
How Gödel’s Proof Works - theafh
https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/
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eindiran
For a detailed perspective on how the proof works, I highly recommend Ernest
Nagel and James Newman's book _Gödel 's Proof_ [0], mentioned in the article.
Alternatively, _Gödel Escher Bach_ by Douglas Hofstader is a classic which
serves as a great (and more accessible) introduction to the proof [1].

[0] [https://www.amazon.com/G%C3%B6dels-Proof-Ernest-
Nagel/dp/081...](https://www.amazon.com/G%C3%B6dels-Proof-Ernest-
Nagel/dp/0814758371)

[1] [https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-
Golden...](https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-
Golden/dp/0465026567)

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ProfHewitt
Actually, the [Gödel 1931] proof does _not_ work as advertised for Principia
Mathematica because the Gödel number of a proposition does _not_ include the
order of the proposition.

Consequently, the rules on orders of propositions mean that the [Gödel 1931]
proposition _I 'mUnprovable_ does _not_ exist in Principia Mathematica where

    
    
                I'mUnprovable <=> ¬⊦I'mUnprovable
    

For details see the following:
[https://papers.ssrn.com/abstract=3603021](https://papers.ssrn.com/abstract=3603021)

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dhy
With today's knowledge, Gödel's theorem should be regarded as a corollary of
the impossibility to solve the halting problem.

In my opinion, this is also how it should be taught in order to convey the
explanation for its truth in the most straightforward way.

After that, one may of course also note that in the old days, without a good
theory of computation, one had to resort to such hacks as Gödel numbers.

A good explanation including all the finer points can be found here:
[https://www.scottaaronson.com/blog/?p=710](https://www.scottaaronson.com/blog/?p=710)

~~~
ProfHewitt
Yes, this is the way to correctly prove "incompleteness", i.e that there are
preferentially undecidable propositions in powerful theories of computer
science.

See the following for a proof:
[https://papers.ssrn.com/abstract=3603021](https://papers.ssrn.com/abstract=3603021)

~~~
ProfHewitt
Of course, I meant to say "inferentially undecidable" instead of
"preferentially undecidable".

Being inferentially undecidable means that it is not the case that for every
proposition, there is either a proof of the proposition or its negation.

Inferential undecidablity implies "incompleteness" in the sense that it is not
the case that every true proposition can be proved.

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SilasX
Just finished a book club for Gödel, Escher, Bach, which gives a lay -- but
pretty detailed and thorough -- explanation of the proof. Here's how I'd say
lay out the key insights as they applied to the parts I didn't understand
before.

You want to use pure math to say "this statement cannot be proven". To do
that, you:

1) Express the concept of statements following from other strings by the
axioms or other theorems.

2) Express the concept of "this is a valid chain of statements, each of which
follows from the previous". (GEB calls it a proof-pair.)

3) Express the concept of "Statement A does not have such a valid chain"
(there exists no such chain with A at the end).

That allows you to say "Statement A cannot be proven."

From that point, it's a matter of extending it to the statement (call it G):
"There exists a statement S such that S has no proof and S statisfies criteria
A/B/C."

...and then constructing the criteria A/B/C such that G is the one statement
that can satisfy it. The details of how you construct the criteria so that you
are specifying G are the Gödel numbering that the Quanta article describes.

~~~
ProfHewitt
Unfortunately, the above proof sketch does _not_ work. Mathematics becomes
inconsistent if a proposition G is allowed such that G<=>¬⊦G.

For a simple proof that allowing such a G makes mathematics inconsistent,
please see the following:

[https://papers.ssrn.com/abstract=3603021](https://papers.ssrn.com/abstract=3603021)

~~~
SilasX
The proposition G in the sketch above just says G can't be proven, not that
it's false, just like the one in GEB.

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avmich
Charles Petzold's "Annotated Turing" is a book about Turing's article, which
provides another approach to this problem. Turing machine, described in the
article, is arguably a more - convenient? generalized? - device for similar
areas.

Yes, Godel's theorem is mentioned more than once.

~~~
ProfHewitt
You might be interested in knowing that there are some nondeterministic
computations that cannot be performed by a Turing Machine.

For a simple example, see the following:

[https://papers.ssrn.com/abstract=3418003](https://papers.ssrn.com/abstract=3418003)

[https://papers.ssrn.com/abstract=3459566](https://papers.ssrn.com/abstract=3459566)

