Tarski3,052407

Ie we have a consistency proof for mathematics zfc

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Unformatted text preview: on. I.e., we have a consistency proof for mathematics (ZFC) relative to that of T. 3 Furthermore, relative consistency proofs arising this way are generally very finitary. AMBITION: Newton/Leibniz calculus: Science and engineering. = Concept Calculus: Everything else. 3. BETTER THAN, MUCH BETTER THAN. We begin with the notions: better than (>), and much better than (>>). These are binary relations. The passage of > to >> is an example of what we call concept amplification. One can also view > and >> mereologically, as x > y iff y is a “proper part of x”. x >> y iff y is a “small proper part of x”. Define z is minimal if and only if ÿ($w)(z > w). Define z is minimally > y ´ ("w)(z > w ´ (y > w ⁄ y = w)). The nonlogical axioms of T(>,>>,=) BASIC. ÿx > x, x > y Ÿ y > z Æ x > z, x >> y Æ x > y. x >> y Ÿ y > z Æ x >> z. x > y Ÿ y >> z Æ x >> z. ($x)(x >> y,z). x >> y Æ ($z)(x >> z Ÿ z is minimally > y). MINIMAL (Political Axiom). There is nothing that is better than all minimal things. EXISTENCE (Exact Bound, Plenitude). Let x be a thing better than a given range of things. There is something that is better than the given range of things and the things that they are better than, and nothing else. Here we use L(>,>>,=) to present the range of things. Existence is like fusion. Here the “range of things” is given by a first order formula in L(>,>>,=) with parameters allowed. 4 AMPLIFICATION (Special Indiscernibility). Let y > given, as well as a true statement about x, using binary relations >,= and the unary relation >> x. corresponding statement about x, using >,= and >> also true. x be the The y, is THEOREM 3.1. Basic + Minimal + Existence + Amplification is mutually interpretable with ZFC. This is provable in EFA. AMPLIFIED LIMIT (Star). There is something that is better than something, and also much better than everything it is better than. BINARY AMPLIFICATION. Let y > x be given, as well as a true statement about x, using the binary relations >,= and the binary relation z >> w >> x. The corresponding statement about x, using >,= and z >> w >> y, is also true. Leads to much higher places in Interpretation Hierarchy than ZFC: THEOREM 3.2. Basic + Minimal + Existence + Amplification + Amplified Limit interprets ZFC + “there is an almost ineffable cardinal” and is interpretable in ZFC + “there exists an ineffable cardinal”. THEOREM 3.3. Basic + Minimal + Existence + Binary Amplification interprets ZFC + “there exists a Ramsey cardinal” and is interpretable in ZFC + “there exists a measurable cardinal”. THEOREM 3.4. Basic + Minimal + Existence + Binary Amplification + Amplified Limit interprets ZFC + “there exists a measurable cardinal with arbitrarily large lesser measurable cardinals” and is interpretable in ZFC + “there exists a measurabl...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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