Saturday, April 18, 2015

Bigger and smaller infinities

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

  1. while there are infinitely many of both, there are more natural numbers than primes.
For the same reason it is also surely the obvious thing to say that
  1. while there are infinitely many of both, there are more real numbers than natural numbers.
So there is nothing counterintuitive about different sizes of infinity. Of course, (1) is 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 much 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.


Michael Gonzalez said...

This is going to reveal my ignorance about transfinite math, but why does proof of a bijection between the primes and the naturals falsify (1)? I thought that part of the concept of an infinite is that it is equivalent to its proper subsets.....

Alexander R Pruss said...

Right: but equivalent sets have the same number of members.

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