[FOM] Explanation/Continuum Hypothesis
Roger Bishop Jones
rbj01 at rbjones.com
Tue May 9 03:56:23 EDT 2006
On Saturday 06 May 2006 14:21, Alasdair Urquhart wrote:
> 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 puzzled here by your use of the term "valid".
As I understand the situation:
(a) the continuum hypothesis is *true* when set theory is
interpreted in Goedel's constructible sets.
(b) the other conception of set referred to be Goedel is one in
which power sets are complete (see below).
This corresponds to what some people call "standard models" and
it corresponds to the continuum hypothesis as rendered in higher
order logics with "standard semantics".
In this case the continuum hypothesis has a definite but unknown
truth value. It is this conception which gives a definite
meaning to the continuum hypothesis in relation to which it is a
genuine, objective and significant, open problem, even for those
(like me) who are not platonists in the popular sense of that
term.
The continuum hypothesis is not "valid" in the usual technical
sense in either case, but is not "invalid" in any formal or
informal sense I can think of unless perhaps so little is done
to settle the context that it must be regarded as neither true
not false.
The distinction is made by Goedel in his paper "What is Cantor's
Continuum Problem" (at least in the later version in Benacerraf
and Putnam) in the following words:
"One class consists of the sets definable in a certain manner
by properties of their elements, the other of sets in the
sense of arbitrary multitudes, regardless of if, or how,
they can be defined".
I don't think the notion of "randomness" appears in this
conception and it seems to me not a good idea to explain it in
that way. The question "which sets are random?" might then
arise and, if this means anything, it is irrelevant, since a set
must appear in the relevant power set whether or not it has any
definite property at all (and of course, especially if its not
random!).
Roger Jones
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