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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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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