FOM:Sha/DAv on Can/Hil & CH? Vagueness
Robert S Tragesser
RTragesser at compuserve.com
Sat Dec 6 04:51:57 EST 1997
Hilbert was quoted: "every DEFINITE mathematical problem must NECESSARILY
be susceptible of exact settlement, either in the form of an actual answer
to the question asked, or by a proof of the impossibility of its solution.
. ."
[1] The proper reply to Shapiro/Davis et al, and the proper expansion of
Feferman's initial remark might be: Neither Cantor nor Hilbert KNEW that
CH? is a DEFINITE mathematical problem.
[2] Furthermore, there is no indication that anyone knows that CH? is
a definite mathematical problem. To know that it is a definite
mathematical problem is to have shown that "either CH? can be answered OR a
proof of the impossibility of its being answered."
Perhaps the occurrence of not wholly unpromising strategies for
answering CH? might take care of the first disjunct. Or at least an
educated suspcion that CH is true or an educated suspicion that CH is false
(but not both sort of educated suspicions). Perhaps even (though not for
me) the metaphysics/epistemology of Cantor (quoted by Feferman from Hallet)
might suffice. We might say, then, that for Cantor, Hilbert, Goedel,
CH? was a QUASI-definite definite mathematical problem (because they
covered the first disjunct, but not the second disjunct Hilbert's
"Nec[answerable or demonstrably unanswerable]" ).
But there is no sign that anyone knows how to take care of the
second disjunct.-- One needs a reasonably cogent idea of how a proof of
impossibility of solution would be mounted in the case of CH?
No such idea is forthcoming, is it? (This circumstance is
rather in contrast to the case cited by Shapiro -- perpetual motion.)
In this sense, CH? is an insufficiently well-formed
mathematical question.
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