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Can you clarify that last point? I don't think it's strictly true. Aside from countable and uncountable infinities, you can have larger and smaller infinites as
by throwawaymath 8y ago
Can you clarify that last point? I don't think it's strictly true. Aside from countable and uncountable infinities, you can have larger and smaller infinites as well. Unless every set S of all rational numbers between any irrational x and irrational y is isomorphic to the set P of all rational numbers, I don't see that this is correct. And I don't immediately see that you can put them into 1-1 correspondence.
- zeeboo 8y agoDo you agree both sets are countable? If so, any two countable sets can be put into bijection by composing their bijections to the naturals.
- throwawaymath 8y agoOh, right. Yeah I guess that makes sense by definition.
- aportnoy 8y agoAny infinite set is either countable or uncountable.
- throwawaymath 8y agoI'm aware, but that's not what I meant.