A good model of the naturalist math that Torgny is talking about is the overflow mechanism in computers. For example in a 64 bit machine you may define overflow for positive integers as 2^^64 -1. If negative integers are included then the biggest positive could be 2^^32-1. Torgny would also have to define the operations +, - x / with specific exceptions for overflow. The concept of BIGGEST needs to be tied with _the kind of operations you want to apply to_ the numbers.
George Brent Meeker wrote: > Quentin Anciaux wrote: > >> You have to explain why the exception is needed in the first place... >> >> The rule is true until the rule is not true anymore, ok but you have >> to explain for what sufficiently large N the successor function would >> yield next 0 and why or to add that N and that exception to the >> successor function as axiom, if not you can't avoid N+1. But torgny >> doesn't evacuate N+1, merely it allows his set to grows undefinitelly >> as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense >> , is a natural number but not part of the set of natural number, this >> is non-sense, assuming your special successor rule BIGGEST+1 simply >> does not exists at all. >> >> I can understand this overflow successor function for a finite data >> type or a real machine registe but not for N. The successor function >> is simple, if you want it to have an exception at biggest you should >> justify it. >> > > You don't justify definitions. How would you justify Peano's axioms as being > the "right" ones? You are just confirming my point that you are begging the > question by assuming there is a set called "the natural numbers" that exists > independently of it's definition and it satisfies Peano's axioms. Torgny is > denying that and pointing out that we cannot know of infinite sets that exist > independent of their definition because we cannot extensively define an > infinite > set, we can only know about it what we can prove from its definition. > > So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical > objects. The first however is more definite than the second, since Godel's > theorems don't apply. Which one is called the *natural* numbers is a > convention > which might not have any practical consequences for sufficiently large > BIGGEST. > > Brent > > > > > > --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to email@example.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 -~----------~----~----~----~------~----~------~--~---