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I've seen a proof that shows that rational numbers are not the same infinity as real numbers.

I'm curious if there is a proof that shows $\mathbb{R}^1$ and $\mathbb{R}^2$ are the same infinity. If so, is there a way to map from one to the other for all points?

VenomFangs
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  • I apologize for the categorization. I took a stab at what I thought this might be related to. Its beyond what I learned in college. – VenomFangs Apr 30 '16 at 14:48
  • I think this might be a repeat of http://math.stackexchange.com/questions/75107/injective-map-from-mathbbr2-to-mathbbr, can someone confirm? – VenomFangs Apr 30 '16 at 14:53
  • Are you asking if they have the same cardinality? – Sigur Apr 30 '16 at 14:56
  • Yes, and if you can map from one to the other. I think the link in my 2nd comment is asking the same thing and there is a solution, so I think the answer is yes. – VenomFangs Apr 30 '16 at 14:57
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    @venomfangs About the link: yes. Since there exist well-known injective maps $\Bbb R\to \Bbb R^2$, the existence of an injective map $\Bbb R^2\to \Bbb R$ yields that there exists a bijection $\Bbb R^2\leftrightarrow \Bbb R$, via Cantor-Bernstein theorem. –  Apr 30 '16 at 14:59
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    If two sets have the same cardinality it means that there exists a bijection between them. – Sigur Apr 30 '16 at 14:59
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    The two sets have the same cardinality. The question you refer to can be used as a main step in the proof. – André Nicolas Apr 30 '16 at 14:59
  • Thanks everyone. I need to confirm this before I posted the next part of my question, which is http://math.stackexchange.com/questions/1765522/mathbbr2-to-mathbbr1-injective-mapping-while-preserving-metric-space – VenomFangs Apr 30 '16 at 15:09

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