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