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Copyright   James R Meyer    2012 - 2025 https://jamesrmeyer.com

The Flaw in Gödel’s proof of his Incompleteness theorem

Page last updated 10 July 2025

 

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.⁠ “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 in the context of this assumed equivalence to the Z function, 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, for the substitution of “fff 0 + ff 0” the Φ function must give a result that has at least 8 prime factors, viz:⁠ 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. See also The Z function assumption.

 

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. Several people have come up with bizarre ideas in attempts to circumvent the flaw, on web-pages such as on Reddit - Gödel’s flaw, where rhetoric, denial of facts and cherry-picking get more upvotes than scholarly logical argument - I detail some of these in the following section.

 

Attempts to evade the contradiction

Various people have come up with weird notions in attempts to evade the inherent flaw in Gödel’s proof. In this section I point out the fallacies in some of these bizarre attempts to rescue Gödel’s proof:

 

Weird idea 1: Gödel didn’t intend an equivalence of his Z function and the Gödel numbering function

This assertion claims that Gödel did not really mean what he actually wrote in his Relation 17:

Z(n) ≡ n N [R(1)]

Z(n) is the number-string for the number n.

 

where Gödel had previously defined that words in italics indicate numbers that are Gödel numbers of the formal sequence that is described by that same word without italics, and a number-string is a sequence of the form fff…0, i.e: Z(n) is the Gödel number of n, in other words:

Z(n) = Φ(n)

where Φ is the Gödel numbering function, where Gödel is clearly assuming that sequences of the form fff…0 are also valid as numbers for number-theoretic relations.

 

It’s not as if there’s any room for nuance in Gödel’s statement in his original German:

Z(n) ist das Zahlzeichen für die Zahl n

and a straightforward word for word translation gives:

Z(n) is the number-string for the number n

which is why all translations of this statement are essentially the same. Furthermore, when van Heijenoort was making a translation of Gödel’s paper, there was frequent communication between them Over the period 1963-1965, see Kurt Gödel, Collected Works Volume V, Oxford University Press, 2013. since van Heijenoort wanted Gödel’s full approval of his translation before publication. In view of the fact that in that communication Gödel made numerous finely detailed suggestions, it’s clear that if Gödel had ever thought that there was anything wrong with the statement above regarding Z(n), he had ample opportunities  (over a period of over 30 years) to clarify what he had actually intended if it was not as he had written it in 1930.

 

Moreover, further evidence that Gödel intended exactly what he wrote in his relation 17 - rather than making some sort of slip-up - can be seen by his use of the number-theoretic function Sb (f  v/y) in Proposition 5 of his paper. This function Sb (f  v/y) is defined by Gödel (his relation 31) to correspond to the substitution of the free variable V of the formal expression F by the formal expression Y where f = Φ(F), y = Φ(Y), and v corresponds to the formal variable V, where both  f  and  y must be Gödel numbers in order that the relationship between the formal expressions corresponds to the relationship between the corresponding Gödel numbers.

 

And Gödel asserts in his equation 3 of his Proposition 5 that (note that here, for simplicity, the Sb function has only one free variable; this makes no difference to the principle involved):

(A) R(x) ⇒ Bew{ Sb[r  u ⁄ Z(x) ] }

where Z(x) takes the place of the third variable y, and which has to be a Gödel number to preserve the aforementioned correspondence between:

  • relationships between formal expressions, and
  • relationships between the Gödel numbers of those expressions,

so there is neccesarily an assumption that Z(x) is a Gödel number, i.e: Z(n) = Φ(n).

 

And we note that an expression such as:

(B) R(x) ⇒ Bew{ Sb[r  u ⁄ Φ(x) ] }

would be assuming that the Φ function is a number-theoretic function, which is clearly not the case. Instead, from the definition of the Sb function, it follows that equation (A) assumes the validity of:

(C) R(x) ⇒ Bew{ Sb[r  u ⁄ w ] }

where w = Z(W) = Φ(W),

where Φ is the Gödel numbering function,

and W is some expression that satisfies the domain of the free variables of both the Z function and the Φ function (i.e: with restrictions on the domain of W).

 

On top of the above, in van Heijenoort’s approved translation, footnote 43 includes the parenthetical addition:

[Precisely stated the final part of this footnote (which refers to a side remark unnecessary for the proof) would read thus: "REPLACEMENT of a VARIABLE by the NUMERAL for p."].

This refers to Gödel’s equation 12, r = Sb(q  19⁄Z(p) ), where the variable y of the number-theoretic Sb (x  v/y) has been substituted, not by p, but by Z(p), thus confirming that, according to Gödel, Z(p) is precisely the same as the Gödel number of p - since in van Heijenoort’s translation the corresponding term for a number given by Gödel’s numbering function is indicated by small capitals, i.e the "NUMERAL" for p is the Gödel number of the “numeral” p, and in that translation a “numeral” is defined as a formal sequence with the format of 0, f0, ff0, fff0, … or a, fa, ffa, fffa, … . This also confirms that Gödel considered sequences so formatted as valid values for his number-theoretic relations.

 

In short, all the evidence demonstrates that the claim that in reality Gödel did not intend what he had actually written is an absurd fantasy denial of the facts of the matter. One will also find instances of cherry-picking, for example, remarking on how hundreds of very clever perspicacious mathematicians have examined Gödel's proof and none of them found anything wrong with it. What they omit to add is the concomitant fact that none of these very clever perspicacious mathematicians ever made any negative remark about Gödel's assertion that the function Z(n) is the Gödel number of n - which, according to the naysayers, is not actually literally correct. Gödel’s Proposition 5 assumes an equivalence of his Z function and the Gödel numbering function, over a restricted domain of the free variables, and as demonstrated above, that results in a contradiction.

 

Further weirdness

Some people try claiming that Gödel could not have intended any sort of equivalence of the two functions on the grounds that the domain of the free variable of Z is objects that are natural numbers, while the domain of the Gödel numbering function is strings of symbols. But every possible substitution of the variable of Z is a string of symbols (where a single symbol is a string of length one), and some of the strings of symbols of the formal system satisfy the definition of natural number, i.e: they satisfy the Peano axioms, hence, over a restricted domain the Gödel numbering function and the Z function both have a domain that satisfies the definition of natural numbers. And as noted above, Gödel approved a translation that includes a direct confirmation of the equivalence.

 

Even further weirdness

In an attempt to show that Gödel did not really intend what he wrote regarding the Z(n) function and his numbering function, some people try this notion: “Just leave out the informal parts of Gödel’s paper” and they claim: “the proof still follows from the formal parts of the paper”. In fact, it doesn’t, since Gödel never furnished any formal proof whatsoever of his Proposition 5, simply giving an informal outline of how he thought a proof might proceed. So if we leave out the informal parts of Gödel’s paper, then there isn’t any proof whatsoever - correct or incorrect - of his incompleteness result.

 

Weird idea 2. Gödel meant that there is another function involved

This claim is often associated with a misunderstanding of the term “NUMERAL” where it occurs in the English translation by Jean Van Heijenoort, where in Relation 17, Van Heijenoort translates as:

Z(n) is the NUMERAL denoting the number n.

 

Here it is important to understand that here the term “NUMERAL” is not used according to its meaning in English - Van Heijenoort uses the term “NUMERAL” in small capitals to indicate the Gödel number of “numeral”, which was previously defined to be the term that applies to any string of symbols of the formal system of the form fff…0, so that what the above line is saying is that Z(n) is the Gödel number of the formal string fff…0 that is n. Van Heijenoort’s choice of the terms “numeral” and NUMERAL is rather unfortunate since the term “numeral” already has its own specific meaning in English and this has led many people to believe that there is some sort of association of the term with the usual English usage - a belief that is completely mistaken (this is why I chose the term “number-string” instead in my translation).

 

The claim is that while the variable of the Z function has the domain of natural numbers, the restricted domain of the Gödel numbering function is formal sequences of the form fff…0 and which, according to the claim, are not numbers, so that Gödel actually meant:

Z(n) = Φ(s)

and that Gödel meant that there is some sort of connection between the n in the Z function and the s in the Gödel numbering function. This, of course, is invoking an additional function that does not appear anywhere within Gödel’s paper, either explicitly or implicitly, so that the claim entails either:

(A) Z(n) = Φ(s), where n = function (s), i.e: Z( function (s)) = Φ(s)

or

(B) Z(n) = Φ(s), where s = function (n), i.e: Z(n) = Φ( function (n))

 

where these imagined functions that are conjured out of nothing are often given a name that includes the word “numeral”.

 

The obvious error in the claim is the fact that the formal sequences of the form fff…0 are, in fact, natural numbers since they satisfy the Peano axioms.

 

Besides that, if the domain of n is natural numbers and the domain of s is not natural numbers, then (A) fails, since there the function (s) cannot be a number-theoretic function since, by the claim, its input values are supposedly not natural numbers, but sequences of the form fff…0, and so it cannot be substituted for the free variable of the Z function. Similarly (B) fails since there the function (n) cannot be an expression of the formal system, since its input values are supposedly natural numbers, and by the claim, natural numbers supposedly do not belong to the formal system, and so it cannot be substituted for the free variable of the Gödel numbering function. You can see an example of this sort of muddled thinking in Mendelson’s Introduction to Mathematical Logic (Chapman & Hall/CRC, 6th edition, 2015). After his Proposition 3.4, he describes a function n that converts “natural numbers” into numbers in the format of the formal system. Then in his Proposition 3.27, he defines a purely number-theoretic function Num( y ) and asserts that this Num( y ) is “the Gödel number of the expression y ”, i.e, that Num( y ) = Φ( y ). But y is not an expression of the formal system. It is a function that cannot be expressed in the formal system; the fact that when its free variable is substituted by a valid value the result is a particular expression is irrelevant - that would be confusing a function and an instance of its results. For some strange reason logicians seem to have difficulty discerning an obvious conflation of levels of language.

 

This conflation of language systems can be easily demonstrated, taking the notion (B) above that:

Z(n) = Φ( function (n))

Then we have that, for example:

function(4) = fffff 0, but we also have that function(4) = function(3 + 1)

But then it is also the case that:

fff 0 *+* f 0 = function(3) *+* function(1) = function(3 + 1)

where *+* represents addition in the formal system. 

 

And so, as before, we then have that:

Z(4) = Φ[  function (4)] = Φ[  function(3 + 1)],

but although:

function(3 + 1) = function(3) *+* function(1) = fff 0 *+* f 0

we have:

Z(4) ≠ Φ[  function(3) *+* function(1)]

and:

Z(4) ≠ Φ[ fff 0 *+* f 0].

Again, we have a contradiction.

 

The absurdity of claims of the above sort is mind-bogglingly obvious, yet the authors of such claims seem to be totally unaware of the nonsensical nature of the claims. The irony is that the idea that the variables of the Z function and the Gödel numbering function are inherently different actually points to the underlying cause of the contradiction - that the Z function and the Gödel numbering function belong to different levels of language, where the Gödel numbering function is, in fact in a meta-language to the Z function. This is analyzed in detail in my paper on Gödel’s proof, see Meta-Language and ‘Number-Theoretic Relations’.

 

Weird idea 3. Gödel’s Proposition 5 does not involve Gödel numbering

This involves the claim that Gödel numbering is not required for Gödel’s Proposition 5, that “Goedel numbering does not intrinsically show up at all” in that proposition. Apart from the patently obvious fact that Gödel did not provide a proof of his Proposition 5, instead assuming it to be obvious, the reality is that the statement of the proposition does require Gödel numbering, being:

Proposition V: To every recursive relation R(x1 … xn) there corresponds an n-place relation-string r …

and by Gödel’s definition of relation-string this is:

Proposition V:To every recursive relation R(x1 … xn) there corresponds a Gödel number r, where r is the Gödel number of a formula with n free individual variables ... …

which unambiguously refers to Gödel numbering, confirming that Gödel numbering is an intrinsic prat of the proposition.

 

Inconsistent systems

Only those who are unaware of basic tenets of mathematics (or those who deliberately ignore them) think that, by carefully choosing the terms and expressions used in an argument in order to avoid any explicit appearance of a contradiction, in some way serves to demonstrate that the system and its assumptions cannot result in any implicit contradiction - that is, that the system is inherently consistent. That, of course is nonsense, and if it can be demonstrated that the assumptions of the system can result in a contradiction, that means that the system is inconsistent and hence any result given by that system is mathematical gibberish. Note that Gödel’s failure to give a detailed argument for his Proposition 5 helps to hide the fact that his assumptions regarding that proposition are intrinsically contradictory.

 

Note that you can see further wacky ideas and my responses in the comments section below.

 

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.”  The convicted paedophile ex-professor Peter Smith, although he is a staunch advocate of Gödel’s proof, acknowledges in his paper, PDF 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 PDF 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 not only content to make the same intuitive assumption, but they also try to avoid any mention of this assumption when they talk about Gödel’s proof, for example see David Pierce on Gödel’s incompleteness proof. But 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 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.⁠ 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.  Also see Gödel’s flawed assumption of equivalence.

 

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.

 

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 Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem

 

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 to check that I have not changed the substance of my demonstration of the flaw in Gödel’s proof, 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 webpage 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

Bertrand Russell remarked:

“If a man is offered a fact which goes against his instincts, he will scrutinize it closely, and unless the evidence is overwhelming, he will refuse to believe it. If, on the other hand, he is offered something which affords a reason for acting in accordance to his instincts, he will accept it even on the slightest evidence.”  Bertrand Russell, in Ch. VI: International relations in Roads To Freedom, G. Allen & Unwin 1918.

 

It might be said that Russell did not go far enough - I would rewrite it as:

If a person is offered a fact which goes against their instincts and beliefs, they will carefully scrutinize the evidence to seek a flaw in it - but if they fail to find any flaw in it, they will not necessarily concede - instead they may simply ignore that evidence, and continue to refuse to believe it. If, on the other hand, they are offered something which in accordance with their instincts and beliefs, they will be inclined to accept it without any detailed examination of any available evidence.

 

And in the 1950s, Leon Festinger coined the term cognitive dissonance, and remarked: Leon Festinger: “When Prophecy Fails: A Social & Psychological Study of a Modern Group that Predicted the Destruction of the World”, Harper-Torchbooks, 1956, ISBN: 0-06-131132-4. The psychologist Carol Tavris has expanded on Festinger’s work and has written a book on the subject (co-written with Elliot Aronson): “Mistakes were made (but not by me): Why we justify foolish beliefs, bad decisions, and hurtful acts”, Harcourt, 2007.

A man with a conviction is a hard man to change. Tell him you disagree and he turns away. Show him facts or figures and he questions your sources. Appeal to logic and he fails to see your point. But suppose he is presented with evidence, unequivocal and undeniable evidence, that his belief is wrong: what will happen? The individual will frequently emerge, not only unshaken, but even more convinced of the truth of his beliefs than ever before.

 

And this is precisely what has happened with Gödel’s proof - I have provided a detailed analysis which the vast majority of people reject, even though they cannot find any error in it, while on the other hand, they are perfectly content to accept Gödel’s Proposition V, a crucial assertion in Gödel’s proof, for which Gödel never provided any detailed proof (and neither has anyone else done so).

 

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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