Bruno Marchal skrev: > On 02 Jun 2009, at 19:43, Torgny Tholerus wrote: > > >> Bruno Marchal skrev: >> >>> 4) The set of all natural numbers. This set is hard to define, yet I >>> hope you agree we can describe it by the infinite quasi exhaustion by >>> {0, 1, 2, 3, ...}. >>> >>> >> Let N be the biggest number in the set {0, 1, 2, 3, ...}. >> >> Exercise: does the number N+1 belongs to the set of natural numbers, >> that is does N+1 belongs to {0, 1, 2, 3, ...}? >> > > > Yes. N+1 belongs to {0, 1, 2, 3, ...}. > This follows from classical logic and the fact that the proposition "N > be the biggest number in the set {0, 1, 2, 3, ...}" is always false. > And false implies all propositions. >

No, you are wrong. The answer is No. Proof: Define "biggest number" as: a is the biggest number in the set S if and only if for every element e in S you have e < a or e = a. Now assume that N+1 belongs to the set of natural numbers. Then you have N+1 < N or N+1 = N. But this is a contradiction. So the assumption must be false. So we have proved that N+1 does not belongs to the set of natural numbers. -- Torgny Tholerus --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---