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?
On
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.
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.