[FOM] Concerning Ultrafinitism.
robblin at thetip.org
Thu Nov 2 17:52:38 EST 2006
Mr. Ozkural, the dichotomy between:
1) There are an infinite number of integers
2) there is a largest integer
is a false dichotomy since there are at least these three other options:
a) There may be no number of integers at all.
There is no reason to suspect that there is a number of Cardinal
Numbers, why should we expect that there is a number of integers?
b) The integers AS A WHOLE may not themselves exist, just as the
Cardinal Numbers AS A WHOLE may not exist.
The integers as defined by successors in counting of "1" where
counting is the practice of identifying the succeeding integer by
adding 1 to the given integer may or may not identify any REAL THINGS.
That is, it is a practice of generating words which themselves may not
have referents. There is no reason, in general, to assume that just
because there is a word for something that the thing it is supposed to
refer to exists. So the fact that we have a word-structure "All the
Cardinal Numbers" does not lead us to deduce that there is any such
thing, neither should it in the case of "All the Integers".
c) The integers may be fundamentally incomplete.
Any given set of integers can be show to NOT be the complete set of
integers. The "complete set of integers" then may not in fact exist -
that is, may not in fact be giveable in a complete intuition. For if
it were so given, it would be possible to produce an integer not in the
given set, thus contradicting the theory that it is the whole set of
Finally a not-very-related mathematical question:
is this a proved theorem (false or true):
Given a unique and infinite decimal expansion of two real numbers X and
Y, there is a non-finite intersection of those expansions of X and Y,
e.g. given a series of integers in the decimal expansion of x:
...21837283940....274829304... there is a non-finite series of integers
in Y identical to it.
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