TITLE: Consistency of Peano axioms (Hilbert's second problem)?
QUESTION [13 upvotes]: (Putting aside for the moment that Wikipedia might not be the best source of knowledge.)
I just came across this Wikipedia paragraph on the Peano-Axioms:

The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen's proof.

And then went back to the article on mathematical induction:

The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms. 

I'm not a math person, but I'm curious - how can something which sounds like it is believed to be consistent, be the base of such widely used proof technique?
(The Wikipedia article on Peano-Axioms also states: "When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a "natural number".)

REPLY [4 votes]: 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.