[FOM] Explanation/Continuum Hypothesis

Alasdair Urquhart urquhart at cs.toronto.edu
Sat May 6 09:21:06 EDT 2006

Paul Studtmann wrote:

> 1. Is there some explanation for the independence of the continuum
> hypothesis from the other axioms of ZFC that goes beyond the claim that
> neither it nor its denial is entailed by the other axioms of ZFC.  [For
> instance, I once read in a book on set-theory that the continuum hypothesis
> is independent from the other axioms because we do not have a clear
> conception of the nature of infinite sets.  I am not suggesting that this is
> a good explanation.]

Kurt Goedel made some remarks about this, which were roughly as follows.
The axioms of set theory are satisfied for two quite different conceptions
of set, namely the notion of a definable collection (the constructible
sets) and also the notion of a random totality of objects --
a generic set can be considered as a random totality in this sense.
The continuum hypothesis is valid on the first conception but not
the second.

I am quoting from memory here -- I think I may have seen it in one
of Hao Wang's two books on Goedel.

Alasdair Urquhart

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