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

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