[FOM] how would a physicist know that we are not living in a Skolem hull?
Nik Weaver
nweaver at math.wustl.edu
Mon Feb 4 23:50:15 EST 2013
Paul Elliott asked for citations concerning the question in the
subject line.
In his paper "The set-theoretic multiverse" (arXiv:1108.4223) Joel
David Hamkins mentions roughly the same issue in support of his proposal
that "Every universe V is countable from the perspective of another,
better universe W." He commentse that "an elementary forcing argument
shows that any set in any model of set theory can be made countable in
a suitable forcing extension of that universe."
It is also something I have written about several times in connection
with a distinction I draw between concepts which are surveyable (in
principle, the objects which fall under the concept could be exhaustively
surveyed) versus those which are definite (any given object definitely
either does or does not fall under the concept, but we do not necessarily
have a clear idea of what it would mean to search through all the objects
falling under the concept). I feel that this is the right way to make
sense of the set/class dichotomy. Here are a couple of relevant passages:
"Can we clearly conceive of a world in which ... all real numbers are
present and surveyable? The Lowenheim-Skolem theorem suggests that we
cannot." (The concept of a set, arXiv:0905.1677)
"But what would it be like to live in a universe containing uncountably
many non-overlapping physical objects? Such a universe is difficult to
imagine ... there is nothing we can say, at least formally, that would
serve to distinguish an uncountable universe from a countable one."
(Axiomatizing mathematical conceptualism, arXiv:0905.1675)
Of course, this line of reasoning leads to the unwelcome conclusion
that only countable collections should properly be thought of as sets.
Nik Weaver
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