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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, also see “Correction to A note on the Entscheidungsproblem.” Journal of Symbolic Logic, 1936 Vol 3, pp 101-102, details at (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.

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Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

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The Lighter Side


Paper on the diagonal proof

There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.

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Other recently added pages

The Myths of Platonism


Goodman’s Paradox


The Platonist Rod paradox


The Balls in the Urn Paradox


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Lebesgue Measure

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

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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).

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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

Moderate Platonism

Descartes’ Platonism

The duplicity of Mark Chu-Carroll

A John Searle Inanity

Man versus Machine

Fake News and Fake Mathematics

Ned Block’s Blockhead

Are we alone in the Universe?

Good Math, Bad Math?

Bishops Dancing with Pixies?

Artificial Intelligence

Cranks and Crackpots

The Chinese Room


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:

Gödel Links


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

Gödel, Minds, and Machines

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