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 BIGGEST.


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