### Infinity can't be reasoned with (or why infinite sets are types)

Now, in the spirit of set theory, I supposedly have this huge set of all the water molecules, but then I ask myself, "which water molecules in X are in a gas state?"

I take it you mean "which water molecules in W are in a gas state?"

So I haven't actually created a giant set, I've only defined a type.

It seems like you've used a label for a collection of things that match your description. I don't see why it's a problem. Because you're not holding them in your hands?

Using the water molecule example, I could define a sub-type, (that is a new type with all the properties of another type, but with additional specifications that limit the possible instances). Let G be the set that inherits all the properties from X and whose members are in the gas state.

Here you're just talking about a subset of W. You've added an additional requirement for inclusion in the new set.

Then I've just defined a sub-type, but again, I can ask a question like "which molecules in G have a kinetic energy greater than 12 zettajoules?" and again I can't answer that question because I don't have access to an actual giant set of the molecules, I only have a type, and to answer that I actually have to go find all the molecules that meet my type description (water molecules in gas state) and measure their kinetic energy.

You have defined a new set.

I don't get what this has to do with infinity anyway. Is the set of all water molecules finite?

Georg Cantor thus did not actually prove that there are different sizes of infinity, what he proved is that there are sub-types and super-types; basically he proved that a square is always a rectangle, but a rectangle isn't always a square.

You can consider bijections between sets that don't have any elements in common. Why does it seem like you're only considering sets and their subsets?

The even numbers is a subset of the natural numbers, but they have the same cardinality. Are the even numbers a "subtype" of the natural numbers? How does this type stuff work?