Disclamer
I am no serious mathematician, just curious
Context
I recently discovered some set theories, ZFC and IZF in particular.
It made me realize that I've studied math a whole year without even knowing what a set really is (it was nothing more than a collection of objects with neither order nor repetition, as in the Naive set theory).
On the one hand I can't blame my teacher because his only goal was for us to pass the entrance exam of some engineer school, and not to turn us into mathematicians,
But on the other hand, it doesn't make any sense to study groups and vector spaces, without a proper definition of what a set is.
Or does it ?
Questions
- Is mathematical consistency really necessary to study mathematics ? (Like having decided if yes or no the LEM holds)
- Can any axiomatic system really be independant ? (As an axiomatic system is defined to be a set of axioms along with the derived theorems)
You will get probably a lot of responses disagreeing with me on this site, because there are a lot of people here that focus a lot on these types of logical foundations. But in my experience, most mathematicians don't pay a lot of attention to this. Yes, logical consistency is important, but most mathematicians take the LEM and the AoC as given. Sets can be worked with on an intuitive level without worrying about a rigorous consideration of what model you are working with.
Of course, some mathematicians do research in the implication of different models of set theory, and are really interested in what different results can be proved under different sets of axioms. Their idea of mathematics looks more like pure logic to me, which I am not really interested in, and they would probably consider me sacrilegious for my lack of interest.