Quentin Anciaux wrote: > 2009/6/3 Torgny Tholerus <tor...@dsv.su.se>: > >> 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.

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