Anecdotal data suggests that a number of people find counterintuitive the Cantorian idea that some infinities are bigger than others.
This is curious. After all, the naive thing to say about the prime numbers and the natural numbers is that
- while there are infinitely many of both, there are more natural numbers than primes.
- while there are infinitely many of both, there are more real numbers than natural numbers.
All in all, rather than taking (2) to show us how counterintuitive infinity is, we should take (2) to vindicate our pretheoretic intuition that cardinality comparisons can take us beyond the finite, even though some of our pretheoretic intuitions as to particular cardinality comparisons are wrong.