(Ind 2) Suppose \(\mathsf{P}\) is a formal one-place predicate and the prime indicates the successor function, then, this is how we expresse some particular instances of the induction principle in a FOL language: \[\mathsf{(P0\land\forall x(Px\to Px^{'}}))\to\forall \mathsf{xPx}.\] (Ind 3) The general principle of induction that applies to any numerical property in a formal language: \[\mathsf{\forall X((X0\land\forall x(Xx\to Xx^{'}))\to\forall x Xx)}\]...

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