Evaluation of the argument
Everything is fine up to step (9). But (9) does not entail (10). Zeno seems to be implicitly
assuming (what I’ll call) the
Infinite Sum Principle
: viz., that
the sum of an infinite number
of terms is infinitely large
. (9), together with the Infinite Sum Principle, entails (10). And from
(10) it follows (by Universal Generalization) that
every
magnitude is infinitely large, which is
the conclusion of the second limb.
The Infinite Sum Principle appears to be correct. But is it? What makes it
seem
correct is the
observation that you can make something as large (a finite size) as you want out of parts as small
as you want, and it takes only a finite number of them to do this! To see that this is so, consider
the following: pick any magnitude,
y
, as large as you like; and pick any small magnitude,
z
, as
small as you like (but
z
> 0). It is obvious that you can obtain a magnitude at least as large as
y
by adding
z
to itself a
finite
number of times. That is:
2200
y
2200
z
5
x
(
x
·
z
y
)
For every
y
and for every
z
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 Fall '09
 JorgeRigol
 Set Theory, 0.999..., Indian mathematics

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