Flaw in Godel's Proof: Accepted by Mathematicians?

[windy miller]

I have heard there is a flaw in Godels proof ? For example the claim here:
https://www.jamesrmeyer.com/pdfs/FFGIT_Meyer.pdf
Is this accepted by other mathematicians or is it a fringe view ?

 

[stevendaryl]

I have heard there is a flaw in Godels proof ? For example the claim here:
https://www.jamesrmeyer.com/pdfs/FFGIT_Meyer.pdf
Is this accepted by other mathematicians or is it a fringe view ?

Godel's theorem has been proved in dozens of ways by hundreds of people. It's certainly possible that his original presentation had mistakes, but the conclusion is certainly not in doubt.

I would go with fringe.

 

[windy miller]

Well actually I am interested in the issue of the robustness of the original proof, was it flawed ? So the fact that "Godel's theorem has been proved in dozens of ways by hundreds of people." is interesting but it doesn't address the issue I am interested. which has nothing to do with how the theorem is viewed today.

 

[stevendaryl]

The key components of Godel's proof are actually pretty simple. The difficulty is filling in the details. But the proof has actually been gone over with a proof-checking/theorem-proving machine, which gives a lot of credence to it: https://www.cl.cam.ac.uk/~lp15/papers/Formath/Goedel-logic.pdf (This is actually done for the theory of hereditarily finite sets, rather than PA, but they are basically equivalent.)

The bare bones of Godel's proof has the following elements:

1. A scheme for coding formulas of arithmetic as numbers. This is clearly doable, since a formula can be written in ASCII, which associates every string of symbols with a number.
2. A formula $P(x)$ in the language of PA with the property that $P(x)$ is true whenever $x$ is the code of a formula that is provable by PA. This is more complicated to show, but we know that you can write proof checkers as computer programs, and we know that every computer program can be translated into a partial recursive function, and partial recursive functions can be defined in arithmetic.
3. A fixed-point operator. For any formula of arithmetic $\phi(x)$ with one free variable, there is a corresponding sentence of arithmetic, $G$ with code $g$ such that $G \leftrightarrow \phi(g)$
4. Putting 3&2 together gives a sentence $G$ such that $G \leftrightarrow \neg P(g)$ ($G$ is true if and only if it is not provable)
5. From that, it follows that if G is provable, then it isn't true, and so PA can prove false sentences.
6. If G is not provable, then it follows that G is true, and so there are true sentences that are not provable in PA.

So we have a weak form of Godel's theorem: PA is either incomplete or unsound (unsound meaning that it proves false statements). To actually get that PA is inconsistent, you need a few other facts about PA:

• For any statement $S$ with code $s$, if $S$ is provable, then so is $P(s)$.
• $G \leftrightarrow \neg P(g)$ is provable in PA (not just true)

So if $G$ is provable, then so is $P(g)$. But $G \leftrightarrow \neg P(g)$. So if $G$ is provable, then so is $P(g)$ and $\neg P(g)$. So PA is inconsistent, since it proves contradictory statements.

 

[windy miller]

Sorry I am not a mathematician or even very familiar with mathematics. I am just a layman interested in the history of ideas which is why I put high school level on the header. I am really just trying to understand if what's was considered proved by some was not considered prooven by others. Most of your reply i wasn't able to follow I am afraid. However I did pick up on that you seem to distinguish between a weak form and a strong form. So was one proved originally and the other not?

 

[fresh_42]

I have heard there is a flaw in Godels proof ? For example the claim here:
https://www.jamesrmeyer.com/pdfs/FFGIT_Meyer.pdf
Is this accepted by other mathematicians or is it a fringe view ?

Just as a side note: This is not an acceptable source and likely for good reasons. You shouldn't waste time on such questionable sources and worst case, will have to unlearn statements! I don't know Gödel's original proof, and experience says, that original papers are better written in books decades later. There are textbooks about logic in abundance and certainly even many for free or small money, if used. These are certainly better sources than colorful websites of crackpots.

This thread is closed.