Church’s Theorem (“An Unsolvable Problem”)
Page last updated 26 Nov 2020
In the paper “An Unsolvable Problem of Elementary Number Theory”, (Footnote: Church, Alonzo. PDF An unsolvable problem of elementary number theory. American journal of mathematics, 1936, Vol. 58, No. 2, pp 345-363.) Alonzo Church claims to prove various theorems regarding number theory by the use of Gödel numbering. One of these is commonly known as Church’s theorem. (Footnote: Note that there are other propositions by Church that are also given the name of “Church’s Theorem”.) However, Church falls into the common error of assuming that certain propositions can be stated within number theory when that is plainly not the case.
In particular, in the proof of his Theorem XVIII, Church claims that functions he refers to as functions H, B and C are recursive and definable in the number system that Church has defined. However, the fact is that H, B and C are defined in terms of the Gödel numbering function, so that the actual definitions of the functions H, B and C are H[GN(u)], B[GN(u), GN(v)], and C[GN(u)] where GN is a Gödel numbering function, and u and v are variables whose domain is a formula of the number system in question.
Church claims that since the functions are recursive, they are definable in the number system in question. This is of course incorrect, since the variables u and v are not variables of the number system in question. This is yet another case of a hidden illegal self-reference, as has already been demonstrated on this website to occur in several other incompleteness ‘proofs’. In common with many such ‘proofs’, the self-reference is slipped in alongside several paragraphs of otherwise unremarkable propositions.
The same error is repeated in Church’s PDF A Note on the Entscheidungsproblem (Footnote: Church, Alonzo, PDF A Note on the Entscheidungsproblem, Journal of Symbolic Logic, 1936 Vol 1, pp 40-41, details at https://doi.org/10.2307/2269326, also see “Correction to A note on the Entscheidungsproblem.” Journal of Symbolic Logic, 1936 Vol 3, pp 101-102, details at https://doi.org/10.2307/2269030 (Note: the correction does not affect the above analysis).) where he refers to:
… the existence of a recursively defined function a of two positive integers such that, if y is the Gödel representation of a well-formed formula Y then a(x, y) is the Gödel representation of the xth formula in the enumeration of the formulas into which Y is convertible.
Again, Church’s actual definition of the function a is a[x, GN(Y )], where the Gödel numbering function GN is not definable in a formula of the number systems that Church refers to.
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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