# Are the axioms of analysis a combination of Peano axioms and set theory axioms?

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.

• You can formalize the set of natural numbers inside of set theory. So, if you take the axioms of set theory (e.g. ZFC) as your basis, then you can prove within your set theory that there is a model of the natural numbers obeying Peano's axioms. So, Peano's axioms are then theorems in set theory. In the same way, as you can construct the field of real numbers inside set theory and the field axioms for this particular field are provable theorems. Does this long comment help? Commented Aug 31, 2023 at 11:14
• You are presumably going to use real numbers in your analysis, including that the real numbers are a complete ordered field. Whether you regard this as using axioms or a model depends on your approach. Commented Aug 31, 2023 at 11:19
• @Cosine yeah thanks! Commented Sep 4, 2023 at 4:38

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:

• $$(\mathbb R,+,\cdot)$$ satisfies the field axioms
• $$\geq$$ is a total order
• If $$x\geq y$$, then $$x+z\geq y+z$$ for all $$x,y,z\in\mathbb R$$
• If $$x,y\geq0$$, then $$x\cdot y\geq0$$ for all $$x,y\in\mathbb R$$
• If $$S\subseteq\mathbb R$$ has an upper bound $$u\in\mathbb R$$ (meaning that $$u\geq x$$ for all $$x\in S$$), then it also has a least upper bound (meaning that the set $$U$$ of upper bounds contains an element $$s\in U$$ such that $$u\geq s$$ for all $$u$$ in $$S$$.

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.

• All of the axioms of real numbers can be stated using first-order logic and the symbols $+,\cdot,\leq$. Except, as far as I can tell, the supremum axiom: we need to introduce sets of numbers. Doesn't this pre-suppose some set theory axioms to be able to work with the real numbers? Commented Aug 31, 2023 at 12:37
• Yes, sets are involved. I'm not entirely sure wether it's possible to work without sets here, but that's not really the point. Many other examples of axioms (like those of group theory) also explicitly reference sets. I thought the question was about this distinction of axioms between the axioms of a theory like group theory, and the axioms of the systems used to construct the objects in question (set theory). Commented Aug 31, 2023 at 12:43
• I'm not too sure what OP intended. But I think it's worth noting that, while "the axioms of real numbers" are typically understood to be the ones you've stated, they aren't a self-contained first-order logical system like Peano arithmetic or ZFC, because they rely on sets. (That's what I understand, anyway; I could be wrong!) Commented Aug 31, 2023 at 12:49
• Yes, that is worth noting! Commented Aug 31, 2023 at 12:58