Torgny Tholerus wrote: > 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". > > I wonder if anyone has tried work with a theory of finite numbers: where BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers?
Brent --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---