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Comment by xyzzyz | Hacker News Reader

I’m a mathematician who spent many hours thinking about Goedel theorem and adjacent topics. The author in a self-deprecating way says that he’s not a mathematician, but just a programmer. Bear no mind to that: this is one of the best popular expositions of Goedel theorem I’ve seen. Everything is very accurately explained, there are no silly mistakes and untruths one often sees mentioned in context of Goedel theorem, and even the original proof is sketched in a very approachable manner, striking a very good balance between getting across all crucial ideas, and skipping most of the distracting technical aspects (which are important and interesting, but I think only to mathematicians, haha). Great job all around.

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Dear xyzzy,

Unfortunately, you missed that the author's proof does not actually prove inferential undecidability (sometimes called "inferential incompleteness") of Russell's Principia Mathematica for the following reason:

     The [Gödel 1931] proposition *I'mUnprovable* does **not** exist in Principia Mathematica because it violates restrictions on orders of propositions that are necessary to avoid paradoxes (such as Russell's Paradox). Gödel numbers (and in the author's case Lisp expressions) leave out the order of a proposition with the consequence that the Diagonal Lemma **cannot** be used to construct the proposition *I'mUnprovable*.

Furthermore, existence of the proposition I'mUnprovable contradicts the following fundamental theorem of provability that goes all the way back to Euclid:

     A theorem can be used in other proofs.

See the following article for further details:

https://papers.ssrn.com/abstract=3603021

Is case anyone is wondering this is MIT professor Emeritus Carl Hewitt

https://en.wikipedia.org/wiki/Carl_Hewitt

A whole bunch of previous posts have be reposted to this discussion (maybe by a bot?).

Instead of plowing through the disjointed repostings,

readers may be better off looking at the articles linked in https://professorhewitt.blogspot.com/.

Also, there is a video here:

https://www.youtube.com/watch?v=PJ4X0l2298k

Wikipedia is contentious and often potentially libelous.

See the following for more up-to-date information:

https://professorhewitt.blogspot.com/

I'm going to need someone to come along and write up a layman's summary of this article. :) Does it change the end result of Gödel? Or is it basically the equivalent of saying what some restricted programming languages are doing, that we can get "good enough" results and not have these problems by restricting what we can do in the language?

Hi GreatQuux!

The article linked below explains why [Gödel 1931] did not prove inferential undecidability of Russell's Principia Mathematica and likewise why the formalization of [Gödel 1931] in Lisp proof being discussed is also invalid:

https://papers.ssrn.com/abstract=3603021

However, the article linked above does have a correct proof of inferential undecidability (also known as "inferential incompleteness").

Would be happy to respond to any questions that you might have.

What do you mean by "inferential" and why are there restrictions on what statements are valid?

Hi Mike!

The word "inferential" has to do with being able to be logically inferred, that is, deduced.

Russell's Principia Mathematica specified that each proposition must have an order to block paradoxes such as Russell's paradox.

See the article linked above for further explanation.

That doesn't explain why there must be such a restriction.

So in layman's terms you're saying the system the author tries to prove incomplete by definition removed the parts that would prove it's inconsistent?

Secondly, in the PM-Lisp it doesn't necessarily prove that theorem a proves b, it just shows that b can be a successor of the formulas in a

Hi Ethn!

Existence of the [Gödel 1931] proposition I'mUnprovable is inconsistent with the following theorem to the effect that theorems can be used in proofs:

      ⊢∀[Proposition Ψ] (⊢Ψ)⇒Ψ

Axioms can be used in proofs, and they are not provable.

If a proposition is not provable, yet raises no issues if regarded as true, then it can be added as an axiom and used in theorems.

It turns out that for powerful theories of Computer Science,

there must exist infinitely many propositions that are

inferentially undecidable, that is, can be neither proved nor disproved.

However, the propositions cannot be specified constructively

and so are not very interesting.

Currently, there seem to be no propositions interesting to

practical Computer Science that are provably inferentially

undecidable.

For a statement so short it would be easy to avoid using jargon...

The following theorem says that every theorem can be used in another proof:

         ⊢∀[Proposition Ψ] (⊢Ψ)⇒Ψ

The above theorem is completely standard mathematical notation.

What character set or input method is used in creating this mathmatical characters?

Just to be clear

      ⊢∀[Proposition Ψ] (⊢Ψ)⇒Ψ

is not jargon. Instead, it is standard mathematics.

It absolutely is jargon. The fact that you explained it by saying it's "mathematics" instead of "English" suggests as much. Jargon means that it's technical language specific to a field and not used in everyday vernacular. What you wrote can't even be typed on a standard keyboard. It's jargon.

I'm pointing it out because its use is not so innocent. Jargon is often used to obfuscate meaning, especially when it means something that would otherwise be easy to say plainly.

This brought a big smile to read, thank you :)

There is an extraordinarily minor typo: “PM-LIsp”

I point it out only because your work here is so good that it feels wrong not to smooth out any small splinters.

Thank you for making this!

Oi, thanks for the kind words and the catch! Updated (should take a few minutes to show up)

I think I maybe noticed another typo... The last two axioms don't have matching parentheses? It looks like one more is opened than closed if I counted right.

What an excellent recommendation to the article. Thank you!

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