I think that resorting to calling the biggest natural number BIGGEST,
rather than specifying exactly what that number is, is a tell-tale sign
that the ultrafinitist knows that any specification for BIGGEST will
immediately reveal that it is not the biggest because one could always
add one more.
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.
>
> 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
>>>>
>>>>
>>>>
>>>
>>>
>>>
>>
>
>
>
>
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Received on Tue Jun 09 2009 - 12:20:17 PDT