2008/11/9 Brent Meeker <[EMAIL PROTECTED]>:
> 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 ?


All those moments will be lost in time, like tears in rain.

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