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I was just wondering the number of all explicit functions , whatever it is continuous or discontinuous, or it has a graph or not.

As far as I know, it seems that $\mathbb{R}^{\mathbb{R}}$ which denotes all functions from reals from reals has a cardinality of $\aleph_2 $, with my attempt like $$card(\mathbb{R}^{\mathbb{R}})=\aleph_1 ^{\aleph_1 } =2^{\aleph_0 \cdot 2^{\aleph_0 }}<2^{2^{2^{\aleph }}}=\aleph_3 $$ If we accepted GCH (the Generalized Continuum Hypothesis) the result followed trivially. But when it comes to "all" functions with no description of domain, I have literally no idea.

Could anyone please help me? Thanks in advance.

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  • $\begingroup$ Isn't $\aleph_0 \cdot 2^{\aleph_0}= 2^{\aleph_0}$? $\endgroup$ – Ranc May 23 '17 at 6:38
  • $\begingroup$ @Ranc Ummm... I've considered this, but I'm not really sure for a elegant proof, so I chose to use a greater tool. $\endgroup$ – BAI May 23 '17 at 6:40
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    $\begingroup$ What is your definition of an "implicit function"? $\endgroup$ – Eric Wofsey May 23 '17 at 6:40
  • $\begingroup$ @EricWofsey I'm sorry I want to say an explicit function at first and my brain experienced short circuit $\endgroup$ – BAI May 23 '17 at 6:42
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    $\begingroup$ $|\mathbb{R}|$ is $2^{\aleph_0}$. The statement $|\mathbb{R}| = \aleph_1$ is the continuum hypothesis, which is indecidable in ZFC. $\endgroup$ – nombre May 23 '17 at 14:25

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