 In high-school calculus (if you've taken that), you apply things like "d/dx" which LOOKS like it is just a fraction, dividing "d" by "d times x". The notation was first dreamed up by mathematicians who were thinking "but if we keep making the d-slices really REALLY small we would move from the discrete approximation to the formula for the correct continuous answer".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. Question: Isn't there an axiom that says "for any real number, there's always a bigger number"? What stopped Patey and Yokoyama from proving Ramsey's Theorem For Pairs/Triples by saying "for any pair which satisfies some relation X, there exists another pair which also satisfies relation X"? > What stopped Patey and Yokoyama from proving Ramsey's Theorem For Pairs/Triples by saying "for any pair which satisfies some relation X, there exists another pair which also satisfies relation X"?Because it's not an axiom? I feel like I'm not understanding your question completely Because there doesn't have to be another pair that satisfies that relation. Counterexamples are trivial: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. Then I misunderstood the article when it said> 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? Search: