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Extra resources for The Philosophy of Mathematics (Readings in Philosophy)

Example text

1o. If F and F 2 are formulae, so is 1(F 1) t (F 2) l, The free occurrences 1 of any variable a. are those of F 1 and those o~F 2 . IfF1 a~d F~ are sentences, then I(F 1) (F 2) l is a sentence and is true 1f and only 1f ne1ther F 1 nor F2 is true. 10 t 10 We could have used the single primitive 'c' (class inclusion) in place of the joint use of •t' and '=' (as occurring between abstracts). We would then have ~ system formulated in a logic based on inclusion and abstractiOn (m the sense of Qume).

However, this is as much as we need of a Tarsk1 sentence (for undecidability results). w. 4 The former will be carried out in this paper and the latter m a forthcommg paper, 'Systems of Protosyntax Self Applied'. 65 norm function, and we finally apply the results, in Section 4, to a systemSA• which is a formal variant of elementary arithmetic. This variant consists of , taking the lower functional calculus with class abstractors, rather than , \ quantifiers, as primitive. This alteration, though in no way affecting the , strength of the system, nevertheless makes possible the particularly simple proof of Tarski's or Godel's theorem, since the arithmetization of substitution can thereby be circumvented quite simply.

3. Semantical Systems with Predicates and Individual Constants. In this section, we consider any semantical system S, of which certain expressions called predicates and certain expressions called (individual) constants are so related that any predicate followed by any constant is a sentence of S. , we consider a 1-1 correspondence g whose domain is the set of all expressions, and whose range is a subset (proper or otherwise) of the individual constants. We shall often write 'E' for 'g(E)'. , E followed by g(E)).

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