FOM: Two "New" Paradoxes.

[Robert S Tragesser]

Is "Galileo's Paradox" (even numbers
exactly count the whole numbers) para-
doxical?  We can extract painful paradoxes from
it if we as it were embed it in space.
(Much simpler and therefore nastier than
the Banach-Tarski "Paradox".)

[1] PARADOX 1:
There are uncountably many finite cardinals.

        Given the infinite string: |||||||...
(which is | || ||| |||| . . . spacing removed).
        Let S be the operation: remove the front stick
from any ||||. . . to which S is appled,  and put
that stick in CACHE.
        Notice that ||||||. . . is a fixed point of S.
        ||||.... , and ||||.... with a front | removed
are absolutely indistinguishable.
        Reiteratively apply S to ||||...,  putting a
stick in CACHE each time.  Take a half second for
the first application,  a quarter second for the 
second,  and so on,  ad infinitum.  At the end of
one second,  what do you have?
        (i) CACHE is filled with a countable
infinity of |'s.
        (2) we still have |||||. . . since it is
a fixed point of S,  S(||||...) =  ||||. . .
                So we continue to reiteratively apply
S at the limit ordinals.
        Since |||. . . is continuously a fixed point
of S,  we can reiterate S up through the uncountable
ordinals.
        But CACHE is now filled with uncountably many |'s,
and ||||... has suffered no loss of identity..
        [Notice that this construction (looks like it)
entails that there
are sets of subsets of the integers of arbitrarily great
cardinality, viz.,  begin with a _random_
                101001010001001101010101001....
(with ||||... written under it) 
and, reiterating S as above, successively remove the first 1, 0.]

Paradox 2. Infinitely sparse matter can fill space by
rearrangement.
        Let there be a countable infinity of sugar cubes
distributed as thinly,  sparsely throughout space.
The sugar cubes can be rearranged so as to completely
fill space.

        For whatever it is worth, Leibniz's Principle
of Sufficient Reason blocks both paradoxes.

Robert Tragesser
Class of '43 Professor in the Hisotry
and Philosophy of Science & Mathematics,
at ConnC.
Current Research Interests:  foundations of
chronobiology,  Galileo's replacing of
Aristotle's logic of opposites by mathematics,
general problems in the philosophy & foundations
of mathematics and logic.