Hi all, (01)
just a short notion about the used terminology. (02)
> 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. (03)
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. (04)
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"? (05)
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. (06)
-Avril (07)
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