In my notes the construction of the least ordinal follows from taking the supremum of the image of the set $$ R=\{ A\in \mathcal{P}(\mathbb{N}\times\mathbb{N}) : A\text{ is a well-ordering of a subset of }\mathbb{N}\},$$ under the order type function. I.e. the supremum of the set of all countable ordinals.
I'm having trouble understanding the notation in the definition of $R$. The rest of the proof makes sense but I'm not sure how $A\in \mathcal{P}(\mathbb{N}\times\mathbb{N})$ can be a well-ordering of a subset of $\mathbb{N}$. Any help would be greatly appreciated.
What is a well-ordering of a subset of $\Bbb N$? It's a set of ordered pairs of natural numbers. In other words, it is an element of $\mathcal P(\Bbb{N\times N})$.
So, $\{(0,1),(0,2),(2,1)\}$ is a well-ordering of $\{0,1,2\}$ that orders the set as $0 \prec 2 \prec 1$. And $\{(n,m)\mid n<m\text{ and both are even, or } m=1\text{ and }n\text{ is even}\}$ is a well-ordering of $\{2k\mid k\in\Bbb N\}\cup\{1\}$.