Do an infinite set and its double have the same cardinality?

Asked Jan 08 '15 at 15:51

Active Jan 08 '15 at 17:56

Viewed 117 times

My question was inspired by this answer.

Suppose $A$ is an infinite set. Does its double, $A\times\{0,1\}$, always have the same cardinality?

In my head I quickly spotted a simple proof that the answer is yes if we assume Axion of choice / Zorn's lemma. I have stumbled upon a couple of related questions, but the question looks so simple that I would like it answered without assuming AC.

edited Apr 13 '17 at 12:20

Community

asked Jan 08 '15 at 15:51

Heimdall

If you want to link to a specific answer, you can find the address if you look at "share" directly underneath the answer. – Hans Lundmark Jan 08 '15 at 17:52
Thanks, @HansLundmark, I've edited my post. – Heimdall Jan 08 '15 at 17:54
1

See http://math.stackexchange.com/questions/393196/the-relationship-of-frak-mm-m-to-ac?rq=1 – Jan 08 '15 at 18:14
Thanks, @OohAah. Rather interesting. I'll have to read about Dedekind-finite sets. – Heimdall Jan 14 '15 at 23:09

Do an infinite set and its double have the same cardinality?

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