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$ | \mathbb Q^2|$ I assume that it has to be continuum, but I have no idea, how to show it

Zauberkerl
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    If $X$ is infinite, then $|X| = |X^2|$. – Dan Rust Mar 28 '17 at 18:37
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    Do you know the mapping from $\mathbb{N}^2\mapsto \mathbb{N}$ that lets you show that $\mathbb{Q}$ is countable? Can you see how you might be able to use that mapping to show that $\mathbb{Q}^2\equiv(\mathbb{N}^2)^2\mapsto\mathbb{N}$? – Steven Stadnicki Mar 28 '17 at 18:38

2 Answers2

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Do you know that $|\mathbb{Z}|=|\mathbb{Q}|$? If not, the argument can be found here. The same argument tells you that $|\mathbb{Q}|=|\mathbb{Q}^2|$, just replace the sequences of integers with sequences of rational numbers. In terms of cardinals, we have

$$\aleph_0^4=(\aleph_0^2)^2=\aleph_0^2=\aleph_0$$

Community
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Stella Biderman
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It will be $\aleph_0$.

$|\mathbb{N}|=|\mathbb{Q}|\\ |\mathbb{N}|=|\mathbb{N}^2|\\ |\mathbb{N}^2|=|\mathbb{Q}^2|\\$

Jaroslaw Matlak
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