[ontolog-forum] form and content

[FERENC KOVACS <f.kovacs@xxxxxxxxxxxxxx>]

Hi Avril,

Thanks for that!

Ferenc

 

----- Original Message -----

From: "Avril Styrman" <Avril.Styrman@xxxxxxxxxxx>

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>; "John F. Sowa" <sowa@xxxxxxxxxxx>

Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>

Sent: Friday, December 11, 2009 10:44 AM

Subject: Re: [ontolog-forum] form and content

> Hi all,
>
> just a short notion about the used terminology.
>
>> Theoretically, Peano's axioms define the common notion of number.  But
>> the number of applications that use integers of arbitrary size (e.g.,
>> the infinite precision Bignum in LISP) are extremely limited.  The
>> overwhelming choice for applications is to use integers modulo some
>> suitable upper value:  2^1, 2^8, 2^16, 2^32, or 2^64.
>
> Always when it is said that a real-life application uses something 
> infinite, it in fact uses only the potential infinite. However, also 
> the 'infinite' in potential infinity is quite misleading, and could be 
> replaced with "as much as can be taken" or something similar.
>
> Also, the notion "arbitrary natural number" only meditates away the 
> problems of the transfinite collection of natural numbers. It does not 
> matter whether Peano's class of naturals or the set theoretic omega is 
> thought of. Consider the class/set/aggregate or whatever sort of 
> completed totality that contains each and every one of the infinitely 
> many natural numbers, where infinite especially means never ending but 
> still completed all the way through. This sort of a collection is 
> called transfinite. If you select "just some" number n from that 
> collection, in the way that all numbers have an equal possibility of 
> getting selected, then the selected number n is so big, that it does 
> not fit in a microchip that is of the size of the known part of 
> Universe, that is, with probability 1. The number n is called 
> arbitrary natural number in the transfinitist parlance. The problem 
> with n is that n is that n is in practice very close to transfinite. 
> Then again, if an arbitrary number is not selected randomly, then what 
> is the meaning of "just some number"?
>
> To conclude, when the term arbitrary is used with real-life 
> applications, it always means a randomly selected number from within 
> some finite range of numbers. The upper limit can be vague, such as 
> the greatest number that fits in a microchip that is of the size of 
> the known part of Universe, but it is still always finite.
>
> -Avril
>
> 
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