What ZF can do and Peano's axiom cannot.

Question

I am interested in how much math can be done from Peano's axioms and what can't.

What is there in the mathematics done with ZF that cannot be done with Peano's axioms?

Answer 1

[I'm a little surprised this isn't a duplicate. But I looked through "[peano-axioms] ZF" on MSE and "[theories-of-arithmetic] ZF" on MO, and couldn't find anything. I also couldn't locate a single article on Wikipedia answering this, although of course there are Wikipedia entries for the various theorems.]

First, a very readable book with lots on this question: Stillwell's Roads to Infinity.

The granddaddy of statements unprovable in PA but provable in ZF is Con(PA), the consistency of PA. Gödel's second incompleteness theorem shows that Con(PA) isn't provable in PA (unless PA is inconsistent, a possibility I will ignore from now on). On the other hand, in ZF we can prove that there is a model of PA, namely $\mathbb{N}$ (i.e., $\omega$ with suitable operations). Also we can show that any theory that has a model is consistent. So ZF proves Con(PA).

Gödel's self-referential statement, "I am unprovable", also is provable in ZF but not in PA. ZF proves it because Con(PA) implies the self-referential statement, and that implication can also be shown in ZF (indeed, in PA). It's not provable in PA by Gödel's first incompleteness theorem.

After Gödel proved his incompleteness theorems, Gentzen gave a proof of Con(PA) using tranfinite induction up through $\varepsilon_0$ ($\varepsilon_0$-induction, for short). Here, $\varepsilon_0$ is the countable ordinal $\omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot}}}}}$. I won't go into exactly what it means to formalize "induction up through an ordinal $\alpha$" in PA, but the upshot is this: you can do it for any $\alpha<\varepsilon_0$, but not for $\varepsilon_0$. So $\varepsilon_0$ in some sense measures the precise "provability strength" of PA.

Goodstein's theorem was proved (by Goodstein) in 1944. The proof uses $\varepsilon_0$-induction. In 1982, Kirby and Paris showed that Goodstein's theorem is in fact unprovable in PA.

Kirby and Paris also introduced the Hydra Game. They proved, again using $\varepsilon_0$-induction, that "the Hydra will always be killed", but that this cannot be proven in PA.

The Paris-Harrington Principle is a modification of the finite Ramsey's Theorem. While PA can prove Ramsey's Theorem, it cannot prove the Paris-Harrington Principle. This fact is known as the Paris-Harrington Theorem.

All the above results have the proof-theoretic ordinal $\varepsilon_0$. At least two results are known with much higher proof-theoretic ordinals: Friedman's form of Kruskal's tree theorem, and of the Robertson–Seymour graph minor theorem.

Finally, all these are examples of things provable in ZF, and expressible but not provable in PA. But as spaceisdarkgreen commented, there are also many things expressible in ZF but not in PA. (There are even things expressible in PA and unprovable in ZF, like Con(ZF)!) However, I won't go into that here.