FOM: the Borel universe/but it wasn't blather -- a problem about meaning

John Case case at
Sun Dec 7 15:58:20 EST 1997

This is a response to 

Date: Sun, 7 Dec 1997 12:25:56 -0500
In-Reply-To: Robert S Tragesser <RTragesser at>
 The emergent very general foundational issue is:
                        When is a (proposed) mathematical problem (fully
and mathematically) meaningful or definite? 

        Stuart Shapiro's quotation from Hilbert ( to the effect that a
question is definite only if it is necessarily the case that it can be
answered or shown to be unanswerable) points
        to a starting point  for resolving that general foundational issue:
to give a rigorous analysis and critique of Hilbert's criterion.


Consider K, the diagonal halting problem set (e.g., for some fixed coding of
Turing machines).  All sentences of the form
                        particular_constant in K
where particular_constant is a definite numeral (e.g., 42), have quite definite
truth values.  However, it seems unlikely WE can list/discover/verify all the
false sentences of that form --- certainly we cannot if we do not surpass the
effectively computable.  Hence, again, I implore us to create agreed upon
standard models to settle things about whether truth values exist and to
avoid confusing this problem with problems of human limitations, problems of
human verification/knowlege-of-which, etc.  Of course for a fragment of sets
sufficient for CH to have a definite truth value, we may not be able to agree
on what, if anything, is the standard model.  However, perhaps, we could agree
on some form of the infinite pictures I proposed (and prove relevant
invariances under which well-orderings, etc), notice some new truths (axioms)
_about them_, and (fantasy fantasy) thereby or otherwise resolve _for those
pictures_ CH.  Leo H., how does this look to you?  Long time no talk to, by the
way. (-8

John Case

PS: Shapiro's first name has the alternate (and equally nice) spelling

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