandy 465:definitions 450:definitions 421:natural law 396:Alan Turing 179:Alan Turing 153:projections 133:composition 101:Alan Turing 6225:Categories 5770:elementary 5463:arithmetic 5331:Quantifier 5309:functional 5181:Expression 4899:Transitive 4843:identities 4828:complement 4761:hereditary 4744:Set theory 4249:Dana Scott 4027:entry by 4011:entry by 3944:Davis 1965 3898:(1937a) , 3015:2017-10-24 2927:References 2379:. p. 2344:2014-07-27 2260:Gandy 1980 2248:Gandy 1980 2236:Gandy 1980 2185:Davis 1965 2169:Davis 1965 2152:Davis 1965 2148:Davis 1965 2114:Davis 1965 2083:Davis 1965 2070:Davis 1965 2036:2016-02-08 1980:Davis 1965 1967:Davis 1965 1954:Davis 1965 1917:Davis 1965 1828:2012-06-23 1789:2014-07-26 1751:Davis 1965 1724:-definable 1649:-recursive 1532:-definable 1453:-definable 1306:2021-12-06 1269:Max Newman 1254:2021-12-05 1027:John Lucas 841:Variations 633:1936, and 631:Kurt Gödel 623:reckonable 617:, and the 615:λ-calculus 512:Thesis I. 485:Thesis I. 463:All three 428:Kurt Gödel 330:a proposal 322:definition 244:See also: 164:λ-calculus 151:, and all 121:Kurt Gödel 6165:Soundness 6101:Metalogic 6041:Supertask 5944:Recursion 5902:decidable 5736:saturated 5714:of models 5637:deductive 5632:axiomatic 5552:Hilbert's 5539:Euclidean 5520:canonical 5443:axiomatic 5375:Signature 5304:Predicate 5193:Extension 5115:Ackermann 5040:Operation 4919:Universal 4909:Recursive 4884:Singleton 4879:Inhabited 4864:Countable 4854:Types of 4838:power set 4808:partition 4725:Predicate 4671:Predicate 4586:Syllogism 4576:Soundness 4549:Inference 4539:Tautology 4441:paradoxes 3789:909679288 3740:cite book 3724:. Dover. 3720:(2003) . 3672:CiteSeerX 3489:116636410 3428:CiteSeerX 3391:Cited by 3312:CiteSeerX 3035:CiteSeerX 2980:CiteSeerX 2916:990755791 2865:456785846 2826:456785846 2755:610918145 2488:CiteSeerX 2399:928791907 2361:(1995) . 2324:See also 2226::382, 536 1777:See also 1719:λ 1675:⟹ 1644:μ 1615:⟹ 1558:⟹ 1517:λ 1479:⟹ 1448:λ 1336:CiteSeerX 1213:Footnotes 1080:citations 1072:talk page 865:classical 782: = n 762: = n 639:canonical 635:Emil Post 404:Emil Post 158:In 1936, 149:successor 137:recursion 119:In 1933, 109:formalize 6026:Logicism 6019:timeline 5995:Concrete 5854:Validity 5824:T-schema 5817:Kripke's 5812:Tarski's 5807:semantic 5797:Strength 5746:submodel 5741:spectrum 5709:function 5557:Tarski's 5546:Elements 5533:geometry 5489:Robinson 5410:variable 5395:function 5368:spectrum 5358:Sentence 5314:variable 5257:Language 5210:Relation 5171:Automata 5161:Alphabet 5145:language 4999:-jection 4977:codomain 4963:Function 4924:Universe 4894:Infinite 4798:Relation 4581:Validity 4571:Argument 4469:theorem, 4152:Students 3946::115ff.) 3847:39499392 3708:(1998). 3694:29843806 3653:(1973). 3454:Archived 3322:Archived 3215:42323521 3171:14181275 3076:Archived 3002:13566703 2881:(1989). 2690:32116031 2620:Elsevier 2514:Archived 2480:Synthese 2292::390ff.. 2116::263ff.. 1969::105ff.. 1384:544–546. 1146:See also 674:computer 646:Hao Wang 566:computor 474:Thesis I 207:informal 190:Church, 81:function 5968:Related 5765:Diagram 5663: ( 5642:Hilbert 5627:Systems 5622:Theorem 5500:of the 5445:systems 5225:Formula 5220:Grammar 5136: ( 5080:General 4793:Forcing 4778:Element 4698:Monadic 4473:paradox 4414:Theorem 4350:General 4031:in the 4015:in the 3987:2317046 3979:2268280 3839:2269059 3607:1990131 3549:2372027 3475:: 1–8. 3450:2031696 3244:2268810 3207:2269326 3163:2371045 3127:1968337 2775:(ed.). 2714:(ed.). 2369:(ed.). 2238::123ff. 1366:5894394 871:or the 778:, put m 642:systems 546:machine 290:History 253: ( 123:, with 83:on the 71:) is a 51:, the 5731:finite 5494:Skolem 5447: 5422:Theory 5390:Symbol 5380:String 5363:atomic 5240:ground 5235:closed 5230:atomic 5186:ground 5149:syntax 5045:binary 4972:domain 4889:Finite 4654:finite 4512:Logics 4471: 4419:Theory 4289:Family 3985: 3977: 3921: 3884: 3865: 3845: 3837: 3787: 3777: 3728: 3692: 3674: 3643:523942 3641: 3605: 3547: 3513: 3487: 3448: 3430: 3361: 3314: 3242: 3213: 3205: 3169: 3161: 3125: 3094: 3074:(81). 3037: 3000: 2982: 2953: 2914: 2904: 2863: 2853: 2824: 2814: 2783: 2753: 2743: 2688: 2597: 2544: 2510:494161 2508: 2490: 2397: 2387: 2019: 1861: 1364: 1358:420992 1356: 1338: 1297: 810:< m 790:> m 774:> m 755:, ... 662:Lambek 454:theses 419:to a " 380:has a 363:Rosser 357:basis. 192:Kleene 139:, and 73:thesis 67:, and 55:, the 43:, the 5721:Model 5469:Peano 5326:Proof 5166:Arity 5095:Naive 4982:image 4914:Fuzzy 4874:Empty 4823:union 4768:Class 4409:Model 4399:Lemma 4357:Axiom 3994:(PDF) 3983:S2CID 3975:JSTOR 3955:(PDF) 3923:73712 3919:S2CID 3903:(PDF) 3843:S2CID 3835:JSTOR 3690:S2CID 3603:JSTOR 3545:JSTOR 3485:S2CID 3457:(PDF) 3446:S2CID 3416:(PDF) 3325:(PDF) 3304:(PDF) 3240:JSTOR 3211:S2CID 3203:JSTOR 3178:(PDF) 3167:S2CID 3159:JSTOR 3139:(PDF) 3123:JSTOR 3079:(PDF) 3068:(PDF) 2998:S2CID 2710:. In 2686:S2CID 2573:(PDF) 2566:(PDF) 2517:(PDF) 2506:S2CID 2476:(PDF) 2446::256. 2365:. In 2338:(PDF) 2331:(PDF) 2312:(PDF) 2214::376. 2199::300. 2171::226. 2154::160. 2072::289. 2030:(PDF) 2001:(PDF) 1822:(PDF) 1815:(PDF) 1753::225. 1362:S2CID 1354:JSTOR 1291:(PDF) 1122:with 818:Claim 629:" of 234:below 232:(see 200:below 6103:and 5844:Type 5647:list 5451:list 5428:list 5417:Term 5351:rank 5245:open 5139:list 4951:Maps 4856:sets 4715:Free 4685:list 4435:list 4362:list 3926:and 3882:ISBN 3863:ISBN 3785:OCLC 3775:ISBN 3746:link 3726:ISBN 3639:OCLC 3511:ISBN 3359:ISBN 3092:ISBN 3009:(PS) 2951:ISBN 2912:OCLC 2902:ISBN 2861:OCLC 2851:ISBN 2822:OCLC 2812:ISBN 2799:cf. 2781:ISBN 2751:OCLC 2741:ISBN 2639:STOC 2595:ISBN 2542:ISBN 2458::284 2395:OCLC 2385:ISBN 2262::126 2250::135 2017:ISBN 1982::40. 1956::44. 1931::99. 1919::89. 1859:ISBN 1295:ISBN 1029:and 913:STOC 680:and 648:and 527:XXX. 308:and 255:1939 224:and 145:zero 5531:of 5513:of 5461:of 4993:Sur 4967:Map 4774:Ur- 4756:Set 3967:doi 3934:doi 3911:doi 3827:doi 3682:doi 3593:doi 3568:doi 3537:doi 3477:doi 3473:166 3438:doi 3232:doi 3195:doi 3151:doi 3115:doi 3045:doi 2990:doi 2678:doi 2498:doi 2484:154 2381:168 2167:in 2068:at 2009:doi 1915:in 1896:at 1749:in 1621:161 1564:160 1346:doi 932:BPP 928:BQP 926:If 897:BPP 812:i+1 802:, m 798:, m 751:, n 747:, n 743:, n 699:": 656:). 446:his 318:Was 304:of 236:). 202:). 39:In 6227:: 5917:NP 5541:: 5535:: 5465:: 5142:), 4997:Bi 4989:In 4057:A 4049:. 4045:. 3981:. 3973:. 3961:. 3957:. 3917:, 3905:, 3841:. 3833:. 3821:. 3805:. 3783:. 3760:. 3742:}} 3738:{{ 3704:; 3688:. 3680:. 3668:48 3666:. 3629:). 3601:. 3589:53 3587:. 3583:. 3562:. 3543:. 3533:57 3531:. 3483:. 3452:. 3444:. 3436:. 3422:. 3418:. 3320:. 3306:. 3238:. 3226:. 3209:. 3201:. 3189:. 3165:. 3157:. 3147:58 3145:. 3141:. 3121:. 3111:33 3109:. 3070:. 3059:; 3043:. 3031:26 3029:. 2996:. 2988:. 2976:36 2974:. 2968:. 2941:; 2937:; 2910:. 2859:. 2820:. 2749:. 2684:. 2674:21 2672:. 2668:. 2637:. 2618:. 2581:^ 2540:. 2536:. 2512:. 2504:. 2496:. 2482:. 2478:. 2393:. 2383:. 2204:^ 2045:^ 2015:. 2003:. 1936:^ 1800:^ 1360:. 1352:. 1344:. 1330:. 1315:^ 1228:, 970:. 958:. 867:) 814:. 736:. 605:. 456:. 384:. 147:, 135:, 115:: 63:, 59:, 6093:e 6086:t 6079:v 5997:/ 5912:P 5667:) 5453:) 5449:( 5346:∀ 5341:! 5336:∃ 5297:= 5292:↔ 5287:→ 5282:∧ 5277:∨ 5272:¬ 4995:/ 4991:/ 4965:/ 4776:) 4772:( 4659:∞ 4649:3 4437:) 4335:e 4328:t 4321:v 4094:e 4087:t 4080:v 4051:1 4021:. 3969:: 3963:2 3940:. 3936:: 3913:: 3890:. 3871:. 3849:. 3829:: 3823:4 3809:. 3791:. 3762:2 3748:) 3734:. 3696:. 3684:: 3645:. 3609:. 3595:: 3574:. 3570:: 3564:2 3551:. 3539:: 3519:. 3491:. 3479:: 3440:: 3424:1 3395:. 3367:. 3294:. 3246:. 3234:: 3228:2 3217:. 3197:: 3191:1 3153:: 3129:. 3117:: 3100:. 3051:. 3047:: 3018:. 2992:: 2959:. 2918:. 2867:. 2828:. 2789:. 2757:. 2720:. 2692:. 2680:: 2641:. 2603:. 2550:. 2500:: 2401:. 2347:. 2314:. 2128:. 2055:. 2039:. 2011:: 1831:. 1792:, 1696:e 1693:n 1690:e 1687:e 1684:l 1681:K 1527:K 1522:- 1494:v 1491:i 1488:r 1485:t 1368:. 1348:: 1332:2 1309:. 1257:. 1124:n 1116:n 1101:) 1095:( 1090:) 1086:( 1068:. 995:. 909:" 901:P 826:i 822:i 808:i 804:2 800:1 796:0 792:1 788:2 784:k 780:1 776:0 772:k 768:k 764:0 760:0 753:3 749:2 745:1 741:0 709:1 705:i 697:1 627:1 516:. 155:. 34:. 20:)

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Knowledge

Church–Turing thesis

Index