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 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.

*false*. Our untutored intuitions are wrong about that case. And that fact should make us

*suspicious*whether (2) is true; given that the same intuitions led us astray in the case of (1), we shouldn't trust them in case (2). However, the fact that (1) is false should not switch (2) from being intuitive to being counterintuitive. Moroever, our reasons for thinking (1) to be false—namely, the proof of the existence of a bijection between the primes and the naturals—don't work for (2).

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.