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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)

Noa Even
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    Twice you wrote a weird (for me) right curly parentheses $;};$ ...what does it mean? And also: $;X\subset\Bbb N\implies |X|\le|\Bbb N|=\aleph_0<2^{\aleph_0};$ ... – DonAntonio May 15 '18 at 11:06
  • fixed it. as I explained, its the set of ALL X's – Noa Even May 15 '18 at 11:09
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    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. – DonAntonio May 15 '18 at 11:13
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    You also probably mean $|X|= \aleph_0$. – Magdiragdag May 15 '18 at 11:14
  • correct. fixed it – Noa Even May 15 '18 at 11:27
  • Your question is not different from the duplicate. It is exactly the same question. The wording is different, yes. But you are asking how many subsets of $\Bbb N$ are infinite, because the subsets of $\Bbb N$ which are infinite are exactly those that have cardinality $\aleph_0$. If you don't see why, then it is a good exercise to figure out this thing before moving forward. – Asaf Karagila May 15 '18 at 11:39
  • Not only the question is the same, the answer you've received and accepted from DonAntonio is exactly the same answer I gave to the duplicate. So I am quite befuddled as to why you disagree that this is a duplicate. – Asaf Karagila May 15 '18 at 11:44

1 Answers1

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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 $\;|P(\Bbb N)|=2^{\aleph_0}\;$

DonAntonio
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