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# On Church’s “An Unsolvable Problem”

In the paper “An Unsolvable Problem of Elementary Number Theory”, (Footnote: Church, Alonzo. “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. However, Church falls into the same 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 “A note on the Entscheidungsproblem (Footnote: Church, Alonzo. “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.

Footnotes:

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## NEWS

### Lebesgue Measure

There is now a new page on a contradiction in Lebesgue measure theory.

### Illogical Assumptions

There is now a new page Halbach and Zhang’s Yablo without Gödel which analyzes the illogical assumptions used by Halbach and Zhang.

### Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

### O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

### Previous Blog Posts

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

– a page with general links:

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

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