The Flaw in Gödel’s proof of his Incompleteness theorem
Page last updated 14 April 2022
I had previously considered that it would not be feasible to give a brief summary of the flaw in the proof of incompleteness (which you can see online at English translation of Gödel’s original proof) by Kurt Gödel, a proponent of intelligent design (see Statements by Kurt Gödel). But in fact it is possible to pinpoint an intuitive assumption that Gödel makes in his proof which is easily demonstrated to be incorrect and logically invalid. Since the remainder of his proof relies completely on this assumption, this renders the result of the proof invalid.
That assumptive assertion is what Gödel calls Relation 17, and which states:
Z(n) = n N [R(1)]
Z(n) is the number-string for the number n. (Footnote: “Z(n) ist das Zahlzeichen für die Zahl n” in the original German. The place in the proof where this occurs can be seen in the online English translation at Gödel’s Proof: Relation 17. )
Gödel previously defined that an italicized word, such as number-string indicates the number calculated by his Gödel numbering function Φ, when applied to a given sequence of symbols of the formal system P (the system for which Gödel is claiming incompleteness). In other words, he is claiming:
Z(n) = Φ(n)
such that their calculated values are equal when their free variables are equal, and the result of his paper has a complete reliance on this assertion, which is an assumption by Gödel for which he offers no logical justification whatsoever.
It is easily shown that no such equivalence can apply, as follows:
The domain of the free variable n of Z is defined to be all natural numbers, i.e. the only values can be substituted for the n are the natural numbers 0, 1, 2, 3, … . On the other hand, the domain of the free variable of Φ is all sequences of symbols of the formal system P, an example being x2(0) ∧ x1∀ (x2(x1) ⊃ x2( f x1)) ⊃ x1∀ (x2(x1)).
Clearly Gödel did not intend the complete domain of Φ to be all sequences of the formal system P, but we can restrict the domain of n in both Z(n) and Φ(n) to be only natural numbers, where the format used for natural numbers in both the formal system and for the Z function is to be in the form of 0, f 0, ff 0, fff 0 … , (the format for natural numbers in the formal system) so that the formats of the numbers that constitute the domain of the free variables of the Z function and his numbering function Φ are precisely identical.
However, even imposing this restriction does not achieve an equivalence of the two functions. This is because there can be infinitely many expressions that have the same value as any given number, and which are also valid values for the substitution of the free variable of the Z function. For example, in the format of the formal system, 3 is fff 0, 2 is ff 0 and 5 is fffff 0, so we have that fff 0 + ff 0 = fffff 0 and both fff 0 + ff 0 and fffff 0 are precisely the same value when substituted for the free variable of Z, and the Z function gives precisely the same value for either of these two substitutions.
But the sequences of symbols “fff 0 + ff 0” and “fffff 0” are not the same value when substituted for the free variable of Φ - the Φ function, by definition, must always give a different value for every different sequence of symbols of the formal system, regardless of any calculated value of that sequence of symbols. For the substitution of “fffff 0” ,the Φ function gives the result:
23 · 33 · 53 · 73 · 113 · 131
i.e, a result with 6 prime factors, while even with a formal system that includes the “+” symbol, the Φ function can only give a result that has at least 8 prime factors: (Footnote: Note: There is no “+” symbol in Gödel’s formal system P, and the concept of addition in that system requires the use of many more symbols, so the result would have many more than 8 prime factors.)
23 · 33 · 53 · 71 · 11x · 133 · 173 · 191
Hence it is quite obvious that there cannot be an equivalence of the Z(n) and Φ(n) functions as Gödel intuitively and incorrectly assumes, since they can result in different values for equal valued substitutions of their free variables.
Gödel uses that incorrect assumption of equivalence as an essential part of his proof of his assertion that he calls Proposition V, which renders his Proposition V has logically invalid and the remainder of his proof that follows is likewise completely invalid.
Perhaps the reader might think that perhaps the above demonstrated error of assumed equivalence is only a curiosity, and perhaps there is some way to circumvent it. But there isn’t. At this point we have to acknowledge that we cannot give a brief summary of the details of how this Proposition V is a confusion of different levels of language by the use of this false equivalence. This is because Gödel did not actually give a fully detailed proof of this crucial part of his incompleteness paper - he simply assumed that it was correct, only giving a rough outline of how a proof of that crucial part might proceed, saying in his paper: “We content ourselves here with indicating the proof of this proposition in outline, since it offers no difficulties of principle and is somewhat involved.” (Footnote:
Peter Smith, although he is a staunch advocate of Gödel’s proof, acknowledges in his paper, Expounding the First Incompleteness Theorem, that:
“Gödel only sketches a proof by induction on the complexity of the definition of the (characteristic function of the) property or relation in terms of definitions by composition and recursion grounding out in the trivial initial functions. The crucial step is just asserted - ‘the processes of definition … (substitution and recursion) can both be formally reproduced in the system P’ … it certainly isn’t spelt out.”
See also Peter Smith’s Gödel Without (Too Many) Tears - Or Not? and A Fundamental Flaw in an Incompleteness Proof by Peter Smith.) This failure to give a fully detailed proof is even more surprising in view of Gödel’s insistence elsewhere on clarity and precision. And today almost every mathematician/logician seems content to make the same intuitive assumption, which is exceedingly strange - in mathematics, one should never let intuition triumph over strict logic. Stranger still is that you might expect that several mathematicians/logicians would have tried to do a fully detailed proof of this crucial part, and published their results - but where are such results? This is why I give in my paper (Footnote: See The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.) a fully detailed account following Gödel’s suggested outline of his Proposition V - and how the incorrect assumption of equivalence of the Z(n) and Φ(n) functions is an indispensable requirement of that proposition. I have now added a section to the paper that gives a brief summary of the underlying illogical assumption of equivalence of the Z(n) and Φ(n) functions that the proof relies on, so that a reader can see in a few pages that the proof is flawed. (Footnote: It is also easy to show that the conflation of levels of language inherent in the assumption of equivalence of the Z(n) and Φ(n) functions leads to a blatant contradiction in Gödel’s Proposition V, see the page Gödel’s Proposition V.)
It might be noted here that Gödel’s incompleteness paper is not the only paper where he made a completely unproven assertion which almost everyone assumed to be correct because of Gödel’s esteemed reputation. In that other paper Gödel made a similar intuitive assertion, but it was eventually proved (some 50 years later) that Gödel’s intuitive assumption stated precisely the opposite of what was actually the case. For details of this see Gödel’s Intuitive Error No.2. Not many mathematicians are aware of this - Gödel’s reputation is still so honored today that most discussions of Gödel’s work skip over this inconvenient detail.
The above is a brief summary of an incorrect assumption, and it is not a detailed exposition of how that incorrect assumption is used in the proof. But experience shows that, regardless of whatever warnings one makes about such lack of details that a brief summary might contain, they are ignored by a multitude of irrational individuals. These persons take delight in spreading misinformation, quoting material from a summary as though it was material from the full explanation. So, rather than attempting to give a brief summary of how the false assumption is used in the proof, I provide a fully detailed account in my formal paper The Fundamental Flaw in Gödel’s proof of his Incompleteness Theorem , which now includes a brief summary of the illogical unfounded assumption that the entire proof relies on, so that the reader can have an idea of the substance of the flaw within a few pages.
A Simplified explanation of Gödel’s proof and the flaw in the proof
If you are not familiar with the details of Gödel’s proof, I suggest that you either read the simplified explanation of Gödel’s proof on this website, or read the novel The Shackles of Conviction. Both include an explanation of the theorem and the flaw in Gödel’s proof of it in a simplified way that makes it accessible to the general reader.
A step by step walk-through guide to Gödel’s Proof
For anyone who wishes to get to grips with Gödel’s original paper, I have created a walk-through guide to Gödel’s original incompleteness proof, and which is intended to be read alongside the paper. Having looked for such a guide and could find none, I thought it odd that there were no such guides, considering how much other material has been written about Gödel’s proof. So I decided to create this guide to assist anyone else who wishes to become familiar with the details of Gödel’s argument.
A Detailed explanation of the flaw in Gödel’s proof
On the other hand, if you are very familiar with the details of Gödel’s theorem, you will probably also want to see the details of the flaw in Gödel’s proof as covered in the paper:
The paper gives a comprehensive demonstration of the flaw in Gödel’s proof of his incompleteness theorem. The link above is a link to the latest version. Please note that the central argument of the paper remains unchanged throughout the revisions. The revisions have been to make the principles easier to understand. After some requests, the latest version includes a brief summary of the illogical unfounded assumption that the entire proof relies on, so that the reader can have an idea of the substance of the flaw within a few pages. If you want to see the previous versions please see the site-map. See also Gödel’s 1934 Undecidability lectures, which, although by then Gödel had three years to reflect on his methods of proof, rather ironically allow a much simpler demonstration of the inherent confusion of language that is involved.
The Flaw and the Substitution function in Gödel’s proof
The web-page Gödel’s Substitution Function describes how Gödel’s incorrect use of a ‘substitution’ function confuses the systems that he is dealing with.
Responses to the demonstration of the flaw in Gödel’s proof
If you would like to see some of the responses that I have received regarding the demonstration of the flaw in Gödel’s proof, please see Responses to the demonstration of the flaw.
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