If $A$ is any infinite set, is $A × A$ equivalent/equinumerous to $A$?

Asked Jul 14 '22 at 17:59

Active Jul 14 '22 at 18:08

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I know the result is true for $\mathbb{N}$ and $\mathbb{R}$ and can even be generalized for $\mathbb{N^n}$ and $\mathbb{R^n}$ where $n \in \mathbb{N}$, but I'm not sure how I can prove this for an arbitrary infinite set $A$ (I'm assuming it's true, but can't provide any proof of this claim). This came up while I was trying to solve some other question, but I'm stuck now.

Any help, feedback or even a solution is much appreciated.

asked Jul 14 '22 at 17:59

Qwen_Marbell

this doesn't always hold, but is equivalent to the axiom of choice. see https://en.wikipedia.org/wiki/Tarski's_theorem_about_choice – peabrainiac Jul 14 '22 at 18:06
See this post as well – Ben Grossmann Jul 14 '22 at 18:06
The statement holds if you assume the Axiom of Choice (in fact, it is equivalent to the Axiom of Choice; this is a theorem of Tarski. – Arturo Magidin Jul 14 '22 at 18:08
To complement the comment of @peabrainiac: This does always hold, because it's equivalent to the axiom of choice. The point is that the meaning of "always" depends on what axiom system you're working in. – Andreas Blass Jul 15 '22 at 01:42

If $A$ is any infinite set, is $A × A$ equivalent/equinumerous to $A$?

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