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.

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Right. So you can't infer a contradiction. Brent > > 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 >> > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---