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An easy way to see that $P_1$ effects a one-to-one correspondence between $\mathbb{Z}_+^2$ and $\mathbb{Z}_+$ is as follows: Define the "successor" of a pair $(x,y) \in \mathbb{Z}_+^2$ to be the pair $(x-1,y+1)$ when $x=0$; otherwise, when $x=0$, the successor is $(y+1,0)$. It is easy to see that every pair has a successor and that every pair except $(0,0)$ is the successor of exactly one other pair. With this definition of successor, the set of pairs of positive integers satisfies the Peano axioms and, hence, is isomorphic to the integers. From the definition of $P_1$ it follows that, if $(x',y')$ is the successor of $(x,y)$, then $P_1 (x',y') = P(x,y) + 1$ and that $P_1(0,0) = 0$. This means that $P_1$ is the isomorphism described two sentences ago.

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