[FOM] Explanation/Continuum Hypothesis
Studtmann, Paul
pastudtmann at davidson.edu
Mon May 8 11:39:02 EDT 2006
I have some questions about the suggestions made concerning the continuum
hypothesis. The questions are purely requests for clarification I don¹t
have any axes to grind but rather want to know what others think.
Tim Chow wrote
>Paul Cohen, in his book "Set Theory and the Continuum Hypothesis," said
>something to the effect that the powerset axiom is a bold new way of
>constructing sets and that one should not expect to be able to reach the
>powerset from below in piecemeal fashion.
Can you explain a bit more as to how this counts as an explanation for one
half of the independence of the continuum hypothesis. Why is the powerset
axiom so bold and what makes a powerset somehow unreachable from below in a
piecemeal fashion? And what precisely is the connection between being
unreachable from below in a piecemeal fashion and failing to be a certain
size?
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.
Were Goedel¹s remarks intended to be an explanation of the independence of
the continuum hypothesis or rather something that one can infer as a result
of its independence? It seems to me that an explanation would require an
explanation as to why the continuum hypothesis holds for one conception of
set and not the other. So, for instance, suppose someone were to ask: what
accounts for the fact that when we think of a set as a random totality the
size of the powerset of a denumerably infinite set is almost entirely up for
grabs? Did Goedel or do you have any suggestions about how one might
answer this?
Paul Studtmann
On 5/7/06 2:15 AM, "Bill Taylor" <W.Taylor at math.canterbury.ac.nz> wrote:
> Alasdair Urquhart <urquhart at cs.toronto.edu> suggests:
>
> ->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.
>
> Are you intending to except AC from the first case (definable collection)?
> If not, how is it clear that AC holds for this?
>
>
> ->The continuum hypothesis is valid on the first conception but not
> ->the second.
>
> Though as I noted here before, CH can be reworded in two ways,
> equivalent under AC but not otherwise, one of which is true
> and one false in the "definable" interpretation. Though the "obvious"
> interpretation is true, as you say.
>
> Bill Taylor
>
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