P. J. Voda, Predicative Justi cation and Development of Theory of Finite Sets, www.fmph.uniba.sk/svoda, 2003. 44  Home/Search Document Not in Database Summary Related Articles Check |
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What Can we Gain by Integrating a Language Processor with a.. - Voda (2003) Self-citation (Voda) (Correct)
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P. J. Voda, Predicative Justi cation and Development of Theory of Finite Sets, www.fmph.uniba.sk/svoda, 2003. 44
Grundlagenstreit in the Theory of Programming - Voda (2003) Self-citation (Voda) (Correct)
....symbol 2 of class membership. The atomic formulas are 1 2 2 and 1 = 2 for any combination of set and class variables. Actually, the two sorted axiomatization can be reduced to the single sort of classes which dispenses with the two sorting axioms (1) 2) We have used this approach in [17] which should be referred to for details of the following sketch of the bootstrap of FSI. The rst axiom asserts that every set is a class: 9X x = X : 1) The axiom of extensionality identi es two classes containing the same sets: 8z(z 2 X z 2 Y ) X = Y : FSI then proves as theorems the ....
....(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 FS Sep is consistent with the set ....
P. J. Voda, Predicative Justi cation and Development of Theory of Finite Sets, www.fmph.uniba.sk/svoda, 2003.
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