Baby Rudin 2.12 vs power set of N

Question

Baby Rudin 2.12 states that union of countable number of countable sets is countable. Doesn’t this contradict that the power set of $N$ is uncountable? Can somebody please explain?

What I mean is: set of 2-element subsets is countable, set of 3-element subsets is countable etc. There’s a countable number of these collections, so why isn’t their union countable?

Hi, please explain your argument that the power set of the natural numbers is countable, so that we can point out the flaw. — Simon, Mar 18 '20 at 23:32
The power set of $\Bbb{N}$ is not the union of a countable number of countable sets. Read about Cantor's diagonal argument. A countable collection of a countable sets can be enumerated using a pairing function. — Rob Arthan, Mar 18 '20 at 23:45
What I mean is: set of 2-element subsets is countable, set of 3-element subsets is countable etc. There’s a countable number of these collections, so why isn’t their union countable? — user623949, Mar 18 '20 at 23:48
But what about the infinite subsets of the natural numbers ? — Simon, Mar 18 '20 at 23:56
I see what you mean. So if we only take finite sets, then it’s countable? — user623949, Mar 18 '20 at 23:59

score 0 · Answer 1 · answered Mar 18 '20 at 23:33

The set of prime number is countable denote it by $\{p_1,...,p_i\}$ Let $A_n$ a family of countable sets $f_i:A_i\rightarrow\mathbb{N}$ an injection, write $g(a_i)=p_i^{f_i(a_i)}, a_i\in A_i$; $g$ is injective.

If $S$ is a set $|S|<P(S)|$ the canonical trick, let $f:S\rightarrow P(S)$ bijection; $T\subset S, T=\{s\in S, s$ is not an element of $f(s)\}$, suppose that $f(t)=T, t\in T$ impossible by definition of $T$, but if $t$ is not in $T$, it is in $T$ by definition, contradiction.

score 0 · Answer 2 · answered Mar 18 '20 at 23:48

0

I don't see what one thing has to do with the other. The power set of $\mathbb{N}$ is uncountable. The union of $\mathscr{P}(\mathbb{N})$ is countable, however. In fact, it's $\mathbb{N}$. The cardinality is the number of elements in the set. The union is the set of elements in any one of the sets in the collection. Two different things.

answered Mar 18 '20 at 23:48

saulspatz

53,131

Please see my added details above. – user623949 Mar 18 '20 at 23:49
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@user623949 You are correct in that there are countably many finite subsets, but what about the infinite subsets? There are uncountably many of those. – saulspatz Mar 18 '20 at 23:52

Baby Rudin 2.12 vs power set of N

2 Answers2