Consistency of Peano axioms (Hilbert's second problem)?

Solution 1:

The answer is relatively simple, but complicated.

We cannot prove that Peano axioms (PA) is a consistent theory from the axioms of PA. We can prove the consistency from stronger theories, e.g. the Zermelo-Fraenkel (ZF) set theory. Well, we could prove that PA is consistent from PA itself if it was inconsistent to begin with, but that's hardly helpful.

This leads us to a point discussed on this site before. There is a certain point in mathematical research that you stop asking yourself whether foundational theory is consistent, and you just assume that they are.

If you accept ZF as your foundation you can prove that PA is consistent, but you cannot prove that ZF itself is consistent (unless, again, it is inconsistent to begin with); if you want a stronger theory for foundation, (e.g., ZF+Inaccessible cardinal), then you can prove ZF is consistent, but you cannot prove that the stronger theory is consistent (unless... inconsistent bla bla bla).

However what guides us is an informal notion: we have a good idea what are the natural numbers (or what properties sets should have), and we mostly agree that a PA describes the natural numbers well -- and even if we cannot prove it is consistent, we choose to use it as a basis for other work.

Of course, you can ask yourself, why is it not inconsistent? Well, we don't know. We haven't found the inconsistency and the contradiction yet. Some people claim that they found it, from time to time anyway, but they are often wrong and misunderstand subtle point which they intend to exploit in their proof. This works in our favor, so to speak, because it shows that we cannot find the contradiction in a theory: it might actually be consistent after all.

Alas, much like many of the mysteries of life: this one will remain open for us to believe whether what we hear is true or false, whether the theory is consistent or not.

Some reading material:

1. How is a system of axioms different from a system of beliefs?

Solution 2:

Mathematicians believe in "the" induction principle for natural numbers not because of set theory or PA or their consistency, they believe in "the" induction principle because of their intuition about natural numbers. Axioms are means for expressing these intuitions.

Mathematicians believe that PA is consistent and in fact sound because it satisfies the standard model of natural numbers (0 and its successors). If you believe in natural numbers then the induction principle would follow automatically for all well defined "properties".

One should be careful here because "the induction principle" is used for various things, e.g. the informal intuition that

if a "property" $P$

• holds for 0, and
• if it holds for natural number $n$, then it holds for its successor $n+1$,

then it holds for all natural numbers.

Here "property" is an informal concept.

Induction is also used to refer to various formal axioms that try to capture this informal intuition e.g. the first-order induction axiom in the theory of arithmetic, the second-order induction axiom of Peano, etc.