[FOM] CH and mathematics
joeshipman at aol.com
Mon Jan 21 16:08:37 EST 2008
Some set theorists (for example, Patrick Dehornoy here:
) believe that recent work of Woodin makes CH a much more definite
Woodin has argued, roughly speaking, that the properties of the sets of
hereditary cardinality aleph_1 are invariant under forcing only if CH
is false; therefore only a set theory where CH is false can settle many
open questions about sets that are quite small and "early" in the
set-theoretic hierarchy. If we believe those questions themselves have
definite answers, then we should accept that CH is false.
I don't necessarily buy this line of reasoning myself, but a lot of
people regard it as genuine progress, so you should address it if you
want to argue for CH's indefiniteness or our inability to settle it.
From: Arnon Avron <aa at tau.ac.il>
For itself, as a unique piece of data, such inability is a very weak
argument. However, when it is about a claim whose definiteness
is independently strongly doubted (and before this inability was
proved), then the proof of such inability is a strong confirmation
of these doubts.
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