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How do mathematicians prove something is infinite?

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There are a couple of different ways that you could do it. The oldest, probably, is proof by contradiction: assume that that there are only finitely many elements and derive a contradiction. read more

Abstract: The errors and defects in the “scientific-mathematical criterions of ‘infinite’ definition” in present classical infinite related science theory system are studied and disclosed; the inevitable relationships among these errors, defects, the quantitative cognizing obstacles to “infinite things” are analyzed and, the scientificity of new definition to “infinite” in new infinite theory system is introduced. read more

The way to prove something about every natural number is to use mathematical induction (which itself is often an axiom, but in some systems can be proved from yet more primitive axioms. That shall not concern us here). Thus: The base case. We must prove that there is no bijection $f$ from $\mathbb Z$ to the set of natural numbers less than 0. read more

You can prove that a set is infinite simply by demonstrating two things: For a given n, it has at least one element of length n. If it has an element of maximum finite length, then you can construct a longer element (thereby disproving that an element of maximum finite length). read more

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