Email: [email protected]
Copyright, 2001
The Russell Paradox considers the set, R, of all sets that are not members
of themselves. As Rucker (1) writes
"Is R a member of R?...And if R is a set and not a member of R, then R
(being a set that is not a member of itself) is a member of R. So, if R is
a set, then we have a contradictory state of affairs with R being in itself
if and only if it is not in itself. So we are forced to conclude that R is,
in fact, not a set".
My own response to this paradox is that R can be a set that is not a member of
itself and still not be a member of R. This is because it is instead a member
of a new set R1, as described below.
Based on the definition of "exists" given in an accompanying paper (2),
anything that "exists" "exists" because it is a whole. A whole, by virtue of
its wholeness or completeness, has an edge which limits and defines that whole
and gives it existence. A whole is something that is fully defined; that is,
the criteria for inclusion (or the "membership list") in that whole (ie, in
that extent) is fully defined. Therefore, in getting back to the Russell
Paradox, in order to even consider set R, it must be redefined as the "set of
all sets not members of themselves and that currently exist at this time, x".
This is because if an element of a set is not yet in existence, then the
membership list of this set cannot be fully defined, and the set itself does
not exist. So, in order to even consider R as existing, it needs to be
redefined as the "set of all sets not members of themselves and that currently
exist at this time, x". Given this, it seems clear that R itself can not be a
member of the "set of all sets not members of themselves and that currently
exist at this time, x" because R did not exist at the time, x, that the
membership list of this set was being fully defined. Additionally, one can't
retroactively stick R back into the membership list of the "set of all sets
not members of themselves and that currently exist at this time, x" because it
would then become a different membership list and a different set, not set R.
However, this doesn't mean that R is not a set. What it does mean is that,
while R can't be a member of the original "set of all sets not members of
themselves and that currently exist at this time, x", it can be a member of a
new set, R1, of the "set of all sets not members of themselves and that
currently exist at this time, x1".
Overall, I would say that the definition of R in the original paradox
needs to be modified to be the "set of all sets not members of themselves and
that currently exist at this time, x".
Another way of looking at this is via the idea of reference frames. Once
one defines the membership list of a set, the set springs into existence, and
one's reference frame now shifts to the "outside" of the set in a different
reference frame where one sees the set as a whole and as being in existence.
The set itself cannot be pushed back into the original membership list because
this list is in a different reference frame, internal to the set. Therefore,
all sets, and all things that exist, are defined as existing relative to a
reference frame.
This same reasoning applies to Godel's Incompletness Theorem, which
states that there will always be some statements within a system, R, that
cannot be proven true or false. The reasoning applies as follows:
1. Assume that system R is a set of axioms, logical rules, symbols etc.
2. Assume that statement S0 is a statement based on the rules in R.
3. Assume first that statement S1A is "statement S0 is true" and that S1A is
true. One cannot make statement S1A until after one knows exactly what
all the rules in R are, that is until R is completely defined and
"exists". Only then can one say that S1A is true.
Now, forget about statement S1A and assume next that that statement S1B is
"statement S0 is false" and that S1B is true. One cannot make statement
S1B until after one knows exactly what all the rules in R are, that is
until R is completely defined and "exists". Only then can one say that S1B
is true.
3. Overall, if R must first exist and be completely defined before one can
even make statements S1A and S1B, then neither statement can be a member of
R because the membership list in R has already been completely defined
before S1A and S1B even came into being. And, if one tries to add S1A or
S1B to R after R has already been formed, then R is no longer truly R but a
new set, R1.
4. So, statements S1A and S1B are not contained within R. They are more
statements about things within R but are not themselves in R. This means
that one cannot say that there are things within system R that cannot be
proven to be true or false. It also means that there are some things that
system R cannot prove to be true or false but that these things are outside
R.
References
1. Rucker, R. "Infinity and the Mind. The Science and Philosophy of the
Infinite", Princetone University Press, 1995, pg. 193-193.
2. http://www.geocities.com/roger846/theory.4.html
Email: [email protected]
Copyright, 2001