S. Feferman, Predicativity, to appear in S. Shapiro ed. Handbook of the Philosophy of Mathematics and Logic, http://math.stanford.edu/sfeferman/papers.html, 2002.  Home/Search   Document Not in Database   Summary   Related Articles

....Our main argument will be: What is good for modern mathematics should be more than sucient for the theory of computer programming. 2 Ontologies for Computer Programming We start with the question: What is the minimal ontology for mathematics The famous indispensability argument of W. Quine (see [5]) is that mathematics is justi ed only insofar as it is needed for the development of natural sciences. Quine thus accepts the original Zermelo set theory (ZF) without the Replace2 ment axiom of Fraenkel whose model is the set V 2 of the cumulative hierarchy of sets. Anything stronger is to be ....

....Cnv 1 (X) hy; z; xi 2 X hx; y; zi 2 Cnv 2 (X) hx; z; yi 2 X 6 where the ordered pair has the standard de nition: hX; Y i = fXg[ffXg[fY gg and where we abbreviate hX; hY; Zii to hX; Y; Zi. Note that hx; yi 2 V . Even the induction axiom, which is to some logicians of impredicative character [3, 5], can be predicatively justi ed (see [17] by interpreting FSI in the weaker theory FS Sep where FS is FSI minus set induction and Sep is the axiom of separation: x X 2 V . Note that Sep is a theorem of FSI by a straightforward induction. The interpretation thus proves by nitary means that ....

S. Feferman, Predicativity, to appear in S. Shapiro ed. Handbook of the Philosophy of Mathematics and Logic, http://math.stanford.edu/sfeferman/papers.html, 2002.

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