Tarski3,052407

The corresponding statement about x using p binary

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Unformatted text preview: ($x)(x >> y,z). x >> y Æ ($z)(x >> z Ÿ z > y). BOUNDED TIME TRANSLATION. For every given range of times before a given time b, there exists a translation time c such that a time before b lies in the range of times if and only the bit at time b+c is 1. Here we use L(>,>>,=,+,P) to present the range of times. The idea is our usual one: the behavior of P over bounded intervals is arbitrary, up to translation. AMPLIFICATION. Let y > x be given, as well as a true statement about x, using P, binary >,= and unary >> x. The corresponding statement about x, using P, binary >,= and unary >> y is also true. AMPLIFIED LIMIT. There is something that is greater than something, and also much greater than everything it is greater than. BINARY AMPLIFICATION. Let y > x be given, as well as a true statement about x, using P, binary >,= and binary z >> w >> x. The corresponding statement about x, using P, binary >,= and binary z >> w >> y is also true. 7 The analogous results hold. 6. PERSISTENTLY VARYING BIT. The objection can be raised that a varying bit realistically has to have persistence. It cannot be varying “densely”. Specifically, if the bit is 1 then it remains 1 for a while, and if the bit is 0 then it remains 0 for a while. Define a persistent range of times in the obvious way. PERSISTENT TIME TRANSLATION. For any time b and persistent range of times before b, there exists a translation time c such that any time before b lies in the range of times if and only if the bit at time b+c is 1. Here we use L(>,>>,=,P,+) to present the range of times. We need to have two additional time principles. ADDITION. y < z Æ x+y < x+z. ORDER COMPLETENESS. Every nonempty range of times with an upper bound has a least upper bound. Here we use L(>,>>,=,P,+) to present the nonempty range of times. We also have our usual AMPLIFICATION, AMPLIFIED LIMIT, BINARY AMPLIFICATION. We get the analogous results. 7. SOME MODELS. We first form an underlying structure (D,>,=). We define pairs (Da,>a), for all ordinals a. Define (D0,>0) = (∅,∅). Suppose (Da,>a) has been defined, and is transitive and irreflexive. Define (Da+1,>a+1) to extend (Da,>a) by adding an exact strict upper bound for every subset of Da even if it already has an exact strict upper bound. For limit ordinals l, define Dl = »a<lDa, >l = »a<l>a. Let D = »aDa, > = »a>a. The new elements introduced at each stage are incomparable. 8 The (eventual) predecessors of x are introduced earlier than x. Each exact upper bound introduced remain valid later. LEMMA 7.1. (D,>) is irreflexive and transitive, satisfies Minimality, and also Existence in second order form. The same claims are true for (Dk,>k), where k is a limit ordinal. Proof: Let A Õ D, with a strict upper bound x Œ Da. Then A Õ Da. Hence an exact upper bound for A is introduced in Da+1, which remains an exact upper bound in D. The same argument works for Dk. For...
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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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