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 -~----------~----~----~----~------~----~------~--~---
