Quentin Anciaux wrote:
> 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 ?

A set of propositions is consistent if it is impossible to infer contradiction.


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