So I've been thinking a lot about the philosophy of mathematics and I'm afraid my lack of formal training in the subject has led me to a lot of questions, so sorry in advance if this is a bad one!
Basically, I understand that mathematics is made up of axioms that we kind of just assume to be true because they work, and that different kinds of maths are made up of different kinds of axioms (like Euclidean and non-Euclidean geometry).
So my question is related to axioms of number theory. What happens to number theory or mathematics generally if we assume that there is a largest number, let us call it m, for example.
Basically, I mean that if you add 1 to m it still equals m, but m << ∞, i.e. it is finite. I know this doesn't work, but I'm curious about how it doesn't work and why it doesn't work? Does it lead to mathematical paradoxes, for example?
Thank you so much, I really appreciate any responses!