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We have the following results

Let $A$ and $B$ be infinite sets s.t. $|A|=|B|$, then $|A\cup B|=|A|$.

I was wondering if we can prove that without the Axiom of Choice or without using cardinal numbers, it's showing that exists the bijection. I'm trying the second part using Zorn's lemma but I've failed.

In the other hand, we have

Let $A$ be an infinite set, then $|A\times A|=|A|$.

I know this is equivalent to Axiom of Choice, but I'm trying a proof without using cardinal numbers.

Please, could you help me whit ideas for that?

MateAndres
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  • How do you want to prove statement about cardinal numbers without using cardinal numbers? Anyways, your statement is not provable without choice. It's not equivalent to choice, but still unprovable without it. – Wojowu Jan 17 '16 at 15:56
  • I suspect the OP is possibly duplicate. – Hanul Jeon Jan 17 '16 at 16:01
  • Regarding the first result: It does not imply choice, but it is not provable without some additional assumption. To see the latter, recall that it is consistent that there are infinite Dedekind finite sets. – Andrés E. Caicedo Jan 17 '16 at 16:15
  • Both questions are duplicates. (There are threads discussing pretty much every aspect. From the use of ordinals, to Zorn's lemma or the necessity of choice in either occasion.) – Asaf Karagila Jan 17 '16 at 17:35
  • Off the top of my quick search: http://math.stackexchange.com/questions/1043350/show-that-an-infinite-set-c-is-equipotent-to-its-cartesian-product-c-times-c and http://math.stackexchange.com/questions/1041731/prove-that-if-a-is-an-infinite-set-then-a-times-2-is-equipotent-to-a – Asaf Karagila Jan 18 '16 at 14:37

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