Can The Null Set Change in Time?

From: OsherD (mdoctorow_at_comcast.net)
Date: 02/04/05 

Date: 3 Feb 2005 23:24:02 -0800

>>From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Can The Null Set Change in Time?
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005.

(This Copyright Notice isn't intended to prevent discussion or
comments by others, but was suggested to me by a lawyer friend who
pointed out that internet postings without Copyright Notices are
public domain for anybody to do anything with that he/she wants
to. Now at least they can't rewrite history.)

The null set of Zermelo-Fraenkel (ZF) set theory and other set
theories is ordinarily regarded as having no members and hence
no properties (other than the property of having no members or
elements). However, it is possible to index a set A by a time
subscript as in A_t or just At here (my computer doesn't have
subscripts), just as it is possible to index any sequence by
(time) subscripts and in particular in probability-statistics
any sequence of random variables through time, etc. If all
elements of the collection of these At, {At}, are regarded as
representing the same set (let's write it A) changing through
time by addition or subtraction of elements or by changing a
property of the set at a particular time, then if the set A is
the physical universe which changes through time by expansion,
it would be curious for the universe to change without the
null set changing.

Another curiosity if not paradox is the question of how to
express a non-existing element which could someday or potentially
exist or which has previously existed, for example in the above
sequence {At}. We could say in English: x is not an element of
A at time t, namely of A_t or At, but x is an element of A_t' for
t' some later time than t. Since x is not in At, with At the
universe at time t for example, then it seems correct to extend
the definition of null set N_t of A_t to include x. In that case,
an element x can be in the null set of the universe A at time t
but leave it at time t'.

This problem has actually arisen in probability-statistics! See
my posting a few minutes ago to sci.stat.math, where I pointed
out that in slightly different language such an extension of
the definition of null set could explain the expansion of our
physical universe by its "outside" (null set) changing from
having no members to having a "potential" member x and so on.

In fact, using A' for set A (complement of set A, that is to say
the part of the universe not in A) as an analog of negation (~)
in classical logic and set union U ("and/or") as an analog of
disjunction V in logic, and set intersection (upside down U, or
simply writing two sets adjacently like AB) as an analog of
conjunction ("and") in logic, we have with P( ) meaning probab-
ility of:

1) P(A-->B) = 1 + P(AB) - P(A)

and putting A = N (the null set) and B = U the universe, we get:

2) P(N-->U) = 1 + P(N) - P(N) = 1

so that the null set has probability 1 ("certainty" up to sets of
probability 0) of influencing the universe. See my postings on
probable influence (PI) over the last two years at sci.stat.math,
either with PI in the titles or Probable Influence or P(A-->B)
types of expressions or the words "Riccati equation" or
"Volterra equation".

Osher Doctorow

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