What is the math definition of infinite in math?

Answer 1

Either define finite first (usually as equivalent to a set of the form {1,2,3,...,n} for an integer n, in which case infinite is not finite; or use a set is infinite if and only if it is equivalent to a proper subset of itself. Equivalent sets A and B are those for which there is a function f taking A into B which is bijective (one-to-one and onto).

For example, the positive integers are infinite because we have the pairing: 1 -> 2, 2 -> 4, 3-> 6, ..., n -> 2n, ...; so {1,2,3,...} is equivalent to its subset {2,4,6,...}