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.
Regards,
Quentin
2009/6/9 Brent Meeker <meekerdb.domain.name.hidden>:
>
> Quentin Anciaux wrote:
>> 2009/6/9 Torgny Tholerus <torgny.domain.name.hidden>:
>>
>>> Jesse Mazer skrev:
>>>
>>>>
>>>>> Date: Sat, 06 Jun 2009 21:17:03 +0200
>>>>> From: torgny.domain.name.hidden
>>>>> To: everything-list.domain.name.hidden
>>>>> Subject: Re: The seven step-Mathematical preliminaries
>>>>>
>>>>> My philosophical argument is about the mening of the word "all". To be
>>>>> able to use that word, you must associate it with a value set.
>>>>>
>>>> What's a "value set"? And why do you say we "must" associate it in
>>>> this way? Do you have a philosophical argument for this "must", or is
>>>> it just an edict that reflects your personal aesthetic preferences?
>>>>
>>>>
>>>>> Mostly that set is "all objects in the universe", and if you stay
>>>>>
>>>> inside the
>>>>
>>>>> universe, there is no problems.
>>>>>
>>>> *I* certainly don't define numbers in terms of any specific mapping
>>>> between numbers and objects in the universe, it seems like a rather
>>>> strange notion--shall we have arguments over whether the number 113485
>>>> should be associated with this specific shoelace or this specific
>>>> kangaroo?
>>>>
>>> When I talk about "universe" here, I do not mean our physical universe.
>>> What I mean is something that can be called "everything". It includes
>>> all objects in our physical universe, as well as all symbols and all
>>> words and all numbers and all sets and all other universes. It includes
>>> everything you can use the word "all" about.
>>>
>>
>> It includes all set, but no all set as it N includes all natural
>> number but not all natural number... excuse-me but this is non-sense.
>> Either N exists and has an infinite number of member and is
>> incompatible with an ultrafinitist view or N does not exists because
>> obviously N cannot be defined in an ultra-finitist way,
>
> That's not obvious to me. You're assuming that N exists apart from
> whatever definition of it is given and that it is the infinite set
> described by the Peano axioms or equivalent. But that's begging the
> question of whether a finite set of numbers that we would call "natural
> numbers" can be defined. To avoid begging the question we need some
> definition of "natural" that doesn't a priori assume the set is finite
> or infinite; something like, "A set of numbers adequate to do all
> arithmetic we'll ever need" (unfortunately not very definite). The
> problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
> except for the case n=N where S{N}=0 and choose sufficiently large N it
> might satisfy the "natural" criteria.
>
> Brent
>
>
>> any set that
>> contains a finite number of natural number (and still you haven't
>> defined what it is in an ultrafinitist way) are not the set N.
>>
>> Also any operation involving two number (addition/multiplication) can
>> yield as result a number which has the same property as the departing
>> number (being a natural number) but is not natural number... Also
>> induction and inference cannot work in such a context.
>>
>>
>>> For you to be able to use the word "all", you must define the "domain"
>>> of that word. If you do not define the domain, then it will be
>>> impossible for me and all other humans to understand what you are
>>> talking about.
>>>
>>
>> Well you are the first and only human I know who don't understand
>> "all" as everybody else does.
>>
>> Quentin Anciaux
>>
>>
>>> --
>>> Torgny Tholerus
>>>
>>>
>>
>>
>>
>>
>
>
> >
>
--
All those moments will be lost in time, like tears in rain.
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Received on Tue Jun 09 2009 - 21:15:35 PDT