Are there any contradictions of the Axiom of Choice (AOC) that are consistent with basic mathematical logic? Has anyone tried to develop a non-AOC theory?

Can something be infinite if there is a definitive number of it? Here's an example: I take a number, the largest I can think of, and never stop adding one to it. The number becomes infinite. Now if you take the number of human beings, and never stop adding to it, is the number of human beings infinite? In contrast, dinosaurs cannot be added to therefore they would not be infinite. Does this make sense?

Can we have a POSITIVE understanding of such concepts as infinity? What I mean is that, whilst I am sure that we can well grasp the concept of finiteness, can we do more than negate it (which would yield not-finiteness), can we understand infinity from the inside instead of by negating everything that lies outside of it? Thanks, Andrea Jasson

If an infinite number of monkeys were at an infinite number of typewriters, would the work of Shakespeare eventually come out?

If we define the science as the study of reality to find the relations among objects through scientific method, is mathematics a science? Or, it's something like logic which can be used in a wide range of sciences, philosophies, and even daily usages.

Do people invent equations, or do they discover them? Examples of the sorts of things I am thinking of are Newton's laws of motion, or Mandelbrot's sets.

If a certain concept is seen as indefinable, such as the concept of infinity, how can it be said to exist?

Is mathematics the only certain knowledge?

Would you agree that numbers are synthetic truths rather than analytic truths? This is because I can imagine a universe, whenever I walk 2 meters foward, space itself 'bends' so I end up 3 meters ahead of where I started. In this universe, when 2 is added you end up with 3. 2+2=5 (or maybe 6).

What is the exact purpose of math? Why is it that math was created? I know that some math has a purpose like finding out how much you may owe someone, but how about Linear Equations or Polynomials? What is the purpose of all this?