Re: infinitely many nn's = infinite nn's?

On Mar 13, 2:32 pm, Phil <toob-head...@xxxxxxxxxxxxx> wrote:

However, do you think, YOURSELF, that infinitely many FINITE values
REALLY CAN sum to a finite value? Remember, the fact that {1/2, 1/4,
1/8, 1/16, ...,} sums to 1 after INFINITELY many elements could simply
be a result of the fact that only finitely many (potential rather than
actual infinity) of these elements are finite, and that infinitely many
of them are infinitesimals.

The answers to your questions are given in the logic NAFL; see

http://arxiv.org/abs/math.LO/0506475

I have already explained in sci.logic threads as to how NAFL deals
with and resolves Zeno's paradoxes. In particular, to answer your
question briefly, there are no infinite sets in NAFL, and
quantification over infinite classes is banned. For example, you
question above as to "How many" intervals are present in the following
sequence of closed real intervals:

{[0,1/2], [1/2,3/4], [3/4,7/8], .....}

is an *illegal* question in NAFL, i.e., it cannot even be formulated
because the above sequence of intervals is not even a legal entity in
NAFL (each of these is an infinite object and to make the sequence of
such intervals a legal entity, you need to quantify over infinite
objects (intervals of reals) which is precisely what you cannot do in
NAFL). There are no infinitesimals in the NAFL version of real
analysis.

Zeno's paradoxes are resolved because there are no open intervals in
the NAFL version of real analysis. The above sequence, with no
smallest interval, can only be possible if open intervals of reals
exist. In the NAFL version of real analysis, the above sequence is
represented indirectly without any quantification over infinite
entities like reals or intervals of reals; one then finds that this
resulting NAFL sequence *must* include an interval of zero length,
[0,0]. So you cannot assert that there is a sequence of such
"infinitely many finite, non-zero intervals" because the sequence must
include an interval of zero length. Secondly you cannot ask "how many"
closed intervals are present in the NAFL version of such a sequence
(which includes the interval [0,0]) because that amounts to
quantification over intervals of reals, whereas NAFL does not even
permit quantification over reals (each of which is an infinite
object), let alone intervals of reals.

In order to clearly understand and appreciate the NAFL resolution of
Zeno's paradoxes, you need to understand NAFL from first principles.
Then you will accept that there is no largest natural number or
infinite natural number. But the NAFL real line must include an object
corresponding to +- 00. So if you consider the sequence of real
numbers

{0.0, 1.0, 2.0, ......}

it *must* include +00. Here it is important that each of the reals in
the above sequence is an infinite object and the entire sequence is
*constructed* (indirectly) in NAFL without any quantification over
reals.

Similarly the sequence of rationals

{1, 1/2, 1/4, 1/8, ....}

does not have a last element (you can directly quantify over rationals
because each of these is a finite object). But if you consider the
corresponding sequence of *reals*

{1.0, 0.5, 0.25, 0.125 ....}

then this sequence *must* include the real number 0.0.

According to you this sequence *must* include infinitesimals, but that
does not hold in NAFL (or you can take zero to be the only legal
infinitesimal in NAFL; there are no non-standard ("infinite") natural
numbers in NAFL. In fact nonstandard analysis does not really resolve
Zeno's paradoxes in the way that you imagine; it will generate its own
set of equivalent paradoxes.

In short, Zeno's paradoxes are resolved because the very statements of
the paradoxes are only possible if open (or semi-open) intervals of
reals exist, but these are illegal in NAFL. Secondly quantification
over infinite entities is banned, so you cannot ask the questions that
lead to paradoxes even concerning entities that are legal in NAFL
(like closed intervals of reals).

And all it would take for that to be true
would be to use ACTUAL rather than potential infinity when determining
not only the number of natural numbers, but also the maximum value that
natural numbers can have. As I said elsewhere, if you use either
potential infinity OR actual infinity for BOTH the number of numbers and
the maximum value of a number, this particular paradox instantly
vanishes. Does that even slightly suggest to you that potential and
actual infinity might NOT be the same thing?

Do you think, in situations like these, that SOMEONE -- preferably a
good mathematician who really knows what he's doing, but someone, in any
case -- should at least look into the possibility that the proofs rest
on incompatible premises? Or do you believe that, despite all the errors
made concerning infinity for 2500 years, that starting about 150 years
ago, we got everything right, everything perfect and error-free, and
there's no need to even verify that no problem exists, no need to even
investigate any paradoxes? It is as much the implicit claim that
mathematicians obtained godlike perfection toward the end of the 19th
century, as much as anything, that does amazes me -- or disgusts me --
about this group.

The resolution of your difiiculties is already available. You haven't
heard about it because of the extreme reluctance, if not outright
unwillingness, of academicians -- in particular, logicians and
phillosophers -- to even acknowledge the existence of my work (let
alone analyze it, criticize it , improve it or dismiss it, etc.).

Regards, RS

.

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