Quentin Anciaux wrote:
> To infer means there is "a process" which permits to infer.. if there
> is none... then you can't simply infer something.

Right. So you can't infer a contradiction.


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

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