Re: Two results of set geometry

Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
stephen@xxxxxxxxxx wrote:
Who said that quantities are points on the continuum?
I did. They can also be considered as partitions of the continuum to the
left of that point.
Okay. But omega is not a point on the continuum. So why bring it up?

For that very reason. I am approaching WM's topic from a few different
areas. He's discussing the nature of omega, and the contradiction
between a diagonal half of a square which is infinitely tall yet only
finitely wide. That violates the definition of a square, because it
violates the meaning of equality.

There are no infinitely tall squares in set theory, so I have no
idea what you are WM are talking about. A square has four sides.

Oh. So, in set theory, a solid square of finite dimensions is not
infinitely tall as far as the number of points on either of the vertical
sides?

A square of finite dimensions is not infinitely tall. Using "number of points"
as a measure of "height" is nonsensical.

The "figures" that WM has been talking about have two sides, the
left side, and the diagonal, as far as I can see. There is no "bottom",
which of course is the simple fact that the two of you seem totally
incapable of understanding.

Incapable of accepting is what I think you mean. When we say two numbers
are equal, and derive this truth inductively, I think WM and I both
agree the equality holds in the infinite case, where such a case exists.

No, you are incapable of understanding is that endless things do not have
an end. There is no last natural number. There is no last row in an
endless list.


Every count is a member of the continuum. I'm a fundamentalist.

Which is all fine and dandy, assuming you actually define what you mean
by "count" and "continuum". You apparently are not using any standard
definition for either.



What is the standard definition of count?

You count something by starting at 1, and ending at some natural number n.

On the other hand, exactly where is your Big'Un located?
Big'un is the count of the real called 1, the Big'unth one after after
the real called 0.
That does not really tell me where Big'Un is located. But it does
raise the question of what is the Big'unth-1 real after the real call 0?
Does that real have a name? Does it even exist?

Sure. It's 1-Lil'Un. That's actually infinitesimal less than 1, so is
not distinguishable on the finite scale. There is no minimal finite
difference possible.
But the real numbers do not include infinitesimals. So when you said that
1 is the Big'unth real number after the real called 0, you were incorrect,
or else you are using your own private definition of "real number".

Well, yes. Once one accepts actually infinite counts, one accepts
infinitesimally different reals. If one wants to apply any count to such
sets, one must establish a unit. Each of those reals is distinguished
from those closest by a unit infinitesimal. It's a different system.
Incompatible, but not wrong.

Who knows if it is wrong, as it is not defined. You cannot say that
Big'Un is the count of the TonyReals from 0 to 1 until you define
what the TonyReals are. And your definition of the TonyReals cannot
include Big'Un.


I define them as points in the interval. Big'un is a marriage of count
and measure, for the uncountable.

Points in the interval? What does that mean? What interval? What points?
Can you define any of your terms?

The real numbers are well defined. The real numbers can be used to describe
points on a line, as typically understood. For example, between any two points
on a line, there is another point on the line. There are no "sub points", or "adjacent
points". However, you claim that your numbers have such things, so you obviously
are talking about a different notion of points on a line.

So what is (1+1-Lil'Un)/2? Is it not a real number? Is it not between
1-Lil'Un and 1?

Yes it is. It's a sub-real. I admit, it's an entirely different system,
and not likely to appeal to those steeped in the standard teaching. Sorry.

Which just further stresses the point that you have totally failed to
unambiguously define the "location" of Big'Un. The "reals" you are counting
apparently have almost nothing to do with the standard reals, which are closed
under addition and division by 2.


Sure, by standard definitions. My reals are points on the line, each
distinguishable point corresponding to a distinguishable real value.

No, your reals are Tony-Points on a Tony-Line. There is not "first" point
after the first point in a line, according to the standard definitions of points
and lines. Yet in your "line" there is a "point" that corresponds to Lil'Un,
the first number after 0.


Big'un as a value is located Big'un unit intervals to the right of 0 on
the continuum.
That is kind of circular. Can you describe the location of Big'un without
using Big'un in the description?


I defined Big'Un already as the count of reals in [0,1).

No, it is the count of TonyReals in [0,1). You have not defined what
the TonyReals are, so I have no idea what Big'un is. Perhaps it is 10,
for all I know.

Consider it the number of points in one linear unit of space.

No, I cannot do that because I do not know what sort of points you are
talking about. You do agree that in the standard definition of points
there is not first point after 0, or last point before 1, on the line from 0
to 1? Every geometry book in the world will tell you that between any two
points there is another point. Yet you claim that there is a first point after
0, and a last point before 1, you obviously mean something different by "points"
than I do.


You asked me
where the value lies on the continuum, and I told you how to find it.
Start at 0, and add a unit interval to the right for each point in the
unit interval.

But you are not talking about any continuum with which I am familiar.
Your "continuum" has "levels", and "adjacent" points with "sub points"
between them. None of these things have been adequately defined, so
your definition of Bug'Un is totally unfounded.

<snip>


Yes, it has levels of infinity and infinitesimality. I'll have to work
of adequately defining those terms.

Until you define them, it is impossible to know if there is any sense to
what you are saying. But you might as well stop using the term "continuum",
because you what are describing bears little resemblance to what the word
means.

Lil'Un is the first infinitesimal greater than 0 on the first level.
Again, infinitesimals are not real numbers. So can you describe what
you actually mean by the "real" numbers between 0 and 1?


Okay, you have a point. A real is a different real from another if it
can be finitely distinguished, quantitatively. So, an infinitesimal
difference does not distinguish two standard reals, and there is no
smallest finite, as there is no largest. However, in order to tie
together measure and count, we can assign a count of points to the unit
interval. That, of course, does require a loosening of the definition of
reals. But that's resolvable. If a "finite difference" means a
difference in a finite number of symbols used to express the numbers,
and we allow infinite bit positions, then the T-riffic numbers can be
taken to express reals, such that 1.000...000>0.999...999. We can do
"real" arithmetic with such numbers.

That does not answer the question of what you mean by the "real" numbers
between 0 and 1. Your Big'Un is not the count of real numbers between
0 and 1. Perhaps it is the count of your T-riffic numbers, but who knows,
as you refuse to actually define what you mean by anything.


How do you define a real such that you can get a count per unit of
space? You have to declare such a unit.

What does that even mean? What do you mean by "count"? How do you
even know that you can define a real such that you can get a count per unit
of space? You are just assuming that there exists some logically coherent
entity that matches these vague intuitions you have, but until you explicitly
define it, you do not know if there is any consistency to your arguments.

Given the standard definitions of "real number", and the standard definitions
of "count", there is no "count per unit of space".

<snip>


As cardinality is defined, in conjunction with Cantor's diagonal
argument, which is flawed, yes.

So you are now back to claiming Cantor's diagonal argument is flawed? Can
you point out the flaw?



Yes. The diagonal argument for the reals does not work in unary,

Does not work in unary? Can you explain how you represent real numbers in unary?

and
should not be dependent on the number system used. The diagonal argument
assumes the list is square,

The list is square? What does that mean? Lists are not squares, and squares
are not infinite.

and shows that it cannot be, thus
demonstrating that the power set is larger than the root set. That's
all. It has more to do with symbolic language than reals. But then, all
math is symbolic language, except the part that's geometry and measure.

I think it has more to do with you simply not understanding the diagonal argument.

<snip>

That is fine, if you precisely define what you mean by "count". But I would
like to see your definition of "count" that allows the real numbers in the
interval [0,1] to have a count, but prevents the naturals from having a count.

It's defined axiomatically, if that's aright, tying measure and count.

What axioms? Can you list the axioms that you are using to define "count"?

Not quite yet in full form. Sorry.

So in other words, you do not have a definition of count, and have no idea
if any of your statements about "count" make any sense.

Stephen



.

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