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

On Mar 13, 4:32 am, Phil <toob-head...@xxxxxxxxxxxxx> wrote:
this is (eventually) equivalent to the example
above using the finite distance between every
real number on the line
segment [0,1], where the sum of INFINITELY
many FINITE distances adds up
to a FINITE distance,

right.

I am asking you whether this seems to you to even
vaguely indicate a potential problem

No, this does not seem vaguely to indicate a potential problem.

(given that normally,

You'll once again at this point be needing to go *** yourself:
YOU DON'T KNOW what's normal and what isn't. "Normal"
means typical, occurring frequently or usually, in the practice
of MATHEMATICANS AND LOGICIANS DOING MATH AND LOGIC,
not "common within the ignorant and insignificant experience of
Phil."

infinity times a finite value ALWAYS results in infinity).

Times a FIXED finite value. Or times a value BOUNDED FROM
BELOW (if we are talking non-negative numbers here, just
to postpone one more complexity). It is NOT NORMAL for, e.g.,
GEOMETRIC (exponential) infinite series to "always" rsum to
infinity. For any natural number n, the infinite sum (with index m
ranging over all natnums) of the series where the mth term is
(n-1)/n ^ m
is n, NOT infinity. All you have to do to prove that is assume
that the sum obeys some basic laws of addition and subtraction.
THAT is the USUAL result for things that decline in size.
That the sum of 1/m was infinite took a LONG time to prove; it
wasn't legitimately just ASSUMED to be infinite; after all,
m^-1.000001 DOES converge, so why shouldn't m^-1.0?
The USUAL behavior for things of this type is that they DO
have a finite sum.


Yes, many such pairs
of proofs exist, but that was NOT my question.
My question is whether
you think the resulting paradoxes

There are no paradoxes about this.
There is just your personal confusion.

The fact that you can get different sums by summing some
series in different orders is NOT paradoxical; it just means that
infinite sums obey some BUT NOT ALL of the same rules as
finite ones (in particular, you don't get to be infinitely commutative
and associative).


In fact, your example is one half of the "infinitesimal paradox" that I
posted both recently and several years ago.

Good scare-quotes (since there is no actually any such thing
as "Phil's infinitesimal paradox").


we can match these elements with a set of
infinitely many infinitesimals g -- if you prefer, from non-standard
analysis, so we do not even appear to be violating
the natural numbers
-- in the following manner:

F = {1/2, 1/4, 1/8, 1/16, ...,}
| | | |
I = { g, g, g, g, ...,}

Where g is defined as having an infinitesimal value such that the
infinite sequence adds up to 1. Therefore, we have two infinite sums,
both with equally many elements, in which the first series consists of
nothing but finite elements, the second series consists of nothing but
infinitesimal elements, and yet the two sums are equal.

That is NOT paradoxical. That IS just HOW INFINITESIMALS WORK.

Furthermore, since the first 3 finite elements sum to
a value quite a bit larger than
the first 3 infinitesimal elements,
the remaining finite elements sum to
a value that is LESS than the sum of an
equal number of infinitesimal elements.

Of course they do -- a LOT less. That "equal number" of infinitesimal
elements is THE EXACT SAME NUMBER AS THE ORIGINAL number
of infinitesimal elements.
You are assuming that the same rules you are applying to rational
numbers will apply equally well to infinitesimals here, AND THEY
WON'T. That, again, is just NOT how infinitesimals work. EVERY
finite sum of infinitesimals, NO MATTER HOW
LONG, is STILL infinitesimal. Just because a result violates
Phil's Ignorant Expectations does NOT make it paradoxical.

Even though EACH AND EVERY member of F is infinitely greater
than ANY member of I, infinitely many F elements sum to a
value that is equal to or even less than an equal number of I
elements.

You don't even need to go to infinitesimals to do that.

Now, does that give you just the slightest pause, the slightest feeling
that the proof showing that there are infinitely many natural numbers,
and the proof that all the natural numbers are finite,

There IS NO proof that all the natural numbers are finite!
THAT IS A *DEFINITION*! THAT *IS* a premise, and ALL the
proofs that talk about natural numbers work EQUALLY on it!

just MIGHT rest
on different sets of premises?

Of course they do. The latter IS a premise; the former is provable
without recourse to that premise, assuming you CAN DEFINE
finite (it's not trivial and you will get different proofs depending
on which different definition you use, unless you invoke the
axiom of choice).

That IS one of the three possible
explanations for the apparently contradictory results,

These results are NOT EVEN APPARENTLY contradictory
to anybody who knows what is going on. It would of course
help if you would acquaint yourself with the DEFINITION of a
contradiction, insteading of just iignorantly and arrognatly
presuming that you know one when you see one.

although the
other possible explanations are that (2) infinity is very weird or (3)
there is a hidden error in my paradox,

The fact that you personally are too stupid to see something
does NOT imply that it is "hidden" in Any General Sense.

meaning that it is NATURAL for a
series of infinitesimals to add up to a value equal to or greater than
an equal number of finite values.

You can make a series of infinitesimals add up to anything
you want, if it is infinite and if you choose them properly.
THAT is natural.

However, do you think, YOURSELF, that
infinitely many FINITE values
REALLY CAN sum to a finite value?

Math is about AXIOMS AND PROOFS,
dip***.
WE CAN PROVE that the sum of the geometric series whose
mth term is a*r^-m is a/(1-r). PROVE it.
So it is NOT a question of what ANYbody might THINK!

Remember, the fact that {1/2, 1/4,
1/8, 1/16, ...,} sums to 1 after INFINITELY many elements

Is just that, A FACT.

could simply be a result of the fact

of the fact that that's how addition, division, multiplication by 2,
and sums of aleph-0 addends ARE DEFINED.

that only finitely many (potential rather than
actual infinity) of these elements are finite,

ALL of the elements are finite, dumbass; they are all
smaller than .51 and bigger than 0. They also all only have
1 divided by A FINITE number of factors of 2, so all of them
are bigger than infinitesimal.

and that infinitely many
of them are infinitesimals.

NONE of them is infinitesimal, DIP***.
EVERY term in the series is 2^-n for some NATURAL n.
It is the result of dividing by 2 some FINITE number of times!

And all it would take for that to be true

would be for Phil to grow a brain, or for the sky to turn lime
green, or for premature babies in general to live to be 3000
years old. Just because "actual infinity" is not a natural
number and not the index of any term in the series does NOT
mean that it Does Not EXIST!

.

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