I cannot figure out what is wrong:

We will attempt to show that $\mathcal{P} (\mathbb{N})$ is countable. We use the following corollary from Rudin's Principles of Mathematical Analysis, p. 29:

Suppose $A$ is at most countable, and, for every $\alpha\in A$, $B_{\alpha}$ is at most countable. Put

$$T=\bigcup_{\alpha \in A}B_{\alpha}$$

Then $T$ is at most countable.

"Proof" 1:

Let $A = \mathbb{N}$ and for every $\alpha \in A$ let $B_{\alpha}=\{S \in \mathcal{P} (\mathbb{N})| \text{the sum of the elements of } S \text{ is } \alpha \}$. $A$ is countable and for every $\alpha \in A$, $B_{\alpha}$ is finite. Therefore

$$\bigcup_{\alpha \in A}B_{\alpha}$$

is countable. But $\displaystyle \bigcup_{\alpha \in A}B_{\alpha}=\mathcal{P} (\mathbb{N})$, so $\mathcal{P} (\mathbb{N})$ is countable.

"Proof" 2:

Let $A= \mathbb{N}$ and for every $\alpha \in A$ let $B_{\alpha}=\{ S \in \mathcal{P} (\mathbb{N}): |S| = \alpha \}$. I think that I can show by induction (if requested) that for each $\alpha \in A$, $B_{\alpha}$ is countable. Thus

$$\bigcup_{\alpha \in A}B_{\alpha}$$

is countable. But again, $\bigcup_{\alpha \in A}B_{\alpha} = \mathcal{P} (\mathbb{N})$

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    $\begingroup$ In both cases, the union of the $B_{\alpha}$ is the set of finite subsets of $\mathbb{N}$ (which is indeed countable, and this is indeed a correct proof of that fact). $\endgroup$ – Qiaochu Yuan Dec 18 '17 at 1:22
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    $\begingroup$ Can the downvoter please explain? $\endgroup$ – Ovi Dec 18 '17 at 1:25
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    $\begingroup$ Yeah, I am tired of the billion questions all involving the same misunderstanding of Cantor, and further many posters (not all) cannot be reasoned with. So I downvote all such questions. Maybe you should read some of the other answers. $\endgroup$ – Rene Schipperus Dec 18 '17 at 1:28
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    $\begingroup$ @ReneSchipperus Nobody can dictate to you how to use your votes, but the Help Center says "Use your downvotes whenever you encounter an egregiously sloppy, no-effort-expended post, or an answer that is clearly and perhaps dangerously incorrect." I don't think my question falls into any of those categories. Additionally, I don't think my question is a duplicate and I fail to find another where the poster had the same misunderstanding (just an oversight rather) as me. $\endgroup$ – Ovi Dec 18 '17 at 1:47
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    $\begingroup$ Rene Schipperus. You are right. How dare anybody who doesn't understand mathematics ask a question about mathematics on a mathematics help site. I'm tired of people not as perfect as I am wasting mine but not their own time in the pointless purpose of attempting to learn something they don't understand. $\endgroup$ – fleablood Dec 18 '17 at 3:27

In both your "proofs", it is not true that $\displaystyle \bigcup_{\alpha \in A}B_{\alpha}=\mathcal{P} (\mathbb{N})$. Indeed, if $S\in\mathcal{P}(\mathbb{N})$ is any infinite set, then $S$ is not in any $B_\alpha$ (by either definition).

What both your arguments show correctly is that the set of all finite subsets of $\mathbb{N}$ is countable.

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    $\begingroup$ Ah got it thanks! $\endgroup$ – Ovi Dec 18 '17 at 1:24
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    $\begingroup$ @Ovi: Furthermore, this fact is important, because similarly the set of all finite sequences of naturals is countable, such as when counting the number of integer polynomials. $\endgroup$ – user21820 Dec 18 '17 at 2:33
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    $\begingroup$ @user21820 Yes, then we have shown that the set of algebraic numbers is countable. $\endgroup$ – Jeppe Stig Nielsen Dec 18 '17 at 11:09
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    $\begingroup$ @JeppeStigNielsen: Right. And similarly that there are countably many computable reals, and also countably many definable reals regardless of how powerful one's foundational system is. $\endgroup$ – user21820 Dec 18 '17 at 11:15
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    $\begingroup$ @user21820 It is consistent that all reals are definable, so it is not provable that there are only countably many definable reals. See this MathOverflow answer: mathoverflow.net/q/44129 (The other statements are correct.) $\endgroup$ – Robert Furber Dec 18 '17 at 17:41

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