FOM: Re: Determinate Truth Values

[Matt Insall]

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