[bradwood]

mathematical intuition? positive numbers when multiplied together always equal numbers larger than either of the multiplicands...

hardly a rigorous proof, I grant you, but surely true?

[r0uv3n]

In fact, the opposite of your statement is true: Assuming the axiom of choice, the cardinality of the product X^2 of some infinite set X is always the same as X.

See https://math.stackexchange.com/a/2464655 for a quick proof