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Why do you usually define m+S(n)=S(m+n) but don't m+S(n)=S(m)+n?
#1
So far I have ever seen the definition of addition of two natural numbers is like:
  • m+0=m
  • m+S(n)=S(m+n)

where m, n are both natural numbers, and S(m) is the successor of m. Today I remembered I have never seen some books or whatever defining m+S(n)=S(m)+n. What makes m+S(n)=S(m+n) better than m+S(n)=S(m)+n?

For example, let's take 1+1=2.

If m+S(n)=S(m+n), then

S(0)+S(0)
=S(S(0)+0)
=S(S(0)).

If m+S(n)=S(m)+n, then

S(0)+S(0)
=S(S(0))+0
=S(S(0)).

Whichever you define what m+S(n) equals, it seems you get same result.
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#2
They are equivalents; choosing one or the other is pretty much tradition.
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