Proposition
2
[1XC] Let \(Aβ β\) and \(P(n)\) be a logical proposition that can be evaluated for \(nβ A\). Suppose the following two assumptions are satisfied:
\(P(n)\) is true for \(n=0\) and
\(β nβ β, P(n)β P(S(n))\)Β ;
then \(P\) is true for every \(nβ β\).
Proof
Let \(U=\{ nβ β:P(n)\} \), we know that \(0β U\) and that \(β x, x β Uβ S(x)β U\) , then from (N5) we conclude that \(U=\mathbb {N}\).