I need to prove that:

$$|{\{X\subseteq\mathbb{N}|\ |\mathrm{X|={\aleph_{0}}}\}}| = 2^{\aleph_{0}}$$. it's allowed to use the fact that $|P(\mathbb{N})|=2^{\aleph_{0}}=|\mathbb{R}|$ - this is the original form of the Q.

It's written terribly, and that's a part of why I failed to prove it. the Q is essentially: prove that the cardinality of the SET of all the infinite sub-sets of $\mathbb{N}$ (referred as $\mathrm{X}$), is equal to $2^{\aleph_{0}}$. further more, I'd rather prove that without using onto/one-to-one function, only if possible, in the tools of elementary set theory (meaning, without ZFC)

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    $\begingroup$ Twice you wrote a weird (for me) right curly parentheses $\;\}\;$ ...what does it mean? And also: $\;X\subset\Bbb N\implies |X|\le|\Bbb N|=\aleph_0<2^{\aleph_0}\;$ ... $\endgroup$ – DonAntonio May 15 '18 at 11:06
  • $\begingroup$ fixed it. as I explained, its the set of ALL X's $\endgroup$ – Jneven May 15 '18 at 11:09
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    $\begingroup$ Again, your set is empty, as there is no subset of the naturals with cardinal equal to $\;2^{\aleph_0}\;$ ...Perhaps you meant the set of all infinite subsets of $\;\Bbb N\;$ ...or something of the like? And edit also the question's header. $\endgroup$ – DonAntonio May 15 '18 at 11:13
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    $\begingroup$ You also probably mean $|X|= \aleph_0$. $\endgroup$ – Magdiragdag May 15 '18 at 11:14
  • $\begingroup$ correct. fixed it $\endgroup$ – Jneven May 15 '18 at 11:27

An idea:

Prove first that the set of all finite subsets of the naturals is countable:

$$\;\left|\left\{\,X\subset\Bbb N\;/\;|X|<\aleph_0\,\right\}\right|=\aleph_0$$

and then use directly that $\;|P(\Bbb N)|=2^{\aleph_0}\;$

  • $\begingroup$ What does the / stand for here? Is it a wobbly replaced for \mid? Is it the leaning mid-line of Pisa? :) $\endgroup$ – Asaf Karagila May 15 '18 at 11:40
  • $\begingroup$ @AsafKaragila Take a wild guess what could it be...:) $\endgroup$ – DonAntonio May 15 '18 at 11:43
  • $\begingroup$ I'd say a fraction. Because you're not a set theorist, and you're unaware of the fact that fractions are not used in set theory like that. :P $\endgroup$ – Asaf Karagila May 15 '18 at 11:43
  • $\begingroup$ @AsafKaragila Yeah...because, you know, you need to be a set theorist to know how sets are written. Right. $\endgroup$ – DonAntonio May 15 '18 at 11:45
  • $\begingroup$ Apparently! :)) $\endgroup$ – Asaf Karagila May 15 '18 at 11:46

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