[cubefox]

The author of this article writes that the theorem cannot be proven in "Peano arithmetic". But that's only true if by that he means "first-order Peano arithmetic", a system which allows for absurd "non-standard numbers". When ordinary mathematicians talk about "Peano arithmetic", they arguably have the second-order induction axiom in mind, not the first-order infinite induction axiom scheme. And they most certainly have the natural numbers in mind, not some possibly absurd "numbers" with infinitely many predecessors. And in this normal version of Peano arithmetic, the theorem can be proven.

[tromp]

> When ordinary mathematicians talk about "Peano arithmetic", they arguably have the second-order induction axiom in mind

When they have the latter in mind, they call it second order arithmetic (or Z2), rather than Peano arithmetic (or PA) [1].

[1] https://en.wikipedia.org/wiki/Second-order_arithmetic

[daxfohl]

The link to "Peano arithmetic" at the top of the Goodstein page takes you to Peano axioms page. That page says Peano axioms are "close to" second-order arithmetic, and it also provides an informal distinction between Peano axioms and Peano arithmetic. But there's no wikipedia page for Peano arithmetic.

So I'm curious if this theorem is unprovable in Peano axioms, or just Peano arithmetic. If the latter, then the link at the top of the Goodstein page is rather misleading, unless you're paying close enough attention to notice the blurb about the distinction between Peano axioms and Peano arithmetic.

[thaumasiotes]

> But there's no wikipedia page for Peano arithmetic.

But there is such a page. It redirects to https://en.wikipedia.org/wiki/Peano_axioms#Peano_arithmetic_... .

[daxfohl]

Oh, I missed that! I'd searched google for the term, and it just returned the top-level Peano axioms page.

Anyway, updated the link on the Goodstein's Theorem page to point to that section specifically.

[cubefox]

Logicians and set theorists use this terminology. But everyone else just uses the second-order induction axiom when talking about arithmetic, without explicitly talking about first or second-order logic. Steven Shapiro made this point in his book "Thinking About Mathematics".

[YetAnotherNick]

According to different page in wikipedia, Peano axioms is second order but "Peano arithmetic" is first order.

> The ninth, final axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to second-order arithmetic. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.

[1]: https://en.wikipedia.org/wiki/Peano_axioms

[cubefox]

Yeah, but this terminology is fairly idiosyncratic. Probably a convention the main author likes to follow.

[throwaway81523]

No it's pretty standard. Peano's axioms from the 19th century had an induction axiom in second order logic, since it quantified over predicates. Peano arithmetic (PA), also called first order arithmetic, came later. It is a first order theory whose induction axioms are an infinite schema. To confuse things further, second-order arithmetic (SOA) is also a first order theory, whose objects are naturals and sets of naturals.

[cubefox]

It's pretty standard also to talk about "first-order Peano arithmetic" and "second-order Peano arithmetic". This is much more clear but inconsistent with the other usage which you describe.

Moreover, non-logicians don't talk about "first-order" or "second-order" logic at all. They just express the induction axiom in plain English, and in this case it is (as Stewart Shapiro argued) equivalent to the second-order axiom.

[throwaway81523]

Yes. I don't remember who called second-order logic "set theory in sheep's clothing". ;)

[cubefox]

Quine. His justification was quite bad though.

[Jaxan]

I think the article mentions this clearly in the introduction: “it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic).”

[cubefox]

Yeah, but this terminological distinction is not really justified. It should be simply between first and second order Peano arithmetic.

[ShamelessC]

That’s clarified almost immediately in the article.

Also, Wikipedia articles can have multiple authors.