Unfortunately, while SOME such formulas worked (the ones you learned in calculus - stuff like the chain rule), other formulas generated using the same kind of reasoning ("just imagine that d becomes infinitely small") come up with answers that are nonsensical or just wrong. How can we know when it's OK to just say "let things get infinitely small" and when it gives bogus answers?
The solution, which is taught in high-school calculus, was the "epsilon-delta" formulation. Instead of saying "let d become infinitely small" (a statement that may just be nonsense), say "I will prove that for ANY small epsilon (greater than 0) I can find a delta > 0 such that for values of x closer than delta, the error will be less than epsilon". That statement doesn't require an infinity to exist anywhere -- it is just a statement about particular finite numbers. And we can build calculus on such principles.
This isn't an EXACT analogy to your question about real numbers, but it uses the same type of reasoning, and I'm hoping the analogy is in terms of mathematical reasoning you are already familiar with.
Because it's not an axiom? I feel like I'm not understanding your question completely
Color the pair blue if its elements are 1 and 0. Color the pair red otherwise.
Color the triplet blue if its elements are 1, -1, and 0. Color the triplet red otherwise.
> When this is done, RT22 states that there will exist an infinite monochromatic subset: a set consisting of infinitely many numbers, such that all the pairs they make with all other numbers are the same color.
I read this as saying "There exists and infinite number of x's which satisfy the relation f(x, y) = blue for all values of y over some arbitrary function f()". What am I missing?