Tarski3,052407

5 this is common in ordinary physical science where

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Unformatted text preview: e cardinal with a normal measure 1 set of lesser measurable cardinals”. There are also some results involving elementary embedding axioms. 4. SINGLE VARYING QUANTITY. We now consider a single varying quantity – where the time and quantity scale are the same, and are linearly ordered. 5 This is common in ordinary physical science, where the time scale and the quantity scale may both be modeled as nonnegative real numbers. The language has >,>>,=,F, where >,>> are binary relations, and F is a one place function. F(x) is the value of the varying quantity at time x. When thinking of time, >,>> is later than and much later than. When thinking of quantity, >,>> is greater than and much greater than. BASIC. > is a linear ordering. x >> y Æ x > y. x >> y > z Æ x >> z. x > y >> z Æ x >> z. ($x)(x >> y,z). x >> y Æ ($z)(x >> z Ÿ z is minimally > y). ARBITRARY BOUNDED RANGES. Every bounded range of values is the range of values over some bounded interval. Here we use L(>,>>,=,F) to present the bounded range of values. AMPLIFICATION. Let y > x be given, as well as a true statement about x, using F, binary >,= and unary >> x. The corresponding statement about x, using F, binary >,= and unary >> y is also true. This also lands at ZFC in the Interpretation Hierarchy. We can strengthen as before: 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 F, binary >,= and binary z >> w >> x. The corresponding statement about y, using F, binary >,= and binary z >> w >> y, is also true. As before, these latter two principles push the interpretation power well into the large cardinal hierarchy. There are versions where we do not assume that the time scale is the same as the quantity scale. Some of these versions use two varying quantities, and there are three separate scales (time, first quantity, second quantity). 6 5. SINGLE VARYING BIT. We now use a bit varying over time. Physically, this is like a flashing light. Mathematically, it corresponds to having a time scale with a unary predicate. In order to get logical power out of this particularly elemental situation, we need to use forward translations of time. We think of b+c so that the amount of time from b to b+c is the same as the amount of time before c. We use >,>>,=,+,P, where P(t) means that the varying bit at time t is 1. Instead of a time scale, we can think of one dimensional space with a direction. P(t) means that there is a pointmass at position t. In the earlier contexts, we did not support continuity. Here we simultaneous support discreteness and continuity. BASIC. > is a linear ordering. x >> y Æ x > y. x >> y > z Æ x >> z. x > y >> z Æ x >> z....
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