Le Friday 16 November 2007 09:33:38 Torgny Tholerus, vous avez écrit : > Quentin Anciaux skrev: > Hi, > > Le Thursday 15 November 2007 14:45:24 Torgny Tholerus, vous avez écrit : > > What do you mean by "each" in the sentence "for each natural number"? > How do you define ALL natural numbers? > > > > There is a natural number 0. > Every natural number a has a natural number successor, denoted by S(a). > > > What do you mean by "Every" here? Can you give a *non-circular* > definition of this word? Such that: "By every natural number I mean > {1,2,3}" or "By every naturla number I mean every number between 1 and > 1000000". (This last definition is non-circular because here you can > replace "every number" by explicit counting.)

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I do not see circularity here... every means every, it means all natural numbers possess this properties ie (having a successor), that means by induction that N does contains an infinite number of elements, if it wasn't the case that would mean that there exists a natural number which doesn't have a successor... well as we have put explicitly the successor rule to defined N I can't see how to change that without changing the axioms. > > > How do you prove that each x in N has a corresponding number 2*x in E? > If m is the biggest number in N, > > > By definition there exists no biggest number unless you add an axiom saying > there is one but the newly defined set is not N. > > > I can prove by induction that there exists a biggest number: > > A) In the set {m} with one element, there exists a biggest number, this is > the number m. B) If you have a set M of numbers, and that set have a > biggest number m, and you add a number m2 to this set, then this new set M2 > will have a biggest number, either m if m is bigger than m2, or m2 if m2 is > bigger than m. C) The induction axiom then says that every set of numbers > have a biggest number. > > Q.E.D. > > -- > Torgny Tholerus Hmm I don't understand... This could only work on finite set of elements. I don't see this as a proof that N is finite (because it *can't* be by *definition*). Quentin Anciaux -- All those moments will be lost in time, like tears in the 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 -~----------~----~----~----~------~----~------~--~---