2009/6/3 Torgny Tholerus <[email protected]>: > > 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.
Hi, No, what you've demonstrated is that there is no biggest number (you falsified the hypothesis which is there exists a biggest number). You did a "demonstration par l'absurde" (in french, don't know how it is called in english). And you have shown a contradiction, which implies that your assumption is wrong (there exists a biggest number), not that this number is not in the set. Regards, Quentin > -- > Torgny Tholerus > > > > -- 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 -~----------~----~----~----~------~----~------~--~---
