2009/6/3 Brent Meeker <[email protected]>: > > Quentin Anciaux wrote: >> 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 > When you arrive at a contradiction it doesn't tell you which assumption > is wrong. > > Brent
Well I agree, but the second assumption depends on the first which is N exists and well defined. If it was, the second assumption is trivially false. Quentin > > > > -- 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 -~----------~----~----~----~------~----~------~--~---
