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. “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 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.
Interested in supporting this site?
As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical terms are correctly formatted.
Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous usernames for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it incorrectly and rewrite it again - still erroneously, or else attack something else on my site - erroneously. After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.
Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.
Note: a password enables editing of comments, an email enables notification of replies