I do not mean branches such as functional analysis
I mean is math we use in elementary school (which I heard uses Peano's axioms) the 'correct' math?
Is there math that uses other axioms? Is Zermelo-Frankel Axioms another type of math? Anything along those lines is my question.
The easiest example is modular arithmetic, which is where you violate a Peano axiom by making your numbers wrap around, as on a clock. So in the ring of integers modulo $12$, the product $5\cdot 11$ is still well-defined but it is not $55$ but instead $7$. You can view this if you like as the remainder after division by $12$; $55$ is $4\cdot12 + 7$.
Interestingly, it turns out that if the modulus is prime, you can have division without fractions: so for example in the mod-$23$ ring, to get a multiplicative inverse of $11$, you don't just need to give up and invent the fraction $\frac{1}{11}$; you can instead use the integer $21$, as $21\cdot11=231$ and the remainder after dividing by $23$ is $1$. So for example to divide $8$ by $11$ you do not have to give up and say "it's just eight elevenths, deal with it" but you can now say "it's $8\cdot21 = 168 = 7\cdot23 + 7,$ so in this ring eight elevenths is exactly the number $7$."
We can also extend the system rather than neglecting part of it. When you invent the real numbers and potentially-infinite decimals, you are defining something with a new axiom that isn't part of the Peano axioms, called the least upper bound property, which says that if you have a sequence of real numbers which has a bound from above, then there is some least upper bound with respect to their normal ordering. That is what makes the real numbers "filled in" in a way that the rationals are not.
There are two main issues here:
Choice of formal system
First you have to choose a logical system.$\def\imp{\rightarrow}$$\def\eq{\leftrightarrow}$ The conventional choice is classical first-order logic, which has boolean operations $\neg,\land,\lor,\imp,\eq$ (not,and,or,if-then,equivalent) and quantifiers $\forall,\exists$ (for every, there is). A first-order language has these as well as equality and a fixed collection of predicate symbols and function symbols. Predicate symbols are used to denote properties, such as if we use "$P(x)$" to represent "$x$ is a person". Function symbols are used to denote an operation that can be done on any object in the domain, such as if our domain is intended to be the natural numbers and we use "$S(x)$" to denote "the successor of $x$".
Classical first-order logic comes with deductive rules that tell you precisely whether a proof is valid or not. There are many equivalent formats of proofs and rules. One intuitive one is called natural deduction (especially Fitch-style). Either way, there is not much you can prove with just a fixed language and nothing beyond the logical deductive rules. All useful formal systems thus consist of not just an underlying logical system but also other rules or axioms governing the objects in the domain.
For example, Peano Arithmetic (PA) originally was based on full second-order logic (which is strictly stronger than first-order logic), and there are only a few axioms for second-order PA. But because first-order logic has much nicer properties, people when referring to PA often mean first-order PA where the second-order induction axiom is mimicked by having infinitely many axioms, each one for a different formula. This PA is over the first-order language with 2-input functions $+,\times$ and the 1-input successor function $S$. Note that in the intended domain every object is a natural number, and so every object has a successor and any two objects can be added or multiplied.
Zermelo-Frankel Set theory [with Axiom of Choice] (ZF[C]) is another first-order formal system, with a different set of axioms, and a different intended domain, namely where every object is a set, and so "$x \in y$" makes sense for any objects $x,y$, and indeed ZF[C] is over the first-order language with just one 2-input predicate symbol "$\in$". Since natural numbers were very useful, ZF includes a special axiom called Infinity that is basically equivalent to asserting that the set of natural numbers exist. Using this axiom (with the other axioms of ZF), we can prove (in ZF) that there is some set that models PA. I'm handwaving here because it would take too long to explain what "models" means.
Some people object to classical logic, and instead stick to intuitionistic logic, which is strictly weaker, meaning that:
Although the real world appears to be modeled accurately by classical logic, which is one assumption underlying all our scientific experiments, there is however a beautiful correspondence between proofs in intuitionistic logic and programs in theoretical computer science.
Meaning and truth
Now the formal system itself does not specify any meaning. To do so we always have to look at the formal system from the outside. When we do so, we might then assign meaning to statements made within the formal system. In the case of first-order logic, we assign the "obvious" meaning to the boolean operations and quantifiers and equality. But really these are circular because you cannot explain "if" to someone who does not know what "if" or an equivalent means. Same with all the intended meanings that we give to predicate symbols and function symbols. (See https://math.stackexchange.com/a/1334753/21820)
It is best to just say that we just lay down the rules and axioms of a formal system, and anyone who accepts that they are meaningful in some interpretation will then have to accept all theorems that can be proven from them, according to that interpretation. It is possible that different people have different interpretations of the same formal system, so they would also have different interpretations of the theorems. There is no way around this, because there is no way to define meaning precisely enough to exclude all interpretation, partly because we ultimately rely on shared understanding of some common language like English to convey our interpretation of mathematical symbols.
But that is precisely the problem when we ask:
Is there a correct math?
Because before we can answer it we have to give meaning to whatever 'math' we do, and then see whether that meaning is 'correct' or not. The term 'math' is too nebulous, and we can 'define' it as working in some formal system, but we still have to give meaning to each statement that is proven in whatever formal system we choose. I have already briefly mentioned that even the formal system chosen depends on philosophical preferences or utility in modeling or both, but both of these criteria depend on interpretation.
Even if we believe there is an absolute truth, we may not be able to access it, and so it is difficult to judge correctness even if our criterion is accurate modeling of the real world. Remember that for a long time Newtonian mechanics passed scientific tests simply because it was a very accurate model for low speeds and low masses/energies (with the notable exception of Mercury's orbit). Einstein's general relativity currently passes all known scientific tests, but who knows if it is yet another approximation? Likewise, today we use RSA encryption for most HTTPS connections on the internet, which is based on Fermat's little theorem, which is a theorem provable in PA. So far we can safely say that it works for 'randomly' chosen 1024-bit semi-primes, but is every theorem of PA really true when the natural numbers are interpreted to be finite strings of digits as represented in physical medium such as computing hardware?
Note that if the universe is actually finite, then PA is clearly false, but we may never be able to know that even if it were so, because even formal systems might break down. The very definition of formal systems rely on the ability to manipulate strings, and perhaps we cannot maintain the integrity of physical representations of strings beyond a certain size...
In any case, we develop and use formal systems that are useful to us, so we do not need to worry too much about whether they fail for extreme situations that we can never encounter. It is of note that the full strength of ZF is hardly needed to prove things that are relevant to our physical world; so people have also investigated what parts of ZF are needed for important facts that have verifiable physical manifestations, such as computability theory, theorems in hard analysis, and some others areas of mathematics. This area of research is called reverse mathematics.
