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Comment by kaba0 | Hacker News Reader

It’s similar to claiming someone made a perpetuum mobile — extraordinary claims going against basically every mathematician/computer scientist since Gödel require extraordinary evidence. And the result is quasi always that the one claiming, looked over some small but crucial detail.

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Veritasium, et. al. overlooked the crucial aspect that

orders on propositions play in maintaining the consistency

of foundations.

Neglecting to include the order of a proposition in in its

Gödel number makes it possible to mistakenly conclude that

the proposition I'mUnprovable exists as a fixed point of a

recursive definition.

See the following for details:

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

Isn’t the order of a proposition included in its Gödel number?

Each proposition is assigned to an increasing prime power, and the increasing list of primes has total order, such that swapping propositions yields a distinct Gödel number.

I think what ProfHewitt means here (based on other writing I've found, as he is frustratingly low on details in these conversations) is "order" in the sense of "first order logic", "second order logic"; not in the sense of "first proposition, second proposition" etc.

His claim is that the proposition "This proposition is not provable", formalized as "P equivalent to P is not provable" is not well formed in a typed logic, as "P"'s type is first-order, while "P is not provable"'s type is second-order. Therefore, his claim is that the proposition is simply not well-typed and therefore not interesting. Godel's proofs were discussing an untyped logic, but according to Hewitt that is not an accurate representation of mathematics.

I don't think anyone in the space agrees with him, though, as far as I could tell from some cursory reading.

[Gödel 1931] was for a system for the foundations of

mathematics with orders on propositions.

The [Gödel 1931] proposition I'mUnprovable is certainly of

historical interest and may perhaps be of philosophical

interest.

However, including the proposition I'mUnprovable in

foundations makes the foundations inconsistent.

[Gödel 1931] encoded the characters of a proposition in its

Gödel number but did not include the order of the proposition.

Could you provide two statements with equal Gödel numbers but distinct proposition orders, and a detail of how you compute the Gödel numbers?

If two propositions have the same characters, then they have the same Gödel numbers.

Could you provide two statements with equal Gödel numbers and the same characters in different orders, and a detail of how you compute the Gödel numbers?

If two propositions have the same characters in the same order,

then they have the same Gödel numbers.

Unfortunately, Gödel number of a proposition leaves out the order of the proposition :-(

If two statements have the same characters in the same order, how are they not the same statement? And why would that make Gödel’s statement invalid in Principia Mathematica?

There might be two propositions of different orders that have the

same characters in the same order.

I know of several computer verified proofs of the incompleteness theorem. See entry 6 of https://www.cs.ru.nl/~freek/100/ for a small catalogue. How does this relate to your objection? Are there bugs in the verifiers? Or did people verify the wrong theorem statement?

Proofs forgot to include order of proposition in its Gödel number.

So you say that the computer verified proofs of the incompleteness theorem of verifying the wrong theorem? Because there can't be bugs in a computer verified proof.

(Of course there can theoretically be bugs in a computer verified proof. If there are bugs in the verifier. But if 4 independent proof assistants verify a proof of a theorem, this is extremely unlikely.)

A proof is only as good as definitions and axioms used in the proof.

All of the computer-verified proofs of existence of [Gödel

1931] proposition I'mUnprovable left out the order of

proposition in its Gödel number :-(

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