It’s easy to show that a claim that Gödel’s incompleteness proof is flawed is wrong - if it really is
This page was written to show how easy it can be to show an error in a claim that Gödel’s incompleteness proof is flawed - that is, of course, if the claim is wrong. So on this page we look at a paper that claims that Gödel’s proof is flawed, and it is explained why the paper is erroneous.
That paper has been provided by a certain Colin Dean. It now appears to have been taken offline.
It has to be said that Dean’s presentation does not help his case, being amateurish in layout, with his comments, arguments, conclusions and references all thrown together in a haphazard fashion, with numerous repetitions.
But here we are interested not in the presentation, but in the argument within it. When we get to page 36 we discover that Dean thinks Gödel’s proof is wrong for six reasons that he gives as:
- Gödel uses the axiom of reducibility
- Gödel uses the axiom of choice
- Gödel constructs impredicative statements
- Gödel miss uses (sic) the theory of types
- Gödel cannot tell us what makes mathematical statements true
- Gödel falls into 3 paradoxes
These are easily shown to be wrong, as follows:
Dean’s principal argument is based on the fact that Gödel’s proof is a proof regarding the particular formal system P that Gödel devised, essentially using the axioms of Peano arithmetic and the concepts behind Bertrand Russell’s Principia Mathematica. (Footnote: Russell, Bertrand, and Alfred North Whitehead, “Principia Mathematica”; 3 Volumes, 1910, 1912, and 1913, Cambridge University Press.)
As Dean points out, that formal system involves certain axioms; the one that concerns him here is the axiom of reducibility. Dean then argues that since one could give reasons why one should not necessarily accept that axiom, then one should not accept Gödel’s proof.
The problem with that argument is that Gödel simply devised that formal system P in order to demonstrate the essence of his argument. There is no reason in principle why Gödel’s argument could not be applied to other formal systems, which need not have that axiom.
The important point to observe is that the core of Gödel’s argument is based on the formal system being able to express straightforward relationships between numbers; those relationships do not require the axiom of reducibility that Dean refers to. Hence there is no reason to accept Dean’s argument that the axiom invalidates Gödel’s proof.
Dean refers to Gödel ‘using’ the axiom of choice, and quotes Gödel as saying:
This allows one, for example, to conclude that even with the help of the axiom of choice (for all types), or the generalized continuum hypothesis, not all propositions are decidable…
Dean is way off the track here. What Gödel said does not mean that Gödel has used the axiom of choice to obtain his result. What Gödel meant was that his proof of incompleteness could be generalised to apply to every formal system which includes a certain amount of arithmetic, even if one was allowed to use the axiom of choice in that formal system.
Dean’s argument here appears to be saying that since Gödel used the axiom of induction in his formal system, and that the axiom of induction is unacceptable, therefore we should reject Gödel’s argument. The problem with that assertion is that if you reject the idea of the axiom of induction in some shape or form you have to reject a huge chunk of mathematics. For example, Euclid’s proof that the number of prime numbers is infinite used the axiom of induction.
On that account Dean’s argument here simply cannot be taken seriously
This is basically the same as Reason 1, except that Dean’s complaint is that Gödel uses Russell’s theory of types in his formal system P. The reason why this cannot be considered to invalidate Gödel’s proof is the same – there is no reason in principle why Gödel’s argument could not be applied to other formal systems which do not use Russell’s theory of types.
Dean claims that Gödel’s proof does not tell us, in Dean’s words, “what makes a statement true”. But that is completely irrelevant, since Dean does not even start to suggest how Gödel’s failure to tell us “what makes a statement true” demonstrates any error in Gödel’s argument.
Dean states as his sixth reason that Gödel’s proof ‘falls into’ 3 paradoxes:
They are, according to Dean:
- The first paradox, according to Dean, is that:
‘[Gödel] says every formal system which satisfies assumption 1 and 2 ie based upon axioms - but [Gödel] said undecidablity (sic) is independent of axioms’
Here Gödel simply means that, regardless of what specific axioms the formal system uses, if those axioms are recursively definable (along with some other conditions) then there will be undecidable statements in that system. There isn’t any contradiction there.
- The second paradox, according to Dean, is that:
Undecidability is independent on nature of [Gödel’s formal system], yet is dependent on the nature of [Gödel’s formal system].’
This is essentially stating the same as the above and the rejoinder is the same.
- The third paradox, according to Dean, is that:
[Gödel’s formal system] is used to prove that a system has statements which cannot be proven true or false, but
[Gödel’s formal system] can only prove this if all its statement (sic) can be proven to be true, but
[Gödel’s formal system] has statements which cannot be proven true or false
thus it cant (sic) prove anything but it is used to prove if systems are undecidadble (sic) thus a paradox
Here Dean is simply wrong. Even assuming that he actually meant that Gödel’s formal system must be able to prove all its statements either true or false, there is no requirement in Gödel’s proof that that should be the case.
As well as the errors already noted, Dean makes several other errors, for example:
He claims that Gödel specifies that he uses the Zermelo axioms. In point of fact, Gödel does not claim to use the Zermelo axioms in generating his result of an ‘unprovable’ statement for the formal system P referred to in the proof.
What Gödel actually claims is that the result of the proof can be extended to include all formal systems that satisfy certain conditions – which he says, includes the Zermelo-Fraenkel axioms. And that simply means that Gödel is claiming that formal systems that are based on the Zermelo-Fraenkel axioms satisfy the conditions that Gödel stated; it does not mean that Gödel used those axioms to produce his result – he didn’t.
In the above it becomes apparent that Dean does not appreciate the difference between finding a fundamental flaw in a proof and quibbling over parts of a proof that can easily be sidestepped by a simple rewriting of the proof. No-one is going to accept such quibbles as a genuine flaw in a proof. To demonstrate an error in a proof, you must show that there is an error that is fundamentally inherent to the argument of the proof, so that there is no way of simply rewriting the proof to avoid the problem.
Finally, it was not at all difficult to demonstrate the errors in Dean’s claim that he has found errors in Gödel’s proof. So why should it be so difficult for any competent mathematician to do the same for PDF the paper on the flaw in Gödel’s proof - which has a similar word count?
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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