2009/6/9 Quentin Anciaux <[email protected]>: > 2009/6/9 Brent Meeker <[email protected]>: >> >> 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. > > then you say it is an axiom, no problem with that.
And your axiom can't just say there is a BIGGEST number without having a rule to either find it or discriminate it or setting the value arbitrarily. BIGGEST must be a well defined number not a boundary that you can't reach... because if it was the case you're no more an ultrafinitist and N is not a problem. >> How would you justify Peano's axioms as being the "right" ones? > > You don't, and either I misexpressed myself or you did not understood. > >> 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. > > No, I have a definition for a set called the set of natural number, > this set is defined by the peano's axioms and the set defined by these > axioms is unbounded and it is called the set of natural number. Any > upper limit bounded set containing natural number is not N but a > subset of N. > > http://en.wikipedia.org/wiki/Natural_number#Formal_definitions > > The set Torgny is talking about is not N, like a dog is not a cat, he > can call it whatever he likes but not N. > But merely what I want to point out is that the definition he use is > inconsistent unlike yours which is simply modulo arithmetics. > > http://en.wikipedia.org/wiki/Modular_arithmetic > > > >> 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 >> >> >> >> >> > > > > -- > All those moments will be lost in time, like tears in rain. > -- All those moments will be lost in time, like tears in rain. --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
