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Does the power set of the natural numbers contain an infinite number of infinite sets? If not, does there exist a power set of an infinite set that contains an infinite number of infinite sets?

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    $\begingroup$ Every infinite set $X$ has infinitely many infinite subsets. For example, $X \setminus \{ x_1 , \dots , x_k\}$ are infinitely many. $\endgroup$ – Crostul Apr 27 '17 at 22:19
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    $\begingroup$ Take, for example, the set of natural numbers, the set of even numbers, the set of multiples of 3, the set of multiples of 4, and so on. There are infinitely many of these sets and each is countably infinite. $\endgroup$ – infinitylord Apr 27 '17 at 22:19
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Yes. For every $n \in \mathbb{N}$, the set $\mathbb{N} \setminus \{ n \}$ is in $P(\mathbb{N})$, and is infinite. And, there are clearly as many of those as there are natural numbers, i.e. infinitely many.

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Yes, two examples of families of infinite subsets of $\mathbb N$:

$$ A_n=\{k\in\mathbb N\mid k\neq n\} $$ and $$ B_n=\{k\in\mathbb N\mid k\text{ is a multiple of }n\} $$

So you can see it's quite easy to find infinitely many infinite subsets of $\mathbb N$. Actually almost all subsets of $\mathbb N$ (and therefore almost all elements of the powerset of $\mathbb N$ are infinite.

More generally (but only interesting if you know what countability for sets means, and that might above the OP's level): the number of finite subsets of a countable set is countable, and the powerset of any countable set is uncountable, so for any countable set, almost all subsets are infinite.

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A trivial example: The set of integers $\Bbb Z$ is an infinite set. Now, $\Bbb Z$ itself is also a subset of its power set $P(\Bbb Z)$. Likewise, every infinite set has trivially an infinite subset in its power set.

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