2009/6/9 Quentin Anciaux <allco...@gmail.com>:
> 2009/6/9 Brent Meeker <meeke...@dslextreme.com>:
>> 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

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 
>> 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.

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