Talk:Entscheidungsproblem

886:(1) Constant function: CLR_A : : clear a hole called "1" of pebbles (aka hole "A") (2) Successor function: INC_A : add a single pebble to the hole labelled "1" (3) Identity function: ldAi, MOVrAi, CPYrAi: From the entire sequence of holes, find the hole, the number of which, is the quantity of pebbles in hole "2" (aka P) Use this procedure. remove all the pebbles from the hole labelled "2" (aka "P" for "pointer"). Start with hole 1 and put the pebbles in one-to-one correspondence with the holes until the pebbles run out: the hole where this happens is the hole of interest. (4) Substitution function: stAi In a manner similar to ldAi, find the hole of interest. Then remove all the holes from pointer-hole #2 (aka P) [either throw them into the pointed-to hole, or count out a similar number from the source S and throw 1152:

numbers to test for “circularity”, starting with “1” and continuing in numerical sequence, (i.e. in modern terms, each number represents a programs to be tested for “unending loops”). He then proves, first, that his universal enumerator/circle-deciding program H = U+D cannot decide whether H (i.e. itself) will or will not end in "a circle" (a never-ending loop). (By assumption: it must never loop, but rather it must go on testing forever. But when it tests itself, its actions are to start with 1 and proceed, number by number, to test each one – by definition this is a circle and violates the premise.) By use of this result, Turing then proves that no universal deciding program E can tell if any machine M "ever prints a given symbol (0 say)." Armed with this second proof he goes on to prove that:

578:. Godel trashes Finsler's argument in an unpublished letter (1970). .others (Hilbert and Bernays 19xx) were (somewhat) accepting, goobut when Finsler attempted to claim priority Godel brushed him off [insert quote here from Breger 1992: "Finsler strives for truth in mathematics, not just for formal derivability" (p. 255) "According to Hilbert and Bernays (1939...; compare also Bernays 1935...) the central idea of Finsler's paper is remarkable and should be given due respect, but his contention is merely analogous to Godel's theorem, and his reasoning cannot be put to use by proof theory" (p. 257)], his proof lapsing into oblivion. van Heijenoort 1967 and more recent commentators yield conflicting/confusing opinions. ] 1200:

that the “Turing simplifications” may work for “analysis of process” but doubts that they will work for “analysis of proof” (cf footnote 6:343) and he wonders if “Turing’s finite numbers of mental states hypothesis” will hold up under scrutiny and whether or not “an equally persuasive analysis be found for all humanly possible modes of symbolization” (footnote 9:344). Post hoped for a “reversal of the entire axiomatic trend of the late nineteenth and early twentieth centuries, with a return to meaning and truth. Postulational thinking will then remain as but one phase of mathematical thinking” (Post 1965:345).

1494:, A K Peters, Ltd. Wellesley MA). Another reason is: all three proofs of the 1930's that were so significant -- Church 1936, Turing 1936-7, Kleene 1943 -- all revolve the unsolvable "decision problem", and they all use similar techniques to arrive at their proofs (conversion of proofs to numbers, their subsequent enumeration (listing) and Cantor's diagonal argument). Godel 1931 (his theorem VI, the so-called first incompleteness theorem) is also a proof of undecidability and follows a similar tack. Is any of this of use to philosphers constructing arguments? I dunno, but I would hope so. wvbailey 1078:”7. The Entscheidungsproblem of a system of symbolic logic is here understood the problem to find an effective method by which, given any expression Q in the notation of the system, it can be determined whether or not Q is provable in the system. Hilbert and Ackerman (loc. cit) understand the Entscheidungsproblem of the engere Funktionenkalkül in a slightly different sense. But the two senses are equivalent in view of the proof by Kurt Gödel and the completeness of the engere Funktionenkalkül . . . .” (brackets in the original, Church 1936 :’’The Undecidable’’ p. 113) 501:"The Entscheidungsproblem is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or satisfiability . . . The Entscheidungsproblem must be considered the main problem of mathematical logic. . .. The solution of the Entscheidungsproblem is of fundamental significance for the theory of all domains whose propositions could be developed on the basis of a finite number of axioms" (this translation, and the original text in German, appears in Dershowitz and Gurevich 2007:1-2). 95: 85: 64: 546:"...in the sense that every valid formula is provable" (van Heijenoort, p. 582). Given that encouraging fact, could there be a generalized "calculational procedure" that would tell us whether a conclusion can be derived from its premises? Davis calls such calculational procedures "algorithms". The Entscheidungsproblem would be an algorithm as well. "In principle, an algorithm for Entscheidungsproblem would have reduced all human deductive reasoning to brute calculation" (Davis 2000:146). 31: 294: 1125:

instructions) very similar to Turing's a-machine. Moreover in a footnote he took a poke at Church's notion of a "definition" for effective calculability: Church’s definition would “mask this identification under a definition the fact that a fundamental discovery in the limitations of the mathematicizing power of Homo Sapiens has been made and blinds us to the need of its continual verification.” (Footnote 8, Post 1936:291 in

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unbounded size) labelled 1, an unlimited source S of pebbles to be put into the holes, and a mechanism that is capable of following a set of instructions in a TABLE with respect to the holes and pebbles. The holes in the ground are the "variables" and will be of only type I. The numbers are the collections of pebbles in the ground. The set of instructions that the mechanism follows will be:

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engere Funktionenkalkül, whether it is decidable in that system; (ii) “ . . . a procedure for determining universal validity, depends upon the non-constructively proved theorem of Gödel that every universally valid expression is deducible in the engere Funktionenkalkül, as well as on the assumption of the converse of this, that every deducible expression is universally valid.”

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that by use/adoption of “recursiveness” as a criterion of solvability, “the unsolvability in question as in the case of the famous problems of antiquity, becomes merely unsolvability by a given set of instruments. . . . The fundamental new thing is that for the combinatory problems the given set of instruments is in effect the only humanly possible set.” (Post 1965:340)

484:), “there exist undecidable sentences ” (Gödel in Davis 1965:6, in van Heijenoort 1967:596). Because of this, “the consistency of P is unprovable in P, provided P is consistent”(Gödel’s theorem IX, Davis 1965:36). While Gödel’s proof would display the tools necessary to resolve the Entscheidungsproblem, he himself would not answer it for reasons described below. 960:”For any such system Σ Gödel constructs a formula φ which is satisfiable, but for which this fact cannot be proved in Σ. As a consequence, given any proposed algorithm α for the Entscheidungsproblem and any system Σ, either it cannot be proved in Σ that α always gives an answer, or it cannot be proved in Σ that its answer is always correct.” (Gandy 1994:63) 196: 169: 460:. Problem of language addressed only cursorily by Finsler (1926) but in great depth (i.e. central argument) by Gödel (1931). "Both Godel's and Finsler's arguments, with their similarities and differences, skirt the Richard paradox without falling into it; both exploit Richard's argument to obtain new and valid conclusions" (van heijenoort 1967:439). 557:” . . . it seemed clear to Hilbert that with the solution of this problem, the Entscheidungsproblem, that it should be possible in principle to settle all mathematical questions in a purely mechanical manner. Hence, given unsolvable problems at all, if Hilbert was correct, then the Entscheidungsproblem itself should be unsolvable" (Davis 1967:108). 759:"In other words, the know-how of a human being is necessary -- a know-how which is not formalized in the axioms...It seems to me that Hilbert...was also aware of this fundamental problem of an axiomatic approach.... Evidentally the know-how which is necessary to understand a certain piece of written formal logic is usually hidden" ((cf Berger 206: 1156:”The Hilbert Entscheidungsproblem can have no solution . . . there can be no general process for determining whether a given formula U of the functional calculus K is provable, i.e. that there can be no machine which, supplied with any one U of these formulae, will eventually say whether U is provable.” (U. p. 145) 456:[Unlike the Russell paradox which involves "sets", but like the Barry paradox, this is a paradox of language, and the use of language. Very simple to state. Careful examination makes the reader Rather uncomfortable because of hidden assumptions and little errors in its original form (a letter). Makes use of 1676:

the reprint is from ser. 2, vol. 42 (1936-7), pp. 230-265, so I don't think the previous explanation of the date is justified from that book. As far as I can tell, the dating is incorrect in MathSciNet, (explaning many people incorrectly referencing the paper from 1937), but it is correct in Zentralblatt.

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undefinability-of-truth theorem that there is no effective (or even arithmetically definable) axiom system strong enough to completely capture the theory of the natural numbers. If you have Hodges' book, could you clarify what he means? Otherwise, I'll try to get to it, but I am busy this week. — Carl

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Most of Turing's paper describes "effective calculability" in terms of "computable numbers are those whose decimals are calculable by finite means" (p. 117), in other words, his a-machines. He sets up an enumeration to be done by his universal machine U; it provides a “circle-deciding machine D” with

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There is something wrong with the section "The Entscheidungsproblem of 1928:". Mathematics is generally believed to be consistent, and although the proof systems for first order logic are complete, the axioms of mathematics are incomplete in a different sense. I have not looked at the sources to see

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An explanation of Church’s example is beyond the reach of this article. Gandy explains (cf Gandy 1994:81-2) that within all the undecidability proofs (Church’s, Kleene’s and Turing’s) the method involves (1) Enumeration -- An algorithmic-like method to list all possible well-formed functions F of a

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In footnote 2, Church states that “definition of effective calculability can be stated in either of two equivalent forms, (1) . . . if it is λ-definable, (2) . . . it is recursive in the sense of " (U. p. 90). Furthermore in this definition he uses the criterion “the algorithm has terminated becomes

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into P; replace the counters into hole A (aka #1). CPYAri ?? (5) Recursion function and mu-operator combined (see Minsky 1967: After each instruction the following will occur. Also, there is a specific IF-THEN operation JZ as follows: Test hole #1 to see whether or not said hole is empty of pebbles.

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It is within the 10th problem where the question of an "Entscheidungsproblem" actually appears, but the question would float about for almost 30 years before it was framed precisely. (The 10th problem itself was resolved in the negative in 1970; its resolution in turn required the tools developed to

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I have fixed the incorrect date on Turing's 1936 paper. The footnote at the end of Bob Soare's 1996 paper "Computability and Recursion": has a good explanation of the correct dates. The reprint of Turing's paper in the collection edited by Martin Davis, (which was referenced before) only says that

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Post was seeking not just a “formulation which includes an equivalent for every possible ‘finite process’ but rather a “description that will cover every possible method for setting up finite processes”; he described this in his part “7. Generated sets of sequences” (Post 1965:402). He acknowledges

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In his 1936 paper Post only proposed his "working hypothesis" of “a process” and criticized Church's "definition", but had proved nothing. rather, Post maintained a more philosophic stance. He hoped to publish a series of "wider and wider formulations . . . The success of the program would for us,

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So why didn’t Gödel just go ahead and prove the undecidability of the Entscheidungsproblem? Gandy supposes that first “one needed to reflect long and hard on the idea of calculability” (Gandy 1994:64). The ever-cautious Gödel was unconvinced that either Church’s λ-definability or even his own forms

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Godel's second restriction, a rather onerous one, is as follows. Because diagonalization arguments (and Godel's argument) operate only on "functions of a single variable" (Godel 1934:Undecidable p.46), the model must be modified so we can create only single-variable formulas. This, as we see below,

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Indeed: What about our Entscheidungsproblem algorithm itself? Can it determine, in a finite number of steps, whether it, itself, is “successful” and "truthful" (that is, it does not get hung up in an endless “circle” or “loop”, and it correctly yields a judgment "truth" or "falsehood" about its own

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pp. 551-553. I see some interesting stuff, but nothing about his contributions to the Engscheidungsproblem, nor about "truth is undefinable". In fact, Tarski's truth is definable by going outside the formal system (theory) into the metasystem (metatheory); Grattan-Lewis in fact gives Tarski credit

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He seems to make a distinction between "number theory problems" of "Church" and what he called "combinatory mathematics" (cf Post 1965:338). Indeed, Post is emphatic about “the fundamental importance to mathematics of the existence of absolutely unsolvable combinatory problems . . . .“ He suggests

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Church sees two forms of the question: (i) “a constructive proof of the unsolvability of what we may call the “deducibility problem” of the engere Funktionenkalkül, that is the problem to find an effective procedure which is capable of determining, about any given expression in the notation of the

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exit-point "at the bottom." Godel's (very important) "immediate consequence" formula #43 Fl(x, y, z) ("Fl" from "Fogle" i.e. "consequence" (cf Meltzer-Braithwaite 1962:33) makes use of a similar notion. To say that a formula "x is an immediate consequence of formulas y and z" means that EITHER (1)

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Our machine will be the Melzak model (holes in the ground), but simplified to the Lambek model (the abacus model, cf B-B-J 2002:45ff), and further modified into the Peano axioms (see Minsky 1967:201, also Shepherdson-Sturgis ??): A line of holes (aka "registers") begins with a hole (of potentially

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By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise. First, was mathematics complete . . . Second, was mathematics consistent. Third . . . was mathematics decidable?" (Hodges p. 91, Hawking p. 1121). As mentioned above, in 1930-1 Kurt Gödel answered the

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Church would in turn defend himself against Post’s criticism by suggesting that “to define effectiveness as computability by an arbitrary machine subject to the restrictions of finiteness would seem to be an adequate representation of the ordinary notion, and if this is done the need for working

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The Cantor diagonalization method requires that all the axioms for all their possible parameters (i.e. CLEAR 1, CLEAR 2, ... CLEAR N, INCREMENT 1, INCREMENT 2, ... INCREMENT N, etc) and all the possible "formulas" such as ADD_TO_A B", "LOAD_B_FROM C", "MULTIPLY_A_BY M", "STORE_A_IN P" etc must be

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In Turing's paper, the text is: "Corresponding to each computing machine it we construct a formula Un (it) and we show that, if there is a general method for determining whether Un (.11) is provable, then there is a general method for determining whether it ever

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IF there was an Algorithm to decide whether a Statement of FOL was universally valid, THEN there would be Algorithm to solve the Halting Problem. An "Algorithm" is identified with a Turing Machine. The Halting Problem cannot be solved (in general) by a Turing Machine. From the Contraposition, we

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I'm neither mathematican nor a philospher and I think it could be mid-to-high, at least from a (mathematics) foundational and historical perspective. In particular it has application to the philosophy of mind (that's the secret reason for my interest). Some writers fault the philosphers for not

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In later years Post remained skeptical of Church’s definition of effective calculability (Kleene’s “Thesis I” of 1943, named “Church’s Thesis” by Kleene in 1952) and Turing’s a-machines (named “Turing’s Thesis” by Kleene in 1952). In a paper Post submitted in 1941 and had rejected (subsequently

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Without knowledge that Church-Kleene-Rosser at Princeton had been working on the same problem, Turing, a Master's student at King's College, Cambridge UK, was approaching his characterization of "effective calculability" from a much different angle. And, the appearance of Church’s proof spurred

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For Gödel a "formula" is a sequence of axioms or a sequence of formulas or a combined sequence of axioms and formulas. So as shown in the example below for "CLEAR x" (x is a register, CLEAR means "empty" or "make equivalent to zero"), the "formula" is actually a sequence of axioms with numbers

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Note that instead of referring to FOL, Turing refers to "functional calculus K in Hilbert and Ackermann's "Grundzüge der Theoretischen Logik (Berlin, 1931), chapter 3", which is about (many-sorted) first-order logic of relations (without equality) ... one should take a look at what functional

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refereed Turing's paper from spring 1936 through the fall; Bernays found errors in Turing's last proof and Turing had to modify it. But this gave Turing time to study Church's paper and add, in August 1936, an Appendix where he sketched a proof that Church's λ-calculus and his a-machines would

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This definition is difficult. It contains many hidden (tacit) assumptions/definitions such as "sequence", "last", "immediate consequence", "preceding". But unlike Finsler, Gödel attempts to clear these issues up in his "rigorous execution of the proof" (Gödel 1934 as the quote appears in Davis

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statement “Arithmetic P is consistent". But as a consequence of his "undecidability Theorem" IV, “the SENTENCE which asserts that x is consistent is not x-provable; in particular the consistency of P is unprovable in P, provided P is consistent (in the contrary of course, every statement is

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to publish; he too had been working in obscurity for a number of years, and he quickly submitted a brief paper in the fall of 1936 that offered as his "working hypothesis" of "effective calculability" a “process” (a man moving through a sequence of boxes marking and erasing per a list of

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My concern is that the question "is mathematics complete" could mean "are the axioms we have chosen for mathematics strong enough to prove all mathematical truths?". This is a very different question than "is first order logic complete". It follows from both Godel's theorems and Tarski's

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I don't think this is accurate. Turing & Church showed that deciding provability of arithmetic statements could be used to decide the Halting Problem, which really means that the Entscheidungsproblem was reduced to the halting problem and not the other way around?

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single variable m, assign a number to m, convert each parametrized-function “F of m” Fm to a Gödel number G(Fm), and put the G(Fm) in numerical order; (2) A successful demonstration that each parameterized-function Fm (and as a consequence its Gödelization G(Fm)) is a

1047:”The purpose of the present paper is to propose a definition of effective calculability which is thought to correspond satisfactorily to the somewhat vague intuition notion . . . and show, by means of an example, that not every problem of this class is solvable.” 1478:

I've changed the importance of this page from "low" to "mid". Actually, I think it rates a "high", but philosophers who are not mathematicians may think differently. The subject of the page is the impossibility of deciding every question that can be posed.

496:. Given a Diophantine equation with any number of unknown quantities and with rational integral coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers. 924:

formulas. Gödel commented on why this is possible in a "Remark: Variables for functions (relations) of two or more arguments are superfluous as primitive symbols since one can define relations as classes of ordered pairs ..."(Davis translation 1965:10).

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The negative solutions (Church's and Turing's) of the Entsheidungsproblem were found by a kind of self-reference (though that's not a phrase I'd use), but Hilbert's Entsheidungsproblem didn't (I shall boldly claim) derive from "self-reference".

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for the names "metalanguage" and "metatheory). To make the point, Grattan-Guiness gives the example directly from Tarski: '"it is snowing" is a true sentence if and only if it is snowing'" (Tarski 1936a, 156)(cf Grattain-Guinness 200:552). Bill

683:, by the proof of his undecidability theorem IV and the consequences thereof (in particular Theorem XI, the second so-called "incompleteness theorem"), answered the first two questions NO. But in order to do this he had to first demonstrate an 378:, without knowledge of Leibniz' work (cf Gratten-Guiness and Bornet 1997:xliii), devised a "calculus" of logic (ironically not accepting the possibility of the logical OR X+X = X discovered by Leibniz (Davis 2000:18) and yet proposing X*X = X; 1142:"But what Church had done was something rather different, and in a certain sense weaker . . . the Turing construction was more direct, and provided an argument from first principles, closing the gap in Church's demonstration." (Hodges p. 112). 981:

The problem was: An answer to the Entscheidungsproblem first required a precise characterization – a definition, an hypothesis/conjecture, or better yet, an axiomization -- of the notion of "definite general applicable prescription" or what

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Godel's formulas and the scheme he used to convert them to numbers is complex. A simpler example might be useful. Perhaps from it we can to see what Godel was up to, and to see if, and if so where, any other tacit assumptions are hiding.

1421:?" By this he meant, did there exist a definite method which could, in principle, be applied to any assertion, and which was guaranteed to produce a correct decision as to whether that assertion was true.” (Hodges 1983:91). In 1930-1 1160:

Explanation of this proof is beyond the scope of this article. It involves Gödelizations of “complete configurations” of a-machines and subsequent attempts to match them to Gödelized “answers”, with failure as the ultimate outcome.

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substituted for the variables. Similar formulas can be written for any arithmetic or logic formula such as "ADD_TO_A B", "LOAD_A_WITH C", "STORE_A_IN B", "AND_TO_A B" etc. But even Godel's axiomatic treatment has "tacit" axioms:

615:?" By this he meant, did there exist a definite method which could, in principle, be applied to any assertion, and which was guaranteed to produce a correct decision as to whether that assertion was true.” (Hodges 1983:91). 1129:). Although Turing’s submission of his paper (received 28 May 1936) gave him a few months' priority over Post, Post's paper published first and Church had to certify that Post's work was independent of Turing's. Church and 737:

Gödel's notion of "a proof" is not what the student learns in geometry. In his "proof" the last axiom or "formula" must follow "immediately from" (as an "immediate consequence" of, ) the preceding axiom(s) and/or formula(s):

782:, that is, work from a tiny set of 5 "number-theoretic formulas/formation-schemata (word used by Godel 1931:12 in U, also Kleene 1952:219). The following is Kleene's list (1952:219) and a similar list was used by Godel: 1334:

first two questions: YES the first order logic was complete (Gödel’s doctoral thesis proved his completeness theorem), but NO: arithmetic’s consistency could not be answered within arithmetic (Peano Arithmetic) itself.

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would follow up with his proposed "a- (automatic) machine" (closer to an axiomization but still not considered by some to be fully axiomatized, cf Gandy 1980, Sieg 1998, Moschovakis 2001, Dershowitz and Gurevich 2007).

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understanding the issues at gut-deep level but nevertheless concocting dubious pronouncements and arguments (e.g. not understanding quantum mechanics re philosophy of mind; also all the examples in Torkel Franzen 2005

673:). (van Heijenoort translation and typsetting 1967:607. "Flg" is from "Folgerungsmenge = set of consequences" and "Gen" is from "Generalisation = generalization" (cf Meltzer and Braithwaite 1962, 1992 edition:33-34) ) 520:"A quite definite generally applicable prescription is required which will allow one to decide in a finite number of steps the truth or falsity of a given purely logical assertion. . ." (Gandy 1994:57 quoting Behmann) 1295:

the notion of primitive recursion (i.e. the arithmetic of Peano axioms and first order logic) was inconsistent (it could be used to “pose” a question about itself that could be answered in both the negative and the

366:, Leibniz dreamed of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements (Davis 2000:3-20). He realized that the first step would have to be a clean 1042:

Indeed, Church's paper (presented to the American Mathematical Society 19 April 1935, published 15 April 1936) showed that an undecidable problem did exist. Church states the purpose of his paper this way:

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Given that Godel was able to display “undecidable sentences ” Gandy can now answer the question posed above about what happens when we submit our Entscheidungsproblem algorithm to itself for testing:

746:(specifically, a proof of the last formula of the sequence) if each formula of the sequence is either an axiom, or an immediate consequence of one or more of the preceding formulas" (Godel 1934:p.41). 1064:

Gödelized answer G(n), (3) Use of Cantor’s diagonal method to derive a contradiction when the function F is the decision procedure D acting on its own number i.e. G(D(D)) =? G(“True”? or “False”?).

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would propose his so-called “working hypothesis” (of a worker moving from room to room writing and erasing marks per a list of instructions) (Post 1936), and lastly a few months later (1936-7)

818:(V) identity or projection function: (∃u)(∀v (u(v)≡ a)). A function u exists, such that for all formulas v used in formula u, the formula v ≡ a can be plucked out of the bunch of them. Here 151: 706:"Some form of diagonalization argument lies at the basis of most proofs, or perhaps of every proof, of the undecidability theorm and of the first incompleteness theorem..."(Franzen 2005:70) 1928: 973:

means, and Gödel was more interested in “nonfinist concepts and methods” (Gandy 1994:64). So, the third question — the Entscheidungsproblem — would have to wait until the mid-1930's.

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By 1922, the specific question of an "Entscheidungsproblem" applied to Diophantine equations had developed into the more general question about a “decision method” for

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on Knowledge. If you would like to support the project, please visit the project page, where you can get more details on how you can help, and where you can join the

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With regards to “completeness”: Godel demonstrates that Peano arithmetic P is powerful enough to express, in P itself (he shows how to do this in footnote 360), the

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I will study this very carefully (I've researched this a bit since your note and agree that what I wrote is misleading). I will propose new wording. Thanks, wvbailey

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which is not part of the article. This seems to be some sort of vandalism, but I found no obvious way to fix it. Similar issue is discussed in this reddit thread:

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The associated link in the Knowledge text also refers to the page on "Reduction" as used in Complexity Theory, which is absolutely not what this is about at all.

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I am very disappointed that there is no mention, either in the article or this talk page, of Alfred Tarski's theorems (in 1936) that truth is undefinable (see

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deduce that there is no Algorithm to decide whether a Statement of FOL is universally valid. So the "Entscheidungsproblem" is reduced to the "Halteproblem"!

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theorems), and Church noted (and Davis reiterates on 1965:108-109, and in 2000:114-5) that this is a “non-constructive” proof i.e. it relies on the use of

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About Post’s “fundamental problem . . . the existence of absolutely undecidable propositions” mentioned above in Post’s footnote 1, Gandy remarks that:

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Hodges describes it this way: "By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise. First, was mathematics

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explains it this way: Suppose we are given a "calculational procedure" that consists of (1) a set of axioms and (2) a logical conclusion written in

476:– his beacon illuminating the way for mathematicians of the twentieth century. The 2nd problem asked for a proof that “arithmetic” is “consistent”. 1190:“1. The phrase “absolutely unsolvable” is due to Church . . . , and yet cannot be proved or disproved in any valid logic" (Post in Davis 1965:340) 527:"Behmann remarks that . . . the general problem is equivalent to the problem of deciding which mathematical propositions are true." (Gandy 1994:57) 1923: 1303:

Tonight I found a good quote in Gandy 1994 that perhaps I will substitute after reading up a bit more on this. Thanks for your look-over. wvbailey

809:(I) Peano axioms: 0 exists and has no successor, IF x' = y' THEN x = y, primitive recursion starting from the 0th case and is over one variable). 1867: 969:

in Gödel 1931, eventually augmented into “Herbrand-Gödel” recursion by the mid-1930's) was adequate to the task. Besides, the question requires

1963: 117: 1212:”This was the problem he hoped one day to solve. It is a problem which most people today have stopped (alas?) thinking about.” (Gandy 1994:90) 1938: 1683: 1417:, in the sense that the statement ‘2 + 2 = 5’ could never be arrived at by a sequence of valid steps of proof. And thirdly, was mathematics 611:, in the sense that the statement ‘2 + 2 = 5’ could never be arrived at by a sequence of valid steps of proof. And thirdly, was mathematics 229: 1948: 1554: 1442:). Thus in Peano arithmetic P there exists at least one statement that cannot be proven “consistent” or “inconsistent”; this fact makes P 1772: 1587: 928:

Another outcome of this second restriction will be the following: "Formulas" (that is, sequences of axioms or other formulas) must have

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If empty go to the instruction marked "=Z" else go to the "≠Z". OR: two holes J1 and J2... no, still must either go to next instruction

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The approach that Gödel took is simular to that taken by Finsler (1926). First, like Finsler, he does not employ the assertion "x is

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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

1233: 801:(V) definition by primitive recursion (2 types, the first one similar to Godel's that starts from 0th case and is over one variable) 108: 69: 219: 174: 1725:

and the German Wikipedians decided to link to that English-language article instead of this one. Please see my proposal at the

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The following is the whole quote from Hodges. I've added some quotes from Godel 1931 and my interpretation of what is going on:

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The "vicious circle" paradoxes and the Entsheidungsproblem derive from the same problem of self-referencing propositions, i.e.

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Indeed, in this context the diagonal argument is used by Gödel (1931), Church (1934), Turing (1936-7), and Kleene (1952).

1409:"By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise. First, was mathematics 424:’s axioms of arithmetic (1899) would, through the efforts of many mathematicians over the next 30 years, evolve into the 1847: 587:

By the 1928 international congress of mathematicians, Hilbert "made his questions quite precise. First, was mathematics

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In 1928 no such characterization existed. But over the next 6–7 years Princeton professor Church and his two students

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of the characterization – should it be a definition, or a hypothesis, or an axiomization? – would come under debate.

457: 301: 179: 878:"c is an immediate consequence of" a and b (of a) if a is the formula (~b)V(c), where c is the formula ∀v(a) <= ?? 937:

y=z implies x OR (2) there exists some bound-variable "v" and x = v for function y "generalized over variable v". .

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of common arithmetic (e.g. "add", "multiply", "raise to a power", "divide" etc). So just what does PROVABLE mean?

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subsequently reworked Frege’s peculiar symbolism into a more "user friendly" format that would appear in 1914 as

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subsequently argued with Boole for the acceptability of the logical OR (Gratten-Guiness and Bornet 1997:xiv)).

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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

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of arithmetic, i.e. a tiny collection of symbols, axioms and formation rules that define "the numbers" and the

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https://www.reddit.com/r/NoStupidQuestions/comments/127b5oj/why_is_the_preview_image_for_american_cuisine_on/

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What has the reference to Whitehead and Russell at the end of the article got to do with the subject matter?

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answered the first two questions: YES the first order logic was complete (Gödel’s doctoral thesis proved his

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This section states: "Turing reduced the halting problem for Turing machines to the Entscheidungsproblem."

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sent a letter to Frege, pointing out a problem with Frege's logic by creating a simple paradox, now called

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https://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Flag_of_Israel.svg/640px-Flag_of_Israel.svg.png

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in the technical sense that every statement . . . could be proved, or disproved. Second, was mathematics

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in the technical sense that every statement . . . could be proved, or disproved. Second, was mathematics

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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

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while it might be true that the first order logic might be complete (Gödel’s doctoral thesis proved his

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algorithm is "true" (i.e. an algorithm that always correctly yields a judgment "truth" or "falsehood"?)

336: 50: 1763:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit 619:

1931: True ≠ Provable: Gödel's undecidability theorem VI (the so-called "first incompleteness theorem")

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https://web.archive.org/web/20100607132453/http://diglib.princeton.edu/ead/getEad?eadid=C0948&kw=

1756: 1679: 1613: 1428: 1290: 1097: 514: 1703: 374:. Much of his subsequent work was directed towards that goal but had no lasting influence. In fact, 21: 1018: 966: 833:"This axiom represents the axiom of reducibility (comprehension axiom of set theory)" (ibid p. 13) 779: 538:, that is -- written in what Davis calls "Frege's rules of deduction" (or the modern equivalent of 473: 393:, a formula language, modeled upon that of arithmetic, for pure thought". About this Frege states: 363: 331: 1617: 618: 116:

on Knowledge. If you would like to participate, please visit the project page, where you can join

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before doing mass systematic removals. This message is updated dynamically through the template

444:. It represented a problem that simply refused to go away and would, 30 years later, bring down 84: 63: 1823: 895:

The Universal program: The TABLE of instructions for U must know the exact construction of the

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to the punch by almost a year (Turing's paper received 28 May 1936, published January 1937).

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I need to read up on this more, compare translations, etc. But this is a stab at it. wvbailey

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I agree. I had a bad time with the one sentence about Godel. Have I struck out the offender?

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1900: The Entscheidungsproblem (the "decision problem") expressed in Hilbert's tenth question

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Nachum Dershowitz and Yuri Gurevich, "A Natural Axiomatization of Church's Thesis", 2007,

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Church wondered : Had he proven case (ii) beyond a doubt? His proof relies upon Gödel’s

1789:, "External links modified" talk page sections are no longer generated or monitored by 1691: 1057: 1002: 726: 421: 379: 1829:

If you found an error with any archives or the URLs themselves, you can fix them with

634:"The general result about the existence of undecidable propositions reads as follows: 488:

answer the Entsheidungsproblem). Hilbert's original description of #10 is as follows:

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1994, “The Confluence of Ideas in 1936”, pages 51-102 in Rolf Herken (ed) 1994-1995,

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Within the year Church would go on to strengthen his result, and in a paper titled

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On Formally Undecidable Propositions in Principia Mathematica and Related Systems I

531: 375: 1075:“The general case of the Entscheidungsproblem 7 of the system L is unsolvable.” 871:"This axiom asserts that a class is completely determined by itself" (ibid p. 13) 1885: 1796: 1322:

Yes, striking out that part is a large improvement. Here is the current version.

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1931-1935: Effective calculability: λ-calculus? Recursion? Machine computation?

340: 1795:. No special action is required regarding these talk page notices, other than 778:, an abstract computational model that we call "M". This M should behave like 224: 201: 90: 623:

The "first incompleteness theorem" appears as "Theorem VI" in his 1931 paper

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The vandalism has been reverted. Please be patient until the cache expires.

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1926: Finsler's (arguable) first demonstration of an undecidable proposition

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how they present it, but as it is stated that para needs some help. — Carl

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change this hypothesis not so much to a definition or to an axiom but to a

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1965:9), or as translated by van Heijenoort "with full precision" (p.599).

990:(cf part "7. The notion of effective calculability" in Church 1936:100 in 1349: 1264: 1450:

cannot be proved within P itself. So in effect, Gödel got “two for one”.

542:). Gödel's doctoral dissertation (1930) proved that Frege's rules were 362:. In the late 1600's, after having constructed a successful mechanical 223:, a collaborative effort to improve the coverage of content related to 549:

In other words: Is there a “decisional algorithm” that can tell us if

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would prove in 1931 that, within what he called “P” (nowadays called

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The Universal Turing Machine: A Half-Century Survey, Second Edition

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behavior and results)? Within this question Russell's paradox (and

448:'s "program" to reduce all of mathematics to symbol-manipulation. 386: 733:"A formula is provable if a proof of it exists" (Godel 1934:p.41) 1186:) he ponders the notion of, and the nature of, undecidability: 1138:

hypothesis would disappear.” (cf Church 1937 in Gandy 1994:79).

1104:, an anathema to mathematicians with an intuitionistic outlook. 771:

A very simple, but very powerful, computational "system" is the

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That language reference is not set. Any sane reason for that?--

953:"enumerable" -- convertable to numbers and countable, to boot. 702:(although his usage is subtle whereas Finsler's is obvious). 494:

10. Determination of the solvability of a Diophantine equation

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preceding formula y. But the point is that either y=z --: -->

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The term (1) takes care of axioms and term (2) takes care of

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free, and I am using modern symbols ∃ and ∀ and modern usage.

1571:. The article is pretty thin and doesn't get into this. Bill 292: 1777:

http://diglib.princeton.edu/ead/getEad?eadid=C0948&kw=

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for additional information. I made the following changes:

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Godel's Theorem: An Incomplete Guide to this Use and Abuse

915:-instruction set; in fact, in our example it will not be. 349:

Holding place, taken from history on Turing machine page

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After reading this I consulted Grattan-Guiness (2000)

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1928: Hilbert further defines the Entscheidungsproblem

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http://research.microsoft.com/~gurevich/Opera/188.pdf

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The origin of the Entscheidungsproblem goes back to

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I deleted the rambling that was here. Knowledge is

112:, a collaborative effort to improve the coverage of 1799:using the archive tool instructions below. Editors 275: 1732:for merging this page's text into that article. — 509:mathematical formula, and H. Behmann stated that 397:"In fact, what I wanted to create was not a mere 1438:provable)”(Gödel’s theorem XI, Godel 1931:36 in 1092:theorem (i.e. his PhD thesis, not his so-called 1009:would propose a "definition" by use of Church's 812:(III) The 4 base axioms of Principia Mathematica 1929:Knowledge level-5 vital articles in Mathematics 1785:This message was posted before February 2018. 911:-instruction set need not be the same as the 405:in Leibniz's sense." (van Heijenoort 1967:2). 8: 1530:The Search for Mathematical Roots 1970-1940 1060:, i.e. to match the Gödelized G(Fm) to the 907:program-instruction will be. Note that the 639:For every ω-consistent recursive class κ of 1755:I have just modified one external link on 903:dedicated registers reside, and where the 742:"A finite sequence of formulas shall be a 570:) is reasserting itself with a vengeance. 272: 163: 58: 805:The Godel list: ≡≠⇒⋀⊗ℵ↑∆←⊆∉∈∸→⊂∀∃ℕ∩∪ǎăℬ⇔ 761:Tacit Knowledge and Mathematica knowledge 630:In Gödel's original notation, it states: 472:posed a set of problems -- now known as 1717:I believe it is because the German word 1656:So, I'm changing the text in the page! 717:Finsler, Gödel expresses his assertions 1919:Knowledge vital articles in Mathematics 1871:2001:14BA:23EF:1A00:2811:49BF:3BC9:B97E 795:(III) identity or "projection" function 165: 60: 19: 1165:1936-1965: Post’s "working hypothesis" 233:about philosophy content on Knowledge. 1934:C-Class vital articles in Mathematics 1637:Yes, this should be other way around. 920:forces us into defining sequences of 7: 1505:Lack of mention of Tarski's theorems 698:". Second, like Finsler he employs 599:?" (Hodges p. 91, Hawking p. 1121). 217:This article is within the scope of 106:This article is within the scope of 1862:The page preview shows this image: 822:is a formula that does not contain 687:assertion (sentence, proposition). 354:Origin of the Entscheidungsproblem: 49:It is of interest to the following 1954:Mid-importance Philosophy articles 1944:High-priority mathematics articles 1549:Reference to Whitehead and Russell 1069:A Note on the Entscheidungsproblem 14: 1759:. Please take a moment to review 1229:, Springer-Verlag/Wien New York, 1071:(April 1936) he would show that: 468:In 1900 the famous mathematician 126:Knowledge:WikiProject Mathematics 1914:Knowledge level-5 vital articles 1204:1965-present: Later developments 694:" but rather, he employs "x is 513:". . . most general form of the 239:Knowledge:WikiProject Philosophy 204: 194: 167: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 1658:There is a T101 in your kitchen 1250:Another major rewrite. wvbailey 1108:1931-1937: A flurry of activity 1052:effectively known” (U p. 100). 798:(IV) definition by substitution 259:This article has been rated as 242:Template:WikiProject Philosophy 146:This article has been rated as 1924:C-Class level-5 vital articles 1894:18:50, 15 September 2023 (UTC) 1879:12:40, 15 September 2023 (UTC) 1853:17:42, 21 September 2017 (UTC) 1511:Indefinability_theory_of_truth 1182:published in 1965 by Davis in 932:entry-point "at the top", and 595:. Third . . . was mathematics 591:. . . Second, was mathematics 463: 330:a place to publish new ideas. 1: 1964:Mid-importance logic articles 1742:01:27, 12 November 2014 (UTC) 1666:23:56, 17 November 2016 (UTC) 1596:17:35, 11 November 2010 (UTC) 1499:21:52, 7 September 2007 (UTC) 1484:19:39, 7 September 2007 (UTC) 120:and see a list of open tasks. 1939:C-Class mathematics articles 1630:08:28, 5 February 2013 (UTC) 1581:13:46, 6 November 2010 (UTC) 1563:12:52, 6 November 2010 (UTC) 1134:compute the same functions. 1021:modified by a suggestion of 341:14:31, 12 January 2007 (UTC) 1949:C-Class Philosophy articles 1712:14:40, 6 October 2013 (UTC) 1543:18:42, 1 October 2009 (UTC) 1523:01:24, 1 October 2009 (UTC) 1460:03:05, 10 August 2007 (UTC) 428:that Leibniz had sought. 1985: 1816:(last update: 5 June 2024) 1752:Hello fellow Wikipedians, 1384:22:25, 8 August 2007 (UTC) 1359:17:58, 8 August 2007 (UTC) 1308:04:11, 7 August 2007 (UTC) 1274:23:17, 6 August 2007 (UTC) 1255:18:18, 7 August 2007 (UTC) 946:y) & (y & y--: --> 458:Cantor's diagonal argument 265:project's importance scale 1969:Logic task force articles 721:within a tightly-defined 300: 271: 258: 189: 145: 78: 57: 1653:calculus K actually is. 1504: 899:-instruction set, where 700:Cantor's diagonal method 669:is the FREE VARIABLE of 452:1905: Richard's Paradox: 451: 432:1901: Russell’s Paradox: 152:project's priority scale 1748:External links modified 1698:de:Entscheidungsproblem 1692:21:32, 1 May 2013 (UTC) 988:effective calculability 276:Associated task forces: 109:WikiProject Mathematics 1959:C-Class logic articles 1909:C-Class vital articles 1671:Date of Turing's paper 1336: 1102:Law of Excluded Middle 1100:and consequently, the 840:(VI) Type-elevation: x 792:(II) constant function 789:(I) successor function 647:r, such that neither v 420:. This, together with 414:Alfred North Whitehead 370:, or what he called a 297: 220:WikiProject Philosophy 1670: 1331: 1147:1936-7 Turing’s Proof 1013:and Herbrand-Gödel’s 945:x ( (z & z--: --> 418:Principia Mathematica 399:calculus ratiocinator 296: 36:level-5 vital article 1797:regular verification 1757:Entscheidungsproblem 1719:Entscheidungsproblem 1429:completeness theorem 1291:completeness theorem 1174:" (Post 1936:291 in 1098:reductio ad absurdum 1038:1935: Church’s proof 986:would come to call " 944:x or y&z --: --> 515:Entscheidungsproblem 322:Talk page cleaned up 132:mathematics articles 1787:After February 2018 1019:primitive recursion 967:primitive recursion 780:primitive recursion 643:there are recursive 426:langua characterica 403:lingua characterica 372:lingua characterica 364:calculating machine 245:Philosophy articles 1858:Page preview image 1841:InternetArchiveBot 1792:InternetArchiveBot 815:(III) Substitution 474:Hilbert's problems 298: 230:general discussion 101:Mathematics portal 45:content assessment 1817: 1723:Decision problem, 1682:comment added by 1633: 1616:comment added by 1601:Negative solution 1357: 1272: 785:The Kleene list: 773:Turing-equivalent 663:belongs to Flg(κ) 568:diagonal argument 536:first order logic 442:Russell's paradox 360:Gottfried Leibniz 339: 319: 318: 315: 314: 311: 310: 307: 306: 212:Philosophy portal 162: 161: 158: 157: 1976: 1851: 1842: 1815: 1814: 1793: 1728:Decision problem 1721:literally means 1694: 1632: 1610: 1435:metamathematical 1347: 1262: 1023:Jacques Herbrand 1017:-- i.e. Gödel's 1015:recursion theory 965:of "recursion" ( 719:"x is PROVABLE" 482:Peano Arithmetic 438:Bertrand Russell 410:Bertrand Russell 389:presented his " 335: 283: 273: 247: 246: 243: 240: 237: 214: 209: 208: 207: 198: 191: 190: 185: 182: 171: 164: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 1984: 1983: 1979: 1978: 1977: 1975: 1974: 1973: 1899: 1898: 1860: 1845: 1840: 1808: 1801:have permission 1791: 1765:this simple FaQ 1750: 1700: 1684:131.215.108.224 1677: 1673: 1639: 1611: 1603: 1569:impredicativity 1551: 1507: 1476: 1448:P is consistent 1440:The Undecidable 1206: 1184:The Undecidable 1176:The Undecidable 1167: 1149: 1127:The Undecidable 1110: 1040: 992:The Undecidable 979: 864: 860: 855: 851: 847: 843: 776:counter machine 727:total functions 621: 584: 576: 466: 454: 434: 391:Begriffsschrift 368:formal language 356: 351: 324: 281: 244: 241: 238: 235: 234: 210: 205: 203: 183: 177: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 1982: 1980: 1972: 1971: 1966: 1961: 1956: 1951: 1946: 1941: 1936: 1931: 1926: 1921: 1916: 1911: 1901: 1900: 1897: 1896: 1859: 1856: 1835: 1834: 1827: 1780: 1779: 1771:Added archive 1749: 1746: 1745: 1744: 1699: 1696: 1672: 1669: 1638: 1635: 1602: 1599: 1555:213.122.67.212 1550: 1547: 1546: 1545: 1506: 1503: 1502: 1501: 1475: 1472: 1471: 1470: 1469: 1468: 1467: 1466: 1465: 1464: 1463: 1462: 1453: 1452: 1451: 1395: 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984:Alonzo Church 976: 974: 972: 968: 959: 958: 957: 954: 950: 949: 942: 935: 931: 926: 923: 914: 910: 906: 902: 898: 894: 893: 892: 889: 884: 877: 876: 870: 869: 868: 867: 839: 838: 832: 831: 830: 829: 825: 821: 817: 814: 811: 808: 807: 806: 800: 797: 794: 791: 788: 787: 786: 783: 781: 777: 774: 769: 762: 758: 757: 756: 752: 745: 741: 740: 739: 732: 731: 730: 728: 724: 720: 716: 711: 705: 704: 703: 701: 697: 693: 688: 686: 682: 672: 668: 664: 660: 656: 652: 648: 644: 640: 637:"Theorem VI. 636: 635: 633: 632: 631: 629: 627: 616: 614: 610: 606: 601: 600: 598: 594: 590: 581: 579: 571: 569: 565: 556: 555: 554: 552: 547: 545: 541: 540:Boolean logic 537: 533: 526: 525: 519: 518: 516: 512: 511: 510: 508: 500: 499: 495: 491: 490: 489: 485: 483: 479: 475: 471: 470:David Hilbert 461: 459: 449: 447: 446:David Hilbert 443: 439: 431: 429: 427: 423: 419: 415: 411: 404: 400: 396: 395: 394: 392: 388: 383: 381: 377: 373: 369: 365: 361: 353: 348: 346: 343: 342: 338: 333: 329: 321: 303: 295: 291: 290: 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B. Rosser 1000: 995: 994:). Even the 991: 980: 970: 963: 955: 951: 940: 938: 933: 929: 927: 918: 912: 908: 904: 900: 896: 887: 885: 881: 823: 819: 804: 784: 770: 766: 760: 753: 749: 743: 736: 722: 718: 714: 712: 709: 695: 691: 689: 684: 678: 670: 666: 662: 658: 654: 650: 646: 645:CLASS SIGNS 642: 638: 624: 622: 612: 608: 604: 602: 596: 592: 588: 586: 585: 577: 560: 550: 548: 543: 532:Martin Davis 530: 517:as follows: 506: 504: 493: 486: 467: 455: 435: 425: 417: 408: 402: 398: 390: 384: 376:George Boole 371: 357: 344: 325: 260: 228: 218: 147: 107: 51:WikiProjects 34: 1646:prints 0" 1293:), but NO: 1223:Robin Gandy 1172:natural law 1114:Alan Turing 1031:Alan Turing 685:undecidable 123:Mathematics 114:mathematics 70:Mathematics 1903:Categories 1848:Report bug 1474:Importance 1444:incomplete 1423:Kurt Gödel 1415:consistent 1287:Kurt Gödel 1011:λ-calculus 681:Kurt Gödel 679:In 1930-1 609:consistent 593:consistent 478:Kurt Gödel 236:Philosophy 225:philosophy 175:Philosophy 1831:this tool 1824:this tool 1730:talk page 1704:Crackwitz 1419:decidable 1122:Emil Post 1027:Emil Post 947:x) =: --> 922:recursive 856:) --: --> 641:FORMULAS 613:decidable 597:decidable 39:is rated 1837:Cheers.— 1680:unsigned 1626:contribs 1614:unsigned 1573:Wvbailey 1535:Wvbailey 1496:Wvbailey 1457:Wvbailey 1411:complete 1381:Wvbailey 1305:Wvbailey 1285:In 1930 1252:Wvbailey 905:starting 848:(x1) ≡ y 696:provable 605:complete 589:complete 544:complete 436:In 1901 332:CMummert 1761:my edit 1618:Froskoy 1062:correct 1025:. 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