Tarski3,052407

Take a to be the vl of the given model and s to be

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Unformatted text preview: st element of S after a’. The statement M[S] |= j(x,>,>> x)) can be viewed as a statement in V(»S) about x,b. Likewise, the statement M[S] |= j(x,>,>> y) can be viewed as a statement in V(»S) about x,b’. By property ii), the two statements must have the same truth value. QED THEOREM 7.7. Basic + Minimal + Existence + Amplification has the same Tarski degree as ZF. I.e., they are mutually interpretable. 10 For Basic + Minimal + Existence + Binary Amplification, we need S of type w satisfying a more powerful form of indiscernibility. LEMMA 7.8. Suppose there is a countable transitive model of ZC + “there exists a measurable cardinal”. There is a countable transitive model A of ZF, and an unbounded S Õ A consisting of limit ordinals, of order type w, with the following indiscernibility property. Let a < b < g be from S. Then S\b and S\g have the same first order properties over A, relative to any parameters from the V(a) of A. Proof: Inside the given model, shoot a Prikry sequence of limit ordinals through the measurable cardinal, l. Take A to be the V(l) of the given model, and S to be the range of the Prikry sequence. QED Let A be a transitive model of ZF, and S be an unbounded subset of A consisting of limit ordinals. Define M[A,S] as follows. The domain and the > is defined internally in M, as proper classes of M, as before. We write these as DM, >M. The >>, which is a binary relation on DM, is then defined as before. We write this as >>S. LEMMA 7.9. M[A,S] satisfies Basic + Minimality + Existence. LEMMA 7.10. Let A,S be as in the conclusion of Lemma 7.8. Then M[A,S] satisfies Binary Amplification. Proof: Let the binary element of of S after Œ DMa’. Let y >M x, and j(x) true in M[A,S], using >M,=, and relation y >>S z >>S x. Let a be the least S such that x Œ DMa. Let b be the least element a. Let a’ be the least element of S such that y b’ be the least element of S after a’. Let j(x) be a statement in M[A,S] about x Œ DM, using >M, and the binary relation z >>S w >>S x. The statement M[A,S] |= j(x,>,binary >> x,=)) can be viewed as a statement in (M,Œ,S\b) about x. This is because the binary relation z >>S w >>S x can be defined from M and S\b. Likewise, the statement M[A,S] |= j(x,>,binary >> y,=) can be viewed as the corresponding statement in (M,Œ,S\b’) about x. By the indiscernibility property, the two statements must have the same truth value. QED THEOREM 7.11. Basic + Minimality + Existence + Binary Amplification interprets ZFC + “there exists a Ramsey 11 cardinal” and is interpretable in ZFC + “there exists a measurable cardinal”. PRINCIPLE OF PLENITUDE From Wikipedia, Plenitude Principle. The principle of plenitude asserts that everything that can happen will happen. The historian of ideas Arthur Lovejoy was the first to discuss this philosophically important Principle explicitly, tracing it back to Aristotle, who said that no possibilities which remain eternally possible will go unrealized, then forward to Kant, via the following sequence of adherents: Augustine of Hippo brought the Principle from Neo-Platonic thought into early Christian Theology. St Anselm 's ontological arguments for God's existence used the Principle's implication that nature will become as complete as it possibly can be, to argue that existence is a 'perfection' in the sense of a completeness or fullness. Thomas Aquinas's belief in God's plenitude conflicted with his belief that God had the power not to create everything that could be created. He chose to constrain and ultimately reject the Principle. Giordano Bruno's insistence on an infinity of worlds was not based on the theories of Copernicus, or on observation, but on the Principle applied to God. His death may then be attributed to his conviction of its truth. Leibniz believed that the best of all possible worlds would actualize every genuine possibility, and argued in Théodicée that this best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature's perfection. Kant believed in the Principle but not in its empirical verification, even in principle. The Infinite monkey theorem and Kolmogorov's zero-one law of contemporary mathematics echo the Principle. It can also be seen as receiving belated support from certain radical 12 directions in contemporary physics, specifically the manyworlds interpretation of quantum mechanics and the cornucopian speculations of Frank Tipler on the ultimate fate of the universe. REFERENCE [1] H. Friedman, Concept Calculus, Preprints, #53, http://www.math.ohio-state.edu/%7Efriedman/manuscripts.html *This research was partially supported by NSF Grant DMS 0245349....
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