Jesse Mazer skrev: > > > > Date: Fri, 5 Jun 2009 08:33:47 +0200 > > From: tor...@dsv.su.se > > To: everything-list@googlegroups.com > > Subject: Re: The seven step-Mathematical preliminaries > > > > > > Brian Tenneson skrev: > >> > >> How can BIGGEST+1 be a natural number but not belong to the set of all > >> natural numbers? > > > > One way to represent natural number as sets is: > > > > 0 = {} > > 1 = {0} = {{}} > > 2 = {0, 1} = 1 union {1} = {{}, {{}}} > > 3 = {0, 1, 2} = 2 union {2} = ... > > . . . > > n+1 = {0, 1, 2, ..., n} = n union {n} > > . . . > > > > Here you can then define that a is less then b if and only if a belongs > > to b. > > > > With this notation you get the set N of all natural numbers as {0, > 1, 2, > > ...}. But the remarkable thing is that N is exactly the same as > > BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other > > natural numbers, so it is then a natural number. But BIGGEST+1 is not a > > member of N, the set of all natural numbers. > > Here you're just contradicting yourself. If you say BIGGEST+1 "is then > a natural number", that just proves that the set N was not in fact the > set "of all natural numbers". The alternative would be to say > BIGGEST+1 is *not* a natural number, but then you need to provide a > definition of "natural number" that would explain why this is the case.
It depends upon how you define "natural number". If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe "unnatural number". -- 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 -~----------~----~----~----~------~----~------~--~---