As a closet pragmatist I've always felt that "existence" is not a binary quality, but fuzzy. Things like 0, 1, 2 and 256 have a "strong existence", whereas 384

As a closet pragmatist I've always felt that "existence" is not a binary quality, but fuzzy. Things like 0, 1, 2 and 256 have a "strong existence", whereas 38476532 has a very weak existence. You can relate this to saying that the strength of existence is inversely proportional to the size of the turing machine required to compute it. In this way, infinity of any form doesn't actually exist. Add to the mix the axiom of infinity and it then does exist, but we're close to the constructivist's world. However, having said that, there's a lot that I do that relies, underneath, on having the more usual model. Allowing the declaration of the existence of all elements of 2^S for any set S is just too useful not to explore. That's the world I usually inhabit. However, it's not clear that getting into those discussions will, in any way, help people to understand the more "regular" viewpoints and reasoning. Encouraging people's intuition can be good, but sometimes it simply prevents them from reasoning clearly. And perhaps that's the most easily defensible aspect of the usual idea (within mathematics) of infinity. Dealing with it forces you not to take things for granted, and to reason carefully about properties. Discovering that 1+aleph_0 = aleph_0 = aleph_0+1 from first principles is useful. Then finding that 1+omega != omega+1 is useful. Mathematical logic and reasoning is so useful, we need places to exercise it. This is one, there are others. But personally, I like the idea that there are different sizes of infinity, and the resulting implications such as the Banach-Tarski theorem, the existence of uncomputables, and the tension between Zorn's Lemma and the well-ordering principle. Although mostly it doesn't really matter.