Is the distributive propriety an axiom?

[mellow violet]

Hello! I've been hearing that the distributive propriety is an axiom. Why? Is a geometric derivation for real numbers not rigorous enough?
This is a question I'd like to know the answer

[hoary shell]

what exactly do you mean by this?

if you're asking about i.e. the peano axioms, it's that

if you're working in the foundations of mathematics, you might ask i.e. "what is a set?"

intuitively people had this notion that a set was just "a collection of objects"

but russel's paradox means you can't define a set like that

so like intuitively, the natural numbers definitely should exist

ZFC provides a framework (and says natural numbers exist)

it turns out that once you know natural numbers exist, you can construct the integers from the naturals, the rationals from the integers, the reals from the rationals etc.

so you only need to know that natural numbers exist

but this is more for foundational mathematics

if you don't work in foundational mathematics and for example you work with fields/rings

one of the axioms of a field/ring is the distributivity property

so distributivity is an axiom

[hoary shell]

also what exactly do you mean by "a geometric derivation for real numbers"?

[mellow violet]

At least it works with arithmetic (I think)

[hoary shell]

i mean that's not rigorous unfortunately

[mellow violet]

This is what I thought

So, even it has a "proof", but is not rigorous enough, right?

According to the Gödel's theorems, there's "infinite axioms", right? I've searched some stuff about it and some guys said that (at least as I've understood it) there are fields of math that there's "infinite axioms" and fields that not

Some of these "infinite axioms" have non-rigorous "proofs" like this one above? Because, for me, saying that a(b+c)=ab+ac is truth, memorize it and that's the end is completely different of saying "this proof, although it's not rigorous, give us an intuition of this fact". I like to accept facts when they are intuitive.

[viral oxide]

This is an intuition, but not a proof

[viral oxide]

Intuition can be dangerous

Here's a little example where intuition can counter truth
https://youtu.be/851U557j6HE?si=w5bva8vFGNZndwUx

[mellow violet]

Thanks!