[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".