Observing the use of mathematical induction in proving the finite version of the axiom of choice, I began to ponder. Why is mathematical induction from Peano axioms employed to prove facts about sets? Shouldn't it be restricted to using only the axioms of set theory?
I've encountered the notion that modern mathematics is built upon axiomatic systems. Group theory seems to rely solely on group axioms, and Euclidean geometry appears to use a limited set of axioms. If that's the case, what are the axioms of analysis?
If you happen to know the answer, I would greatly appreciate your insight.
An axiom is a statement that we assume to be true. Often it is added: "without proof". That's true in some way, but also misleading in some situations.
For most modern mathematicians, the axioms of set theory (specifically ZFC set theory) are true axioms in the sense that they are never proved, just assumed to be true, and that everything else is built on them.
Axioms like Peano's axioms, the axioms of group theory, etc., are only axioms in the sense that the theory built up in some subset of maths, like group theory, is built on the assumption that some given structure satisfies the axioms. For instance, if we assume that a pair $(G,\ast)$ satisfies the axioms of group theory, then we can prove things like: "If $G$ is a finite set and the prime $p$ divides its cardinality with multiplicity $n$, then there is a subgroup with $p^n$ elements" (one of Sylow's theorems). The theory is only about such pairs which satisfy the axioms, so in that sense, they are assumed to be true for all the relevant theorems. That's why they are called axioms.
However, if we want to apply the theory, we need a group to begin with, and we need to know that it is, in fact, a group. For instance, group theory is often applied to symmetry groups. The simplest such group is the symmetric group of a set $M$: Let $M$ be a set, and $S$ the set of all bijections $M\to M$. Then the pair $(S,\circ)$, with $\circ$ being function composition, is the symmetric group of $M$. But is it actually a group? Yes it is, and this can be shown by proving the group axioms using set theory, like: the composition of bijections is a bijection, there is an inverse to every bijection, the composition of a map with the identity map results in the same map again, etc.. So for the application, the axioms need to be proved first, and then afterwards the theory, which was built with the axioms being assumed as true, can be used to reason about the object for which the axioms were proved to be true.
In summary it is important that whatever we are doing in math, there are some things we need to assume and can't prove, unless we want to start an infinite descent of "prove it!". These things are called axioms. When working on something specific, we often assume things about the objects we're going to work with, and call those axioms because for the time being, we do assume them to be true. But when working with specific instances of such objects, we do need to make sure that they actually satisfy the axioms, and we have to take a step back to more foundational theories, mostly set theory, and use the axioms of that theory to prove the axioms of our more specific theory.
Now to analysis:
The axioms of analysis are really the axioms of the real numbers. They can be formulated similarly to group theory axioms:
A quadruple $(\mathbb R,+,\cdot,\geq)$ of a set $\mathbb R$ with binary operations $+,\cdot$ and a relation $\geq$ on $\mathbb R$ is called a real number system if:
And analysis is essentially about things that can be done using such systems and derived systems (like the complex numbers, or real vector spaces with a norm). Also, an important theore mof the theory is the following: Any two systems of real numbers are isomorphic, so the whole theory is really about just one object that has different ways to be constructed. Most of these do start out with a system of natural numbers, so the Peano axioms are kind of involved. But they are not an axiom of analysis, since they are not necessary for the construction of the reals (one could start with the integers, which have an axiomatic characterization without any reference to Peano). However, one can prove that the intersection of all inductive subsets of $\mathbb R$ (a subset $U$ is inductive if it contains $0$ and $x\in U$ implies $x+1\in U$) together with the successor function $S(x):=x+1$ satisfies Peano's axioms.