Do they prove axioms from the very beginning in order to take them on faith later?

Question

Mathematics

I read on Wikipedia that

Axiom (ancient Greek ἀξίωμα - statement, position) or postulate - the initial position of a theory, accepted within the framework of this theory as true without requiring proof and used in proving its other provisions, which, in turn, are called theorems [1 ].
The need to accept axioms without proof follows from an inductive argument: any proof is forced to rely on some statements, and if each of them requires its own proofs , the chain will turn out to be infinite. In order not to go to infinity, you need to break this chain somewhere - that is, to accept some statements without proof, as initial ones. It is these statements, accepted as initial ones, that are called axioms [2].

So Euclid came up with an idea, he tested it, looks, it always converges, and then declares that this is an Axiom? But what about the proof?
Sorry for the stupid questions, but it's still interesting how it turns out.

Answer 1

An axiom is what a further theory is based on. An axiom is an axiom because it is taken without proof.
Euclid said: parallel lines do not intersect. And then builds its geometry on it.
Lobachevsky (if I'm not mistaken about the surname) said that parallel lines intersect and built his own geometry on this.
Those. an axiom is like a foundation for something.

Answer 2

No, everything is exactly as it is written in the wiki.
Moreover, there may be several different sets of axioms (see Lobachevsky geometry) that lead to different results. But they are all correct and suitable for different occasions.
Well, and then by experience they find out that the axioms, for example, of arithmetic, correspond well to what happens to objects in the real world.

Answer 3

Axioms cannot be proven. But you can do one cool thing. Namely, to build a model of the theory. In other words: find in a neighboring theory that you "trust" (for example, the theory of real numbers or Euclidean geometry) such "points" and "lines" that they meet all the axioms. And these axioms need to be proved in order to show that, for example, R² with "points" (x,y) and "lines" ax+by+c=0 is indeed a model of Euclidean geometry.
Yes, and mathematicians often, but incorrectly, say: “A vector space is a collection of the main set X, a number field K, operations x + y and x k such that it meets the axioms ...” Actually, the requirements, not the axioms, and these "axioms" need to be proven to prove that, for example, R² is a vector space over the field R .
Yes, but what about Euclid? And Euclid himself probably didn’t know what a cool thing he came up with. In addition, an exhaustive axiomatic of Euclidean geometry was invented ≈1900. Scary as hell, six basic concepts ... But this is often not required to solve problems - you need to somehow define the object of study and start proving theorem after theorem. Most of us, even university graduates, do not know either the theory of the real number, or Peano's axiomatics for arithmetic, or the axiomatics for set theory ...

Answer 4

No. To prove something - you must always be based on something. Either on other proven theorems, or, in the end, on axioms.
Quote from wiki:

The need to accept axioms without proof follows from an inductive argument: any proof is forced to rely on some statements, and if each of them requires its own proofs, the chain will turn out to be infinite. In order not to go to infinity, you need to break this chain somewhere - that is, to accept some statements without proof, as initial ones.

Answer 5

If we take pure mathematics, then neither one nor the other. Axioms are not proven and are not taken for granted. They are simply used to build a theory. To do this, you do not need to believe in them, just as you do not need to "believe" in the rules of chess in order to play them.
If you look from a practical, applied point of view, then the axioms allow you to determine the area of ​​applicability of the theory. If we describe some part of reality in terms of a theory, and if the axioms of that theory correspond to true statements about reality, then that theory is valid in this particular case. Here, the applicability of the axioms cannot simply be accepted, it must either be proved from a previously accepted theory, or be confirmed empirically.