User:Wvbailey/Criticism of Godel's incompleteness theorems

A number of writers have presented arguments against Godel's incompleteness theorems.

Professional criticism of Godel's incompleteness theorems

More a question of disappointment that the theorems did not extend to (were not applicable) to a wider (i.e. universal), or absolute notion of undecidability. Post understood the notion of undecidability with respect to a formal system, but Finsler did not:

Finsler (1926) presents a precursor argument in his formal proofs and undecidability that, at a first viewing, seems remarkably similar to Godel's argument in Godel 1931. However, closer inspection shows that it proceeds informally, in particular without the specification of a formal system in which to define the notion of "provable" .

There is therefore nothing in the incompleteness theorem to show that there are true arithmetical statements that are unprovable in any absolute sense." (Franzén p. 25)

> But Post seemed to believe that Church had demonstrated an "absolutely unsolvable" problem as opposed to a problem unsolvable with respect to a specific formal system "S". He expressed his notion of "... chang[ing] this hypothesis [Church's thesis] not so much to a definition or to an axiom but to a natural law. Only so, it seems to the writer, can godel's theorem concerning the incompleteness of symbolic logics of a certain general type and Church's results on the recursive unsolvability of certain problems be transformed into conclusions concerning all symbolic logics and all methods of solvability." (Post 1936 in Davis 1965:291)

He persisted. From Post 1941 unpublished until reprinted in Davis 1965:340ff:

"[Footnote 1]. The phrase "absolutely unsolvable" is due to Church who thus described his problem in answer to a qurey of the writer as to whether the unsolvability of his elementary number theory problem was relative to a given logic [Church 1936]. By contrast, the undecidable prpostitions in this and related papers are undecidable only with respect to a given logic. A fundamental problem is the question of the existence of absolutely undecidable propositions, that is, propositions which in some a priori fashion can be said to have a determined truth-value, and yet cannot be proved or disproved by any valid logic ....

The writer cannot overemphasize the fundamental importance to mathematics of the existence of absolutely unsolvable combinatory problems.... [Here he points out that e.g. recursive unsolvability is "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." (p. 340)

In footnote 6, Post mentions Turing 1937 and his own Post 1936. But:

In this connection we must emphasize the distinction between a formulation which includes an equivalent of every possible "finite process", and a description which will cover every possible method for setting up finite processes." (p. 343)

> Godel's letter (never sent) commenting on the unlikelihood of something like this:

"As for work done earlier about the question of (formal) decidability of mathematical propositions, I know only of a paper by Finsler published a few years before mine (I believe in Math. Zeits. 25 (1926), p. 676) He also applies a diagonal procedure in order to construct undecidable propositions. However, he (Finsler) omits exaclty the main point which makes a proof possible, namely restriction to a definite (some well defined) formal system The distinction between truth and demonstrability is This "proof" (which incidentally was not known to me when I wrote my paper) is therefore worthless and the result claimed, namely the existence of "absolutely" undecidable propositions is very likely wrong. in which the proposition is undecidable. For, he had the nonsensical aim of proving formal undecidability in an absolute sense. this leads to the nonsensical definition given in the first two parags. of set. 5, p. 678 (su;;oemented) by the 2nd parag. of sect. 9, p. 680; and this leads to the flagrant inconsistency that he decides the ("formally) undecidable (") proposition by an argument ((sect. 11, p. 681)) which, according to his own (definition in) the two passages just cited is a formal proof while on the other hand he asserts on p. 681 it/s formal undecidability. If Finsler had confined himself to some well defined formal system S, his proof (by changing replacing the nonsensical section 11 with a proof that the (proposition in question) is expressible in S) could be made corret and applicable to any formal system. I myself did not know his paper when i wrote mine, and other mathematiciians or logicians probably disregarded it because it contains the obvious nonsense just mentioned."(strikeouts in original, p. 11, letter to yossef Balas ca Spring 1970)

To quote Dawson, "Godel's results appeared to him a smere specializations of those he had already obtained. ... he noted in particular that the antidiagonal construction Finsler had employed would, if applied to a "truly formal" system, define a sequence that lay outside the system itself." (p. 405)

>>Post's simple sets ?? "in a sense extremely undecidable" cf p. 70 in Franzen.

References

• Jean van Heijenoort, 1967 From Frege to Godel: A Source Book in Mathematical Logic, 1979-1931, Harvard University Press, Cambridge, MA, ISBN 0-674-32449-8 (pbk).
• M. Randall Holmes, on-line/preliminary review: David booth and Renatus Ziegler, eds. Review of "Finsler Set Theory: Platonism and Circularity," at http://math.boisestate.edu/~holmes.
• M. Randall Holmes 1997,Book Review: Finsler set theory: Platonism and Circularity", Mathematical Gazette, Vol. 81, No 490 (Mar. 1997), pp. 159-162
• "The Reception of Godel's Incompleteness Theorems", John W. Dawson, Jr., PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1984, Volume Two: Symposia and Invited Papers. (1984), pp. 253-271. [1]