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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.
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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.
 Fall '08
 JOSHUA
 Math, Calculus, Set Theory

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