نظریه ابتدایی اعداد
However, two remarkable theorems assure us, on the contrary, that neither the set shortage nor the sentence shortage will have any bearing on the definition of validity as Jong as our object language is reasonably rich: rich enough for elementary number theory.
The demand that the object language be rich enough for elementary number theory is a moderate demand.
Elementary number theory comprises only what can be said about positive integers in terms of plus, times, identity, truth functions, and quantification; no sets.
(I) If a schema comes out true under all substitutions of sentences of elementary number theory, it is satisfied by every model.
Elementary number theory is in effect equivalent still to a certain amount of set theory, as is the theory of finite sequences, but either way it is a modest part: the theory of finite sets.
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