# prove that $|{\{X\subseteq\mathbb{N}|\ |\mathrm{X|={\aleph_{0}}}\}}| = 2^{\aleph_{0}}$ [duplicate]

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)

• 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 – Jneven May 15 '18 at 11:09
• 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
• You also probably mean $|X|= \aleph_0$. – Magdiragdag May 15 '18 at 11:14
• correct. fixed it – 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 $\;|P(\Bbb N)|=2^{\aleph_0}\;$

• What does the / stand for here? Is it a wobbly replaced for \mid? Is it the leaning mid-line of Pisa? :) – Asaf Karagila May 15 '18 at 11:40
• @AsafKaragila Take a wild guess what could it be...:) – DonAntonio May 15 '18 at 11:43
• 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 – Asaf Karagila May 15 '18 at 11:43
• @AsafKaragila Yeah...because, you know, you need to be a set theorist to know how sets are written. Right. – DonAntonio May 15 '18 at 11:45
• Apparently! :)) – Asaf Karagila May 15 '18 at 11:46