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Copyright   James R Meyer    2012 - 2024 https://www.jamesrmeyer.com

On Web Forums and Gödel’s proof

While the internet does contain much useful information, it is unfortunate that it also is a source of much misinformation. And it would appear that many forums attract persons who have more enthusiasm than knowledge and understanding.


This page is included because it was discovered that certain persons had indicated that they thought that a certain forum supposedly showed errors in the paper on Gödel’s incompleteness proof: The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.


And a certain Russell O’Connor has a web page (O’Connor blog: here) where he indicates that he believes that this is the case.


At one stage I contributed to a forum discussion, but after a time it became obvious that no-one on the forum was interested in rational argument, so I stopped contributing. I made a final post on the thread in Feb 2009, see Google groups sci.math forum where I set down the facts of the matter.


There was a further contribution which I address here to set the record straight. I had said in the forum that a contributor named Jack Markam, who also uses the alias MoeBlee, was incorrect to state that Gödel meant that Z(n) is the combination of symbols of the form 0f0ff0fff0, … for the number n. Markam/MoeBlee in a later posting attempted to support his contention by:


Gödel wrote:

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

That is found on page 604 of Van Heijenoort’s ‘From Frege To Gödel’, regarding item 17, as part of Van Heijenoort’s translation of Gödel’s paper ‘On Formally Undecidable Propositions’. And how ironic that Meyer mentions that his criticism of my remarks is based on translations by Meltzer and Hirzel, which, though perhaps unknown to Meyer, are translations NOT approved by Gödel, unlike the Van Heijenoort translation that WAS approved by Gödel.


The fact is that the actual translation used is utterly irrelevant - the differences in the translations in this respect concerning the function Z(n) are differences only in the assignation of names and are completely immaterial - so it doesn’t matter whether Gödel approved the translation or not.


So, yes, indeed, in Van Heijenoort’s translation, Gödel does indeed state:

Z(n) is the NUMERAL denoting the number n ”,

where the word NUMERAL is in capitals.


But what Markam/Moeblee fails to realise, is that where the word NUMERAL is in capitals, it has a completely different meaning to the normal meaning of numeral, and this fact is specifically stated earlier in that translation of Gödel’s paper. In the translations of Hirzel (Footnote: Hirzel’s translation:
We will denote the classes and relations on natural numbers which are associated with the meta-mathematical concepts, e.g. ‘variable’, ‘formula’, ‘proposition-formula’, ‘axiom’, ‘provable formula’ etc., in the above mentioned manner, by the same word in small caps. The proposition that there are undecidable problems in system P for example reads like this: There are PROPOSITION-FORMULAE a, such that neither a nor the NEGATION of a is a PROVABLE FORMULA.
Hirzel’s translation is available online at PDF Hirzel’s translation of Gödel’s Incompleteness paper.)
and Van Heijenoort, (Footnote: Van Heijenoort’s translation:
The relations between (or classes of) natural numbers that in this manner are associated with the metamathematical notions defined so far, for example, “variable”, “formula”, “sentential formula”, “axiom”, “provable formula”, and so on, will be denoted by the same words in SMALL CAPITALS. The proposition that there are undecidable problems in the system P, for example, reads thus: There are SENTENTIAL FORMULAS a such that neither a nor the NEGATION of a is a PROVABLE FORMULA.
Van Heijenoort’s translation is not available online; it can be found in the book: From Frege to Gödel: A Source Book in Mathematical Logic, pub Harvard University Press, details From Frege to Gödel: A Source Book in Mathematical Logic: here.)
Gödel states that by the Gödel numbering system, every combination of symbols of the formal system has a corresponding Gödel number and that when a word appears in capitals, that word is not meant to be read as denoting the concept of the word in its normal usage, but is to be read as the Gödel number that corresponds to that concept. So for a number-theoretic relation, if the numbers are Gödel numbers, then there is a corresponding relation between formal system symbol combinations - the formal system symbol combinations that correspond to those Gödel numbers.


So, the fact is that in Van Heijenoort’s translation (and Hirzel’s), the word ‘numeral’ and the word ‘NUMERAL’ are actually two different words, with quite distinct meanings - which is quite different to normal English usage. It is easy to see why someone with a superficial knowledge of Gödel’s paper might make the incorrect assumption (as does Markam/relations/functionsMoeBlee) that they both have the same meaning, whereas that is quite clearly not the case.


And so, for example, when Gödel states of his relation 46, which is Bew(x), that (Van Heijenoort’s translation):


that is not an assertion that x, which is a variable for a number, is actually a provable formula of the formal system. That, of course, would be absurd. No, it is an assertion that x is a Gödel number that corresponds to a formula of the formal system that is provable by the system; the point being that if, for any number x, Bew(x) is correct, then the formal formula that corresponds by Gödel numbering to the number x will be provable in the formal system. (Footnote: Provided the Gödel numbering system and the relations are correctly defined.) In Meltzer’s translation the same applies, except that Meltzer uses italics to indicate the difference rather than capitals. (Footnote: Meltzer’s translation was freely available online for many years but it now seems that many sites want you to pay for it, and hide it behind a pay wall, but you may find it online at PDF AltExploit - Meltzer’s translation and PDF Minnesota State University - Meltzer’s translation.)


In Van Heijenoort’s translation, a ‘numeral’ (small letters) is actually defined as a combination of signs that has the form of 0f0ff0fff0, …

but a ‘NUMERAL’ (in capital letters), has an entirely different definition, which is “the Gödel number”,

so that the phrase:

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

is defined as: “Z(n) is the Gödel number of the number n ”.


As for Meltzer’s translation, in fact that translation states:

Z(n) is the number-sign for the number n ”

with number-sign in italics. And in Meltzer’s translation, a number-sign is similarly defined, so the word number-sign (in italics) in Meltzer’s translation is also defined as “the Gödel number”, so that in Meltzer’s translation, the phrase:

Z(n) is the number-sign for the number n ”

is defined as

Z(n) is the Gödel number of the number n ”.


Now, Markam/MoeBlee admits that he does not have an in-depth knowledge of Gödel’s proof, and clearly he is unaware that the differences in the translations have no bearing whatsoever on Gödel’s assertions regarding the function Z(n), which is a straightforward function of number theory. Nevertheless he has the audacity to claim that I do not understand Gödel’s proof, saying:


What is so poignant is that Meyer claims to understand Gödel’s work … yet he is quite confused about Gödel’s paper and doesn’t understand it even as well as someone, such as me, with a fair but not expert understanding as merely a hobbyist at not much more than an undergraduate level. And even on a point of basic scholarship, which HE makes such a big deal of, he’s dead wrong: My remark is completely faithful to a translation, UNLIKE those he relies upon, that Gödel APPROVED.


Poignant? The simple fact is that it is Markam/MoeBlee, despite his claims to the contrary, who does not have sufficient knowledge or understanding of Gödel’s paper. The fact remains that nothing that Markam/MoeBlee has said indicates any error in my paper detailing the fundamental flaw in Gödel’s paper.


As for the claim that I don’t understand Gödel’s proof, in fact I have created a walk-through guide to Gödel’s original incompleteness proof, and which is intended to be read alongside the paper. I suggest that you take a look at it; it may help you to decide whether it is reasonable to suggest that I do not understand the proof. When I decided to make that guide, I did an intensive search to see if there was anything similar already published anywhere, but I found no detailed guide at all - so I constructed my guide from scratch - which I would claim is hardly indicative of someone who does not understand Gödel’s proof.


The fact is that each of the three translations mentioned assert that Z(n) is the Gödel number of the number n. And this gives rise to the crucial point that Markam/MoeBlee (among others) prefers to ignore. That assertion is logically untenable.


The Gödel numbering function (which here we call GN(x)) is a function that can only take values of x that are symbol combinations of the formal system, whereas the Z(n) function is a number-theoretic function that can take values of n that are numbers, without those numbers being in any specific format. That is to say, for example, if we let n = 4, and if we allow that fff0 = 4. then we have Z(4) = Z(fff0).


Gödel’s assertion is that for all n as a natural number, Z(n) = GN(n), without any format being specified for n. So that we could have Z(fff0) = GN(fff0). But by applying the same logic, that logic indicates that we could also have ZN(4) = GN(4). But we cannot have GN(4), since the value for x in the Gödel numbering function GN(x) must always be in the format of the formal system. Always - there is no other possibility.


Here we see how Gödel fails to apply logically rigorous analysis. His vaguely expressed assertions tend to hide the fact that in so doing he is confusing different languages, resulting in assertions that are logically absurd. In fact, I had made this very point repeatedly in the forum to which Markam/Moeblee was replying, a point that he failed to address, conveniently ignoring the facts of the matter.

It might be noted here, that Markam/Moeblee also had claimed that what Gödel really meant, rather than what he actually stated, was: (Footnote: I leave the reader to ponder why Gödel’s proof, for so long acclaimed as a pinnacle of mathematical logic, should now need propping up by attempts to rewrite what he actually stated.)

Z(x) is the Gödel number of the ‘numeral’ for x ”, that is, that:

Z(x) = GN(the ‘numeral’ for (x))

where ‘numeral’ means of the form 0, f0, ff0, fff0, …


I had already pointed out that that expression was logically invalid in the same way as the expression:

Z(n) is the Gödel number of the number n ”.

This is so, since it could not be an expression of the meta-language of Gödel’s proof - because Z(x) is a number-theoretic function and is an expression of the sub-language of number-theoretic relations, which means that the variable x in Z(x) cannot at the same time be a variable of the meta-language of Gödel’s proof. And that this means you cannot use:

Z(x) = GN(the ‘numeral’ for (x))

as a valid step in Gödel’s proof, since that proof is expressed in a language that is a meta-language to both the sub-language of number-theoretic relations, and to the sub-language that is the formal system. It is an expression that is logically absurd. See also the web-page Gödel’s Proposition V which discusses Gödel’s use of the Z function in detail.


And the fact remains that nothing that Markam/MoeBlee has said indicates any error in my paper.


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