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I'm looking for cardinality of $P_{fin}(\Bbb{N})=\{x|x\subset\Bbb{N}$ and $x$ finite$\}$. I was told in my classes that it's $\aleph_0$, but how to prove it?


marked as duplicate by Asaf Karagila cardinals Jan 27 '15 at 17:14

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  • $\begingroup$ Show there is a bijection between $P_{fin}(\mathbb N)$ and $\mathbb N$. $\endgroup$ – davcha Jan 27 '15 at 9:41

Hint: Show that there is a natural bijection between $P_{\text{fin}}(\mathbb{N})$ and $\mathbb{N}$; one particularly nice one is more apparent when you work in binary.

  • $\begingroup$ Is it something like: Give me a subset and I'll give you 1... where ... is a number that on i'th position has 1 if i is in set and otherwise 0? $\endgroup$ – qiubit Jan 27 '15 at 9:49
  • 1
    $\begingroup$ Yes, that will do nicely! All you have to do to finish the argument is to show that this candidate map is bijective. In this case, it's not too difficult to describe the inverse of the map explicitly. $\endgroup$ – Travis Willse Jan 27 '15 at 9:51

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