> Mathematics need not directly model the 'real' world or match our physical intuition There's so many other ways to demonstrate this. For example, it's possib

> Mathematics need not directly model the 'real' world or match our physical intuition There's so many other ways to demonstrate this. For example, it's possible to map the integers to the rational numbers one-to-one, which is a good definition for "as many as" when dealing with infinite sets, even though there are infinitely many rational numbers between any two rational numbers. Moreover, there are as many integer multiples of a trillion as there are integers total. This is to do with set cardinality: https://en.wikipedia.org/wiki/Cardinality https://en.wikipedia.org/wiki/Cardinality ... which again comes back to the axiom of choice: > If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions: > * Any set X with cardinality less than that of the natural numbers [...] is said to be a finite set. > * Any set X that has the same cardinality as the set of the natural numbers [...] is said to be a countably infinite set.[10] > * Any set X with cardinality greater than that of the natural numbers [...] is said to be uncountable. (... with some mathematical notation that doesn't work here elided)