Countable Sets: Cantor's Theorem & Galileo's Paradox

[Hippasos]

Hi!

Can You please confirm that there is no contradiction between Cantor's theorem of countably infinite power sets and Galileo's paradox?

- Thanks

 

[disregardthat]

Don't think of infinite sets as something "huge", and don't think of different countable sets as "equally huge", because it will only cause confusion. There are not "more" integers than perfect squares, because "more" doesn't make sense when it comes to infinite sets, unless you define what you mean by "more".

We use cardinality as one type of measure of infinite sets. Countable sets are simply sets that can be indexed by the natural numbers, that is: put in a bijective correspondence with the set of natural numbers. In this context there are "equally many" perfect squares as natural numbers, simply because they can be put in a bijective correspondence.