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.
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
> 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
> which might not have any practical consequences for sufficiently large
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