Comparing 2^∞ and ∞^2: Which is Larger?

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[steveharry]

Amazing video teacher. Simple and more informative.

[k-theory]

This is a nice idea, but unfortunately it is wrong. There is no generally accepted definition of 2^oo nor oo^2, and so it is meaningless to try and compare such things. It's possible to take the limits lim_{x -> oo} 2^x, and lim_{x -> oo} x^2, yes, but they are both infinite, hence equal.

It is true that one can define arithmetic operations on cardinal numbers, and it's a theorem that for any cardinal k, we have k^2 <= 2^k, and, in particular, aleph_0^2 < 2^{aleph_0}, but this requires rigorous proof using technical definitions. See, for example, Introduction to Set Theory by Hrbacek and Jech.

[ur_dadlol]

Ok but this is not an appropriate proof. You’ll need to prove for every value to prove the statement is true if you’re using this way. You should solve it algebraically.

[Jesuisunknown]

There are infinite prime and non prime numbers