EDB — 292

[292]Having fixed the parameter \(h∈ ℕ\), we define the operation \(h+⋅\), which will be a function \(f_ h : ℕ→ℕ \) given by \(f_ h(n)=h+n\), using a recursive definition: we wish to express the rules

• \( h+0=h\) ,

• \(∀ n∈ ℕ, h+S(n)=S(h+n)\) .

To this end, set \(A=ℕ\), and \(g(n,a)=S(a)\), we rewrite the above as recursive rules for \(f_ h\)

• \( f_ h(0)=h\) ,

• \(∀ n∈ ℕ, f_ h(S(n))=g(n,f_ h(n))=S(f_ h(n))\) .

This defines recursively \(f_ h\). Considering then the parameter \(h\) as a variable, we have constructed the addition operation, and we define the operation ”+” between natural numbers as \(h+n=f_ h(n)\).