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I am going through Analysis V.1, Terence Tao. In his definition of Addition, screenshot given below, how did he deduce that (N++)+M := (N+M)++ ?? I am not able to understand the steps.

Definition of Addition Definition 2.2.1, Volume 1 Analysis, Terence Tao

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    Thats a recursive definition of addition of natural numbers. If you know what n+m is, then you define inductively (on n) what $(n++)+m$ is. You could also view it as function $f_m:\mathbb N \rightarrow \mathbb N:m \mapsto n+m$. Then you define $f(0)=m$ and given that value you definie $f(n++)$. Thus $f(n++):=f(n)++$, which is a natural way do to so.2012-11-19

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You have to remember that in developing formal rules for addition, we are taking the properties in which we are already naturally familiar and changing them into a rigorous definition.

Informally, we perceive $n++$ as $n+1$. Therefore the recursive definition simply says that $(n+1) + m = (n+m)+1$ You must remember that we already have in mind what properties we want addition to have and that we are simply providing a definition.

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    He doesn't prove it. He _defines_ addition as the above recursive rule. Up to this point, there is not such thing as addition, so there is nothing _to_ prove.2012-11-19