Le 16-nov.-07, à 09:33, Torgny Tholerus a écrit :
>>>> >> 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.) > >> >>> 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. What do you mean by "every" here? You just give us a non ultrafinitistic proof that all numbers are finite, not that the set of all finite number is finite. Bruno > > Q.E.D. > > -- > Torgny Tholerus > > > > http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---