[irishsultan]

Consider the following sets: the primes (let's call it P), the odd numbers (let's call that O), the odd numbers except for 3 (and let's call this O3).

Using your definition O would be equal in size to P, P would be equal in size to O3, but O3 would be smaller in size than O.

[godelski]

Ah thanks! That's an obvious counter example!

But as a followup, are there definitions that rely the rate at which sets approach infinity? P clearly "fills" its set more slowly than the even integers whereas the evens and odds "fill" the set in similar times. This would obviously mean 2Z, 3Z, and 3Z - {3} would be the same size (unless we invoke the disjoint requirement), but these could be used categorically like Big O notation (which can be refined).

Would this an even useful metric? Are set theorists even interested in differentiating these infinities?

Edit: I also gave a bit more motivation in the reply to your sibling comment. Short is that if we do "Z - 2Z" we can get the two sets of positive odds and negative odds without {0}. It seems reasonable that since we can do this decomposition and match in the normal manner that since there is a remainder that one would be larger than another but this also does not clear up the example you provided with primes.

[cubefox]

I always thought that a "growth rate" definition of sizes of infinity makes more sense than Hume's principle. Then the odd numbers approach infinity half as fast as the natural numbers. But that only works for ordered quantities.