This preview shows page 1. Sign up to view the full content.
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
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
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,
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
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.
View Full Document