> From what you said earlier, BIGGEST={0,1,...,BIGGEST-1}. Then > BIGGEST+1={0,1,...,BIGGEST-1} union {BIGGEST} = {0,1,...,BIGGEST}. > Why would {0,1,...BIGGEST} not be a natural number while > {0,1,...,BIGGEST-1} is?

If {0, 1, ... , BIGGEST-1} is a natural number, then {0,1,...,BIGGEST} is too, and then so is {0, 1, ... , BIGGEST+1}, etc. There's no such thing as a largest natural number: that's the whole point of the construction. The set of all natural numbers is an infinite set, unbounded above. The set N has no largest element within it: it is the set of all finite ordinals. N (usually called omega when treated as an ordinal) has no predecessor, because it is formed by taking the limit of all the ordinals below it, *not* by applying the successor function "x+ = x U {x}". This is the way well-ordering works...it's not symmetric. So any set described {a, b, ... , z} in the standard way is not N. N is not the successor of any natural number; rather, it contains them all. This allows us to talk about (and prove things about) all natural numbers. This isn't an arbitrary mathematical choice. Without infinite sets, we would be unable to rigorously prove things by induction, which is necessary for a wide array of basic arithmetical proofs. This is because a finite set of natural numbers cannot be closed under successor (or addition or multiplication, for that matter). If you relied on only finitely many numbers, your functions could take natural numbers and hand you back something that isn't a number at all. This makes even basic math untenable. Taking the closure of {} under successor is the solution. (There are non-standard models of the natural numbers that contain numbers other than the elements of N, but these are not well-ordered.) Anna --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---