They are susceptible to a decision procedure as illustrated by the example of checking for fixed n and any given triple of integers x,y,z whether or not the sum of the n-th powers of the first two is equal to that of the third. The amazing often elusive power of the universal quantifier brought home by G del's incompleteness proof, discussed in the next section, is again manifest in the intrinsic difficulties with which the conclusive proof of Fermat's theorem is fraught. Based on a naive concept of Truth, every true theory of a definite structure is complete and consistent in both senses, a pretty useless observation. P is a principle that adults may consider a definition of the set N, while children will in my experience take it for granted. But, if you want to articulate it in the language of the first order predicate calculus you run into trouble.
P is a typical sentence of second order logic.
W of the elementary language of arithmetic. But there is a subtlety here. G is the sentence that truthfully claims its own unprovability. Much deep work is required to establish this result rigorously. Occasionally proofs by mathematical induction are confused with arguments based on so-called 'inductive reasoning', a term used in philosophical discussions of logic and the sciences yet another reason for all these elaborations.