2008/11/9 Brent Meeker <meekerdb.domain.name.hidden>:
>
> A. Wolf wrote:
>>> I can if there's no rule of inference. Perhaps that's crux. You are requiring
>>> that a "mathematical structure" be a set of axioms *plus* the usual rules of
>>> inference for "and", "or", "every", "any",...and maybe the axiom of choice too.
>>
>> Rules of inference can be derived from the axioms...it sounds circular
>> but in ZFC all you need are nine axioms and two undefinables (which
>> are set, and the binary relation of membership). You write the axioms
>> using the language of predicate calculus, but that's just a
>> convenience to be able to refer to them.
>>
>>> Well not entirely by itself - one still needs the rules of inference to get to
>>> Russell's paradox.
>>
>> Not true! The paradox arises from the axioms alone (and it isn't a
>> true paradox, either, in that it doesn't cause a contradiction among
>> the axioms...it simply reveals that certain sets do not exist). The
>> set of all sets cannot exist because it would contradict the Axiom of
>> Extensionality, which says that each set is determined by its elements
>> (something can't both be in a set and not in the same set, in other
>> words).
>
> I thought you were citing it as an example of a contradiction - but we digress.
>
> What is your objection to the existence of list-universes? Are they not
> internally consistent "mathematical" structures? Are you claiming that whatever
> the list is, rules of inference can be derived (using what process?) and thence
> they will be found to be inconsistent?
>
> Brent
Well I reverse the question... Do you think you can still be
consistent without being consistent ?
If there is no rules of inference or in other words, no rules that
ties states... How do you define consistency ?
Regards,
Quentin
--
All those moments will be lost in time, like tears in rain.
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Received on Sat Nov 08 2008 - 20:26:50 PST