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Attempts to Refute Cantor's Uncountability Proof?
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Ross A. Finlayson
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Joined: 30 Apr 2005
Posts: 873

PostPosted: Fri Jul 14, 2006 1:46 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

David R Tribble wrote:
Quote:
Ross A. Finlayson wrote:
There's only one theory with no axioms.


Jonathan Hoyle wrote:
Correct. It is the one with no theorems.


Ross A. Finlayson wrote:
No that's not what it is.

Consider Goedel, vis-a-vis incompleteness, and the physicists' notion
of a "Theory of Everything". Apparently, those people never heard of
Goedel, or didn't agree that his results about incompleteness hold in
their case, because they talk about a "Theory of Everything."

You're confused. The physicists' TOE delas with unifying gravity and
quantum physics, and while it uses a lot of complex math, it has
nothing to do with set theory.

There is
no "Theory of Everything" in ZF or other regular set theories. There
is no universe in ZF. Quantify over sets, in ZF: it's not a set. So,
it's a non-sets theory.

If you mean there can be no "set of all sets", yes, that is well known.

There's only one theory with no axioms. It has all the theorems. Any
other is inconsistent or incomplete, just ask Goedel. Your regular set
theory is incomplete, via Goedel, and inconsistent, via universal
quantiification, not to mention paradoxes in them, generally paradoxes
of unrestricted comprehension or the Liar, or about
symmetry/antisymmetry. Incomplete means inconsistent, of a universal
theory.

There's only one theory with no axioms, the null axiom theory.

I refute.

It still sounds like gibberish.
Could you show us a theorem in this theory with no axioms?

I think that the axioms of ZF besides regularity are theorems. While
that is so, some interpretations of what the objects are lead to
differences among what you'd expect primitive objects to be. There are
some more theorems than that, that may provide a physical basis.

About a "Theory of Everything", superstrings are mathematical
infinitesimals. Technicolor is a notion that there are particles
comprising quarks and particles comprising those etc ad infinitum, yet
while that is so something along the lines of the atom is a nilpotent
infinitesimal, in a sense. The particle/wave duality is not so far
removed from, say, various vacuous statements about null vis-a-vis the
universe. Colder than absolute zero is hotter than the sun, i.e.,
infinity = negative one. That might seem rather unobvious, or not.
Points are polydimensional, for example real numbers in or on the real
number line.

The universe is everything. It's all-encompassing. Were there a
parallel "universe", it would be part of the universe also. There's
only one universe, by definition, and conveniently in expression, the
universe contains itself, it's irregular, not well-founded, as it were,
in the views of some cosmologists. The universe is infinite.

There is no universe in ZF. Infinite sets are equivalent because
they're infinite. Basically infinite sets aren't regular.

So, there is no universe in ZF. Where there are only sets in set
theory, yet no sets, there is nothing. It's only truth is in vacuously
being the null axiom theory.

I borrow the book mentioned about universal sets in set theory, I'll
look at it. That is to say: there is a "sets of all sets" considered
as parts of set theory. It is agreed that there is no set in ZF.

Infinite sets are equivalent.

Makes sense to me.

Ross
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Fri Jul 14, 2006 7:41 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

David R Tribble wrote:

Quote:
Aatu Koskensilta wrote:
It's apparent you just don't know what recursive enumerability
means - why not take the time to find out?


Hatto von Aquitanien wrote:
This is roughly what I had in mind.
http://en.wikipedia.org/wiki/Recursively_enumerable_set
"There is an algorithm that "generates" the members of S. That means that
its output is simply a list of the members of S: s1, s2, s3, ... If
necessary it runs forever."

And for the sake of talking about generating .3, .33, .333,..., it is
quite sufficient.

Are you saying that this eventually generates 0.333... (1/3)?

Why would you think such a thing? Is it because you can set up the pattern
and then apply strong induction? Is there some way of communicating or
arriving at that idea which doesn't involve some kind of generalization
from an iterative process?

Quote:
The only addition I need to make in order to get my original
point of it producing a bijection with a subset of N is that I need to
keep track of the loop count.

And what is that loop count after you've generated 0.333... ?

I don't have a fast enough CPU, nor enough RAM to answer that. What we do
when we set up an induction hypothesis and then show it hold for some start
value as well as for n+1 for arbitrary n is to conclude that _if_ we could
do this infinitely, then.... Such open ended conclusions are useful when
evaluating certain kinds of questions such as whether a series converges.
We have a tendency to accept .333... as being as good a specification of a
number as is 3, for example. That is to say "for all intents and
purposes, .333... is as good a specification of a number as is 3." But is
it? Are there some attributes which we should not attribute to .333...
which we attribute to 3 by virtue of its being a number?

More importantly - and this matters in my area of interest - what
metaphysical assumptions go into this idea of .333...? That may sound like
some kind of pseudo-scientific drivel to veteran mathematicians who believe
they have exercised the spirit of hidden assumption from their psyche, but
to me .333... still means 'cut it into pieces of size s=1/10, glue the
whole s-size pieces onto the assembly to the left, take the remainder and
cut it into pieces of size s^2, glue the whole pieces onto the assembly to
the left, then do the same to the remainder....'. That uses the concept of
cutability. We started off talking about countability and were supposed to
show that there are sets of countable things which are uncountable[*]. Can
we simply introduce that concept of cutability into our proof without
examination or acknowledgment?

[*]Yes 'there are sets of countable things which are uncountable' is exactly
what I intended to say.
--
Nil conscire sibi
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Fri Jul 14, 2006 7:46 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Virgil wrote:

Quote:
In article <1152828016.052334.224610@35g2000cwc.googlegroups.com>,
"david petry" <david_lawrence_petry@yahoo.com> wrote:

We're not talking about the computability of the number, but the
computability of the digits of the number. The problem is, how do you
compute the digit preceding a very long string of nines or zeros?

Unless the string of 0's or 9's is completely endless, you do it the
same way as for any other digit, you may just need to be a little more
careful.

The word here is not 'careful', what is required is greater _precision_
_pre_ - _cise_ . Measure before cutting.
--
Nil conscire sibi
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Nathan
science forum addict


Joined: 29 Apr 2005
Posts: 97

PostPosted: Fri Jul 14, 2006 2:56 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:

Quote:
More importantly - and this matters in my area of interest - what
metaphysical assumptions go into this idea of .333...? That may sound like
some kind of pseudo-scientific drivel to veteran mathematicians who believe
they have exercised the spirit of hidden assumption from their psyche, but
to me .333... still means 'cut it into pieces of size s=1/10, glue the
whole s-size pieces onto the assembly to the left, take the remainder and
cut it into pieces of size s^2, glue the whole pieces onto the assembly to
the left, then do the same to the remainder....'.

This is the sort of intuitive understanding of the meaning of .333...
that students tend to have starting in elementary school and continuing
maybe into the second semester of calculus. But you're supposed
to outgrow it. You're supposed to learn to understand the meaning of
the limit of a sequence, particularly the idea of the sum of an
infinite series as the limit of the sequence of partial sums.

Not everybody does learn to understand this, which explains the
..999...=1 threads.
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Fri Jul 14, 2006 3:23 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Nathan wrote:

Quote:
Hatto von Aquitanien wrote:

More importantly - and this matters in my area of interest - what
metaphysical assumptions go into this idea of .333...? That may sound
like some kind of pseudo-scientific drivel to veteran mathematicians who
believe they have exercised the spirit of hidden assumption from their
psyche, but to me .333... still means 'cut it into pieces of size s=1/10,
glue the whole s-size pieces onto the assembly to the left, take the
remainder and cut it into pieces of size s^2, glue the whole pieces onto
the assembly to the left, then do the same to the remainder....'.

This is the sort of intuitive understanding of the meaning of .333...
that students tend to have starting in elementary school and continuing
maybe into the second semester of calculus. But you're supposed
to outgrow it. You're supposed to learn to understand the meaning of
the limit of a sequence, particularly the idea of the sum of an
infinite series as the limit of the sequence of partial sums.

Not everybody does learn to understand this, which explains the
.999...=1 threads.

Is that what happens to all the smart people in college? They have their
common sense psychologically pounded out of them. They either give lip
service to the orthodox canon, accept it as truth, or transfer to computer
science.

If you looked at the rest of what I posted, I pretty much said what you did,
except I did not accept it without reservation.
--
Nil conscire sibi
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Shmuel (Seymour J.) Metz1
science forum Guru


Joined: 03 May 2005
Posts: 604

PostPosted: Fri Jul 14, 2006 9:30 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In <1152794173.455331.251310@75g2000cwc.googlegroups.com>, on
07/13/2006
at 05:36 AM, matt271829-news@yahoo.co.uk said:

Quote:
If you accept
that two infinite sets have the "same number" of elements when their
members can be put into one-to-one correspondence, then you
inevitably have the "paradox" that a set (points in larger circle)
can have the "same number" of elements as a subset of itself

And, in fact, that is precisely the definition of an infinite
(transfinite) set.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spamtrap@library.lspace.org
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David R Tribble
science forum Guru


Joined: 21 Jul 2005
Posts: 1005

PostPosted: Fri Jul 14, 2006 10:03 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Virgil wrote:
Quote:
One can establish a bijection between the points of R^2 and the interior
of a circle in R^2.


Hatto von Aquitanien writes:
Quote:
As a side note. I had a friend raise a very simple objection to the notion
that every finite region of R^2 has the same number of points. His
argument was that if we draw a circle and confine a bunch of points in it,
then draw a larger circle around it. there are points in the larger circle
which are not in the smaller one. It would seem to follow that there are
more points in the larger circle than in the smaller one.


Michael Stemper wrote:
Quote:
Let's define two disks. (A disk is the set of points enclosed by a circle.)
[...]
I can define a function that maps disk 1 to disk 2 as follows:
[...]
Therefore, every point in disk 2 has a corresponding point in disk 1
and vice versa. Therefore, there is a bijection between them.

Here's a little diagram I created a few weeks ago to illustrate the
same concept, except showing a bijection between the points
of two 3-D spheres:
http://david.tribble.com/img/hprojection1.png

The larger sphere has 8 times the volume and 4 times the surface
area of the smaller sphere, but both have the same cardinality (c).
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Nathan
science forum addict


Joined: 29 Apr 2005
Posts: 97

PostPosted: Fri Jul 14, 2006 10:35 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
Nathan wrote:

You're supposed to learn to understand the meaning of
the limit of a sequence, particularly the idea of the sum of an
infinite series as the limit of the sequence of partial sums.

Not everybody does learn to understand this, which explains the
.999...=1 threads.

Is that what happens to all the smart people in college? They have their
common sense psychologically pounded out of them. They either give lip
service to the orthodox canon, accept it as truth, or transfer to computer
science.

No, there's another option you left out; the one I gave.
They can also learn to understand the concept of a limit.
As I said, this is what is supposed to happen.
This doesn't require that they "accept it as truth", as some sort of
dogma.

There may indeed be those who "give lip service to the canon",
because they really don't grasp it. There certainly are those who
"accept it as truth"; it can be very hard to help certain students
to learn to understand concepts and not just to do mechanical
calculations. Some in that category do transfer to
computer science, where they still tend to have trouble
understanding concepts.

And no, this doesn't necessarily happen in college. I said
"around the second semester of calculus", which for a lot of
"the smart people" means in high school.
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David R Tribble
science forum Guru


Joined: 21 Jul 2005
Posts: 1005

PostPosted: Fri Jul 14, 2006 10:35 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Ross A. Finlayson wrote:
Quote:
Consider Goedel, vis-a-vis incompleteness, and the physicists' notion
of a "Theory of Everything". Apparently, those people never heard of
Goedel, or didn't agree that his results about incompleteness hold in
their case, because they talk about a "Theory of Everything."


David R Tribble wrote:
Quote:
You're confused. The physicists' TOE deals with unifying gravity and
quantum physics, and while it uses a lot of complex math, it has
nothing to do with set theory.


Ross A. Finlayson wrote:
Quote:
About a "Theory of Everything", superstrings are mathematical
infinitesimals. Technicolor is a notion that there are particles
comprising quarks and particles comprising those etc ad infinitum, yet
while that is so something along the lines of the atom is a nilpotent
infinitesimal, in a sense.

Uh, no. Superstring theories treat particles as vibrations on the
surfaces of n-dimensional manifolds approximately the size of
the Planck scale, somewhere in the neighborhood of 10^-35 meters.
That's a rather large distance to consider "infinitesimal".
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David R Tribble
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Joined: 21 Jul 2005
Posts: 1005

PostPosted: Fri Jul 14, 2006 10:48 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Hatto von Aquitanien wrote:
Quote:
"There is an algorithm that "generates" the members of S. That means that
its output is simply a list of the members of S: s1, s2, s3, ... If
necessary it runs forever."

And for the sake of talking about generating .3, .33, .333,..., it is
quite sufficient.


David R Tribble wrote:
Quote:
Are you saying that this eventually generates 0.333... (1/3)?


Why would you think such a thing? Is it because you can set up the pattern
and then apply strong induction? Is there some way of communicating or
arriving at that idea which doesn't involve some kind of generalization
from an iterative process?

Yes. 1/3 = sum{i=1 to oo} 3x10^-i, which is a sum of infinite terms,
not an iterative process.


Hatto von Aquitanien wrote:
Quote:
The only addition I need to make in order to get my original
point of it producing a bijection with a subset of N is that I need to
keep track of the loop count.


David R Tribble wrote:
Quote:
And what is that loop count after you've generated 0.333... ?


Hatto von Aquitanien wrote:
Quote:
I don't have a fast enough CPU, nor enough RAM to answer that. What we do
when we set up an induction hypothesis and then show it hold for some start
value as well as for n+1 for arbitrary n is to conclude that _if_ we could
do this infinitely, then.... Such open ended conclusions are useful when
evaluating certain kinds of questions such as whether a series converges.

Yes, the series 0.3, 0.33, 0.333, etc.., does indeed converge to 1/3.
But the (countable) sequence of these values does not actually contain
the limit value 0.333... (1/3).


Hatto von Aquitanien wrote:
Quote:
We have a tendency to accept .333... as being as good a specification of a
number as is 3, for example. That is to say "for all intents and
purposes, .333... is as good a specification of a number as is 3." But is
it? Are there some attributes which we should not attribute to .333...
which we attribute to 3 by virtue of its being a number?

1/3 = sum{1 to oo} 3 x 10^-1 = 0.333...
3 = 3 + sum{1 to oo} 0 x 10^i.
I don't see any significant difference in the two as "numbers",
that is, I don't see how "3" is a "better" specification of a number
than "0.333...".

We only have a "tendency to accept" 0.333... as a good "specification"
of 1/3 because we can prove that they are exactly one and the same
number. Why would you think otherwise?
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Hatto von Aquitanien
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Joined: 19 Nov 2005
Posts: 410

PostPosted: Fri Jul 14, 2006 11:15 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Nathan wrote:

Quote:
Hatto von Aquitanien wrote:
Nathan wrote:

You're supposed to learn to understand the meaning of
the limit of a sequence, particularly the idea of the sum of an
infinite series as the limit of the sequence of partial sums.

Not everybody does learn to understand this, which explains the
.999...=1 threads.

Is that what happens to all the smart people in college? They have their
common sense psychologically pounded out of them. They either give lip
service to the orthodox canon, accept it as truth, or transfer to
computer science.

No, there's another option you left out; the one I gave.
They can also learn to understand the concept of a limit.
As I said, this is what is supposed to happen.
This doesn't require that they "accept it as truth", as some sort of
dogma.

There may indeed be those who "give lip service to the canon",
because they really don't grasp it. There certainly are those who
"accept it as truth"; it can be very hard to help certain students
to learn to understand concepts and not just to do mechanical
calculations. Some in that category do transfer to
computer science, where they still tend to have trouble
understanding concepts.

And no, this doesn't necessarily happen in college. I said
"around the second semester of calculus", which for a lot of
"the smart people" means in high school.

Perhaps you can explain to me how one arrives at 1/3 = .333... without
appeal to an iterative process extended ad infinitum.
--
Nil conscire sibi
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Fri Jul 14, 2006 11:27 pm    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <iJOdnfATh6UuuCXZnZ2dnUVZ_qGdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

Quote:
Nathan wrote:

Hatto von Aquitanien wrote:
Nathan wrote:

You're supposed to learn to understand the meaning of
the limit of a sequence, particularly the idea of the sum of an
infinite series as the limit of the sequence of partial sums.

Not everybody does learn to understand this, which explains the
.999...=1 threads.

Is that what happens to all the smart people in college? They have their
common sense psychologically pounded out of them. They either give lip
service to the orthodox canon, accept it as truth, or transfer to
computer science.

No, there's another option you left out; the one I gave.
They can also learn to understand the concept of a limit.
As I said, this is what is supposed to happen.
This doesn't require that they "accept it as truth", as some sort of
dogma.

There may indeed be those who "give lip service to the canon",
because they really don't grasp it. There certainly are those who
"accept it as truth"; it can be very hard to help certain students
to learn to understand concepts and not just to do mechanical
calculations. Some in that category do transfer to
computer science, where they still tend to have trouble
understanding concepts.

And no, this doesn't necessarily happen in college. I said
"around the second semester of calculus", which for a lot of
"the smart people" means in high school.

Perhaps you can explain to me how one arrives at 1/3 = .333... without
appeal to an iterative process extended ad infinitum.

By appeal the usual 'delta-epsilon' type of definition of the limit of a
sequence of partial sums?

This does not require appeal to extensions ad infinitum but only on
finite extensions being of circumscribable effect.
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 15, 2006 12:27 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Virgil wrote:

Quote:
In article <iJOdnfATh6UuuCXZnZ2dnUVZ_qGdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

Perhaps you can explain to me how one arrives at 1/3 = .333... without
appeal to an iterative process extended ad infinitum.

By appeal the usual 'delta-epsilon' type of definition of the limit of a
sequence of partial sums?

This does not require appeal to extensions ad infinitum but only on
finite extensions being of circumscribable effect.

"finite extensions being of circumscribable effect". Can't say I really
know what that means. "Circumscribable" means I can draw a circle around
something. What are you drawing a circle around? IOW, what exactly is it
that you are confining? You can give me some expression for delta which
satisfies an arbitrary epsilon, but that seems to assume that I can prove
something about every possible term in a potentially infinite sum. Whether
I am to understand that the sum has actually been calculated or merely
contemplated is irrelevant to the question of whether the idea of repeating
a process ad infinitum is involved.

More to the point, these partial sums are the very evil I am supposed to
have banished from my psyche by the second semester of calculus.
--
Nil conscire sibi
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

PostPosted: Sat Jul 15, 2006 1:34 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

In article <W9qdneqXJ7s1qyXZnZ2dnUVZ_r2dnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

Quote:
Virgil wrote:

In article <iJOdnfATh6UuuCXZnZ2dnUVZ_qGdnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

Perhaps you can explain to me how one arrives at 1/3 = .333... without
appeal to an iterative process extended ad infinitum.

By appeal the usual 'delta-epsilon' type of definition of the limit of a
sequence of partial sums?

This does not require appeal to extensions ad infinitum but only on
finite extensions being of circumscribable effect.

"finite extensions being of circumscribable effect". Can't say I really
know what that means. "Circumscribable" means I can draw a circle around
something. What are you drawing a circle around?


Epsilon is the radius and the potential limit value is the center. For
each epsilon greater than zero, only finitely many partial sums are
allowed outside this circle if the series is to converge to that limit.

Quote:
that you are confining? You can give me some expression for delta which
satisfies an arbitrary epsilon, but that seems to assume that I can prove
something about every possible term in a potentially infinite sum. Whether
I am to understand that the sum has actually been calculated or merely
contemplated is irrelevant to the question of whether the idea of repeating
a process ad infinitum is involved.

As convergence depends on the finiteness of a set depending on epsilon,
no appeal to an endless sequence of calulations is required or implied.
Quote:

More to the point, these partial sums are the very evil I am supposed to
have banished from my psyche by the second semester of calculus.

Any problems you have with managing your psyche would come more under
the heading of psychology than mathematics.
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Hatto von Aquitanien
science forum Guru


Joined: 19 Nov 2005
Posts: 410

PostPosted: Sat Jul 15, 2006 2:44 am    Post subject: Re: Attempts to Refute Cantor's Uncountability Proof? Reply with quote

Virgil wrote:

Quote:
In article <W9qdneqXJ7s1qyXZnZ2dnUVZ_r2dnZ2d@speakeasy.net>,
Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:

"finite extensions being of circumscribable effect". Can't say I really
know what that means. "Circumscribable" means I can draw a circle around
something. What are you drawing a circle around?


Epsilon is the radius and the potential limit value is the center. For
each epsilon greater than zero, only finitely many partial sums are
allowed outside this circle if the series is to converge to that limit.

that you are confining? You can give me some expression for delta which
satisfies an arbitrary epsilon, but that seems to assume that I can prove
something about every possible term in a potentially infinite sum.
Whether I am to understand that the sum has actually been calculated or
merely contemplated is irrelevant to the question of whether the idea of
repeating a process ad infinitum is involved.

As convergence depends on the finiteness of a set depending on epsilon,
no appeal to an endless sequence of calulations is required or implied.

Finiteness of a set? How can I show that all possible values of a
potentially infinite series of partial sums will lie within some
confinement without implicitly contemplating all members of that series via
some form of extrapolation?

--
Nil conscire sibi
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