FOM: Re: Determinate Truth Values
Matt Insall
montez at rollanet.org
Tue Sep 5 17:53:28 EDT 2000
For some reason, I am having the hardest time getting this message posted to
FOM.
Something strange is happening to it between my computer and the FOM list.
Please forgive the recent blank post from me with the same (or similar)
title.
Martin Davis:
I find the notions: "subset" and "power set" crystal clear. Likewise for
omega in the sense of the von Neumann finite ordinals. Since CH is a very
specific assertion involving these notions, I regard it as having a
determinate truth value.
Matt:
So do I. I have never really been sure why it was ever thought that these
were any less clear than, say, the extension principle. Clearly, the
extension principle is independent of certain other axioms of set theory,
and in fact, researchers in various disciplines have denied certain axioms
of ZF for various purposes. As I recall, Aczel, among others, denies, for
instance the axiom of foundation. Personally, I find the idea that a set
might be a member of itself, as well as any other cycle of membership
irritating enough to prefer that such objects not even be referred to as
sets. However, if we are going to deny statements like CH because it does
not follow from the axioms of ZF and some people do not like its
consequences, then I see no reason for failing to question each axiom of ZF
in the same manner. At each step in teaching set theory, I must appeal to
my intuition to explain why I accept the next axiom, when we have shown it
to be independent from, but, thank goodness, consistent with the other
axioms. There is a sense in which Formalists should accept CH (I
personally
feel compelled to deny it). The reason is the following: Historically,
Cantor stated CH as part of his set theory. He clearly originally
``wanted'' it to be true. He apparently thought it was provable. That
turned out to not be the case. So what? Neither is AC. But it was part
(if a ``hidden'' part) of Cantor's set theory, and was adopted by many when
it was found to be consistent with ZF. Once CH was found to be consistent
with the other axioms, it should have been considered part of set theory.
At least, it is part of what one may call ``Cantorian Set Theory''. Now, I
have said before why I feel compelled to deny CH, but again, that is
essentially a ``philosophical position'' at this time in history, as I see
it. One way that one choice or another can eventually be justified is in
terms of its consequences. In particular, if ZF+CH is ever found to have a
consequence in certain models of physical reality that contradicts the
results of experiments, then it may be appropriate at that time to at least
decide that CH is unsuitable for physics, if nothing further can be said.
It would then behoove applied mathematicians to adopt ZF+not(CH), perhaps.
Martin Davis:
My reasons for the bet are much weaker. In principle, I have no problem
with the possibility that although CH has a determinate truth value, the
human race may never determine it. After all, there are many such
propositions.
Matt:
I essentially agree. (If I were to disagree at all, it would be to say
that
I would merely modify the last statement to read ``After all, there
probably
are many such propositions.'') In fact, my reason for believing this is
actually the same fact, I guess, that leads some to believe there are
well-formed formulas of set theory that have no determinate truth value.
In
particular, it is because any consistent, complete extension of ZF is
necessarily not recursively enumerable, and, as I understand it, we
currently have no clear reason to believe that humanity will ever find a
proof method that can generate in finite time a non-r.e. set of
consequences
from a recursive set of axioms. (Even if a proof method were developed and
generally accepted that could theoretically produce a non-r.e. set from an
r.e. set of axioms, this would likely require unbounded time, so
feasibility
questions would still remain.) Currently, there is a project called
``Consequences of the Axiom of Choice'' which continues to be carried on,
because there are still some who worry that acceptance of it goes too far,
in some sense. This project may be a model for future projects involving
various proposed axioms, such as CH, V=L, and the large cardinal axioms.
(I
realize that those who are close to these problems are already ``working on
these projects'', but what I expect is that compendia will be assembled,
perhaps now electronically, and provided to the mathematical community,
which will collect together in one place what is known about each of
several
individual axioms, depending upon the level of interest each generates
because of connections realized by researchers such as professor Friedman,
etc.) Eventually, enough mathematicians will have examined each such axiom
so that we will determine which ones are preferable, for reasons similar to
those that I am convinced will parallel the reasons given for the current
fairly overwhelming acceptance of AC.
I will go further than Professor Davis in one regard. He said, ``In
principle, I have no problem with the possibility that although CH has a
determinate truth value, the human race may never determine it.'' I, in
fact, in principle have no problem with the possibility that although CH
has
a determinate truth value, even if the human race sometime in the future
decides as a whole that that truth value of CH has been determined, and
never reverses its decision as to what that truth value happens to be, for
whatever reason, the whole human race could be wrong.
Dr. Matt Insall
http://www.umr.edu/~insall
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