Re: Skolem's 'Paradox'

In article <o9bdj19pjmqi68r1vehb5aroj1qfguvgsv@xxxxxxx>,
David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:

>On 24 Sep 2005 17:18:10 -0700, "Keith Ramsay" <kramsay@xxxxxxx> wrote:
[SNIP]

>>To have sequences of proper
>>classes one has to have something that can have a proper class as
>>a member, and this is not allowed in GB. But it's sort of an
>>arbitrary restriction, isn't it?
>
>Yup. Always seemed clear to me that axiomatic set theory
>didn't resolve various paradoxes so much as simply agree
>to ignore them. The "real" Russel's paradox remains:
>Whether allowed by some specific axiomatization or not,
>we can consider the class of all classes that do not
>contain themselves as an element...
>
>What's always seemed like the actual resolution is this:
>In the _actual_ paradox we're talking about the "real"
>class of all "real" classes that "really" do not contain
>themselves as an element. This assumes that there is a
>"real" element-of relation that is defined (true or false)
>for all "real things", and it seems to me that there is
>no such relation.
>
>>If one is ready to accept these
>>definitions individually, it only makes sense to accept them as
>>a sequence. It just lies outside of what ZFC or GB allows.

ISTM the fact that axiomatic set theories do not allow (proper) classes
to be members of any higher collection is not arbitrary. For example,
it can be motivated by cardinality considerations:

Consider a theory where quantification is only over sets, and classes
are a shorthand for predicates on sets. Then under the usual
interpretation (outside the theory) of "all predicates", there are
vastly more predicates than there are sets. Therefore almost all
predicates must correspond to proper classes. And the Russell class is
one of those.

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Relevant Pages

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    (sci.logic)
  • Re: Nice Set Theory
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    (sci.logic)
  • Re: Nice Set Theory
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    (sci.logic)