How an axiomatic system is made?

Question

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call theorem, proposition, lemma, corollary, ect.

My question is how can an axiom be made? How an axiomatic system is arranged so that a whole theory can be build from that?! Does it have to be an 'inspiration' to formulate axioms, or you get them by working on problems which need true assumptions without proving them?

Answer 1

The thing is that at some point you have to assume something. It's not a matter of needing to prove something that can't be proved. It's about trying to find the most basic assumptions possible that you can take and build from there.

Answer 2

Axioms are the formalizations of notions and ideas into mathematics. They don't come from nowhere, they come from taking a concrete object, in a certain context and trying to make it abstract.

You start by working with a concrete object. After some investigation, you distill the axioms from that concrete objects, often you make mistakes and you need to add, or you can remove, some of the axioms.

For example, you work with the real numbers. But then you realize that most of what you need is in fact just a field which satisfies certain properties. These become axioms, for example, real-closed fields.

Or another example, you work with "collections" and you name them sets, at first they are just sets of numbers or sets of functions, but then you realize that you can talk about sets of these sets and sets of sets of these sets, and so on. You work and work with them, and you get some general idea about what properties these "sets" should have. For example, if you have a set, then collection of all its subsets should also be a set. So you write this formally and you have the power set axiom. And slowly you shape axioms for set theory.

Answer 3

In the case of mathematics, the axioms were defined by "working backwards", more or less. Math as a system was already vaguely defined as a need in early civilizations, which evolved into something more. As it grew into its own system of logic, mathematicians recognized its need for a formal set of axioms and definitions, by which rigorous proofs and derivations of all of math may follow, and this was the work of most early-mid 20th century mathematics. In general though, mathematics as a system of formal logic is arbitrarily defined, and there's no significance to the addition or removal of an axiom other than it's what we're used to in math and math has come to be quite popular with a logical tradition to upkeep. So if you were to create a different axiomatic system, or a "new math" even, all you'd have to do is create a formal set of axioms, of which are to be assumed true in your system, and then follow from there with first-order logic and rigorously define terms as needed.