2009/6/9 Brent Meeker <meekerdb.domain.name.hidden>:
>
> 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 00 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.
> 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
>
>
> >
>
--
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Received on Tue Jun 09 2009 - 22:07:05 PDT