W. Ackermann, Die Wiederspruchsfreiheit der allgemeinen Mengenlehre, Mathematische Annalen vol 100, 1937.  Home/Search   Document Not in Database   Summary   Related Articles

....straightforward induction. The interpretation thus proves by nitary means that FS Sep is consistent with the set induction axiom. The next step in the development of FSI is to establish an isomorphism between the intended models of FSI and ACA 0 . This can be done by the Ackermann s bijection [1] : H 7 N satisfying (0) 0 and (x [fyg) 2 (y) x) whenever y = 2 x. By working in FSI we can de ne a relation over sets and prove that is satis es the recurrence: x y 9s2ynx8t2xny t s asserting that for sets x and y we have x y i the largest bit in which x and y di er ....

....of natural numbers. With the coding, say by the bit relation 2 1 , one then has a rather annoying duplication: x 2 Y i x 2 1 y for the code y of the nite class Y . Incidentally, FSI is easily seen to be equivalent to VBG minus the axiom of in nity (for a similar equivalence of PA and ZFC see [1]) The theory FSI has a simple axiomatization and minimal ontological commitments: just the natural numbers which are identi ed with hereditarily nite sets and the classes generated by the Weyl G odel operations. We can show it closed under a strong scheme of well founded clausal de nitions ....

W. Ackermann, Die Wiederspruchsfreiheit der allgemeinen Mengenlehre, Mathematische Annalen vol 100, 1937.

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