I might be wrong , but I think Cantor tried to say something like x approaching infinity vs x= infinity , and I think he almost proved that infinities can be different (IDK if he was successful ).

Infinities are different. There are an infinite number of prime numbers. There are an infinite number of real numbers. But the number of primes and reals aren't equal.

One case where the sets aren't equal if one of the sets contains the other set. For example, the number of odd numbers is equal to the number of even numbers because the sets have no common elements, but the perfect squares and positive integers are not equal because the positive integers contains the set of perfect squares \(\textit{as well}\) as the set of non-squares. However, this can be kind of confusing because for every natural number \(n,\) there is a number \(n^2.\)

But in fact, the sets you mentioned have equal size. Your last statement pretty much disproved what you said; if we could find a 1-1 correspondence between two sets, then they are equal. The 1-1 correspondence between positive integers and perfect squares is exactly \(n\iff n^2\), so the sets are equal.

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aredifferent. There are an infinite number of prime numbers. There are an infinite number of real numbers. But the number of primes and reals aren't equal.Log in to reply

One case where the sets aren't equal if one of the sets contains the other set. For example, the number of odd numbers is equal to the number of even numbers because the sets have no common elements, but the perfect squares and positive integers are not equal because the positive integers contains the set of perfect squares \(\textit{as well}\) as the set of non-squares. However, this can be kind of confusing because for every natural number \(n,\) there is a number \(n^2.\)

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But in fact, the sets you mentioned have equal size. Your last statement pretty much disproved what you said; if we could find a 1-1 correspondence between two sets, then they are equal. The 1-1 correspondence between positive integers and perfect squares is exactly \(n\iff n^2\), so the sets are equal.

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Here's an interesting read about this. It's also darn funny!

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