FOM: Knowing something to be essentially undecidable
JSHIPMAN@bloomberg.net
JSHIPMAN at bloomberg.net
Fri Dec 5 12:12:43 EST 1997
Neil Tennant wonders if my claim that a statement which can be known to be
essentially undecidable is meaningless is a vacuous claim, because he doesn't
see how we could *know* that a mathematical proposition is essentially
incapable of being decided. One way in which we could know such a thing is if
an analysis of the mathematical notions involved in the claim reveals a
fundamental indefiniteness. For example, the statements "GCH holds above the
first huge cardinal" or "GCH fails arbitrarily far up" may plausibly come to be
recognized as "essentially indefinite" (i.e. having no "fact of the matter"
because we are not sure what we are talking about when we talk about such large
sets). Another way in which we could know something is essentially undecidable
is if we can show it cannot be completely formalized and hence is "inherently
vague". For example, "mathematics is consistent" was considered by Hilbert to
be a definite statement susceptible to investigation, but then Godel showed
that you can't fix the meaning of "mathematics" and hope to decide this. (See,
Torkel, there is a distinction between "vague" and "indefinite").-Joe Shipman
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