# Continuity and uncountability

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Continuity and uncountability
Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis.
The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem. Rational numbers are countable, the line they make contains holes. Real numbers are uncountable, the line they make is continuous. So, continuity must be created by the uncountability of the points of a continuous line. One can imagine that uncountable points are so numerous on the real line that real numbers are squeezed together.
Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean?
PDF Continuity and uncountability

Continuity and uncountability
Peng Kuan 彭宽
27 September 2016
Abstract: Discussion about continuity of line, how continuity is related to uncountability and the continuum hypothesis.
What is uncountability for?
The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem. Rational numbers are countable, the line they make contains holes. Real numbers are uncountable, the line they make is continuous. So, continuity must be created by the uncountability of the points of a continuous line. One can imagine that uncountable points are so numerous on the real line that real numbers are squeezed together.
Georg Cantor called the set of real numbers continuum, so he probably thought of creating continuity with discreteness when inventing uncountability. But, what does continuity really mean?
Cauchy’s continuity
I haven’t found existing definition for continuity of line but definitions for continuous function instead. For example, using a Cauchy’s sequence s=(x_i | x_i∈R)_(i∈N) which converges to a point a, the continuity of a function f (x) at point a is defined as follow:
lim┬(i→∞)⁡〖x_i 〗=a⇒ lim┬(i→∞)⁡f(x_i )=f(a) (1)
The real line is the function f=0, satisfies this definition at any real numbers and is continuous. I call this definition Cauchy’s continuity.
However, if a and xi are rational numbers, the converging sequence will be entirely in ℚ and Cauchy’s continuity will allow the set of rational numbers to be continuous, which is wrong. So, Cauchy’s continuity is inadequate to define continuity of line.
Geometric continuity
Figure 1
Line is a geometric form that represents the form of real objects, for example conductive wire, water pipe, trajectory of planets etc. To illustrate the continuity of line, imagine the lines in Figure 1 as a conductive wire interrupted between points A and B. When electric current flows in the wire and the interruption, electrons move in the conductive medium of the wire and make an electric arc through the air in the interruption. To cross the interruption an electron must quit the conductive medium from point A, pass through the air and enter the point B. Following this image, continuous line is a mathematical medium in the form of line in which a point can move without quitting. An interruption is a location where a moving point must quit the medium.
So, I propose the following definition of continuity:
A line is continuous between 2 points C and D if the space between them is zero. Equivalently, the line is continuous between C and D if a moving point can go from C to D without crossing other point else than C and D. If all points of a line satisfy this condition, then the line is everywhere continuous.
C and D are said to be in contact and adjacent to each other. In the following, this kind of continuity will be referred to as geometric continuity.
Real line
Is the real line geometrically continuous? No interruption can be found on the real line, but the condition of geometric continuity is not satisfied. Take 2 different real numbers a and b and bring them close to each other, no matter how close they are, they are always separated by infinitely many other numbers. If an imaginary electron goes from a to b, it must cross many other points else than a and b. So, the real line is interrupted between a and b but not geometrically continuous.
Also, being not in contact with other point, a is an isolated point. As a can be any real numbers, all real numbers are isolated and the set ℝ is discrete. So, ℝ is not a continuum.
Constructing continuous line
Figure 2
Why are real numbers discrete? Let us see Figure 2. The points on the right are in contact to each other and they are continuous. The distance between the centers of adjacent points is denoted by d and the width of points by w. These points are continuous because w=d.
On the left, the distance between the points is still d but the width of points is smaller, w<d, this makes them discrete. However, we can shrink the distance d on the left to make the points continuous again.
If the points were real numbers, the width of points equals zero and, however small the value of d is, the points are always separated by a distance because d>0. Therefore, that the width of points is zero is the reason that makes real numbers discrete. This also proves that uncountability is unrelated to continuity. Indeed, real numbers are uncountable and discrete at the same time.
On the other hand, if one puts a real number s in contact with another number r, they will occupy the same point because their widths are zero. If t is put in contact with s, the 3 numbers r, s and t will occupy the point of r. We can repeat this operation uncountably many times, we will obtain only one point, not a line. So, uncountably many points of zero width do not make continuous line.
Figure 3
Figure 4
So, to construct a geometrically continuous line the constructing points must have nonzero width, that is, w>0. What would be the value of w? Let us deconstruct the continuous line in the interval [0,1] by splitting, as shown in Figure 3 and Figure 4. The first splitting point is ½, then the resulted 2 segments are split at ¼ and ¾. And then, the 4 resulted segments are split at ⅛, ⅜, ⅝ and ⅞. The spitting goes forever and we obtain an infinite sequence of splitting points ssplit=(aiℝ )iℕ and an infinite sequence of segments.
The segments are in contact with one another, securing continuity. Their length equals the infinitesimal number ε=1/2^∞ . These segments are the constructing blocks of the original line, each one starts at its splitting point aiℕ and has the length .
Remark: The construction of geometrically continuous line proves that the controversial infinitesimal number  really exist, otherwise, continuity cannot arise.
General model
Figure 5
For a general line in space such as the one shown in Figure 5, a constructing segment is determined by 6 quantities: 3 coordinates for starting position, 2 angles for direction and  for length. This segment, S in Figure 5, will be referred to as infinitesimal vector-segment and is the constructing blocks for general line.
Real numbers are discrete points that are 0-dimensional objects. In the contrary, infinitesimal vector-segment has nonzero length and is a one-dimensional object. So, we have the following property:
One-dimensional geometrically continuous line is constructed only with one-dimensional objects.
Consequently, 0-dimensional points cannot construct one-dimensional line, even they are uncountably many. In general, continuous objects in higher dimension are not constructed with objects of lower dimension. For example, 2-dimensional surfaces are constructed with infinitesimal surface2 and
n-dimensional volumes with infinitesimal n-volumen.
Uncountability
How did Georg Cantor link uncountability to continuity? In fact, he constructed the continuum ℝ in two steps: 1) ℝ is uncountable; 2) Uncountability of ℝ creates continuity for the real line.
Figure 6
He concentrated himself on proving that ℝ is uncountable. The first proof he gave was based on nested intervals [a0, b0], [an, bn], as shown in Figure 6. Because anbn when n, Georg Cantor claims that the limit of an and bn is a number not included in the lists a0a and b0b, thus real numbers are uncountable.
However, does the limit of an and bn really exist? A limit is a real number which must be fully determined, that is, all the digits from 1st to th are fixed, for example . When n increases, the first m digits of an and bn get fixed and make a number that seems to converge. The first m digits of the limit may equal this number, but the limit’s last digits, from m+1st to th, will never be determined. In fact, when n increases, an and bn both vary and the points within the interval [an, bn] are all undetermined. So, the limit that Georg Cantor claims cannot exist and this proof is invalid.
In addition to this flaw which is explained in «On Cantor's first proof of uncountability», Georg Cantor’s later proofs, the power-set argument and the diagonal argument, contain also flaws, which are explained in «On the uncountability of the power set of ℕ» and «Hidden assumption of the diagonal argument». So, all 3 proofs that Georg Cantor provided fail and uncountability possibly does not exist.
About the second step Georg Cantor did nothing but simply claim that ℝ is a continuum; probably he assumed that uncountability really created continuity. But it is shown above that uncountability is not related to continuity. So, uncountability has lost its utility and becomes useless except for itself.
Continuum hypothesis
The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers which is a discrete set and the real numbers which is a continuum. The idea behind this hypothesis is that there cannot be set that is discrete and continuous at the same time. Georg Cantor tried hard to find such set; the Cantor’s ternary set is probably one of his attempts.
However, it is shown that uncountability is not proven, then the cardinality of real numbers is questionable. Anyway, ℝ is not a continuum and the continuum hypothesis makes no longer sense.

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Some questions.

The integers are countable (by definition) but they do not possess the property that between every pair of integers there exists another integer.

For example there is no integer between 2 and 3.

So if you line up the integers you can reasonably state there are 'holes' in the line.

The rational numbers are made by ratios of the integers and do posses the property that between every pair of rationals there exists another rational.

In fact this means that between every pair of rationals there is an unending sequence of rationals.

So you must prove the assertion there are holes in the rational lineup.

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The real line is made of real numbers which are points. Points are discrete objects, but lines are continuous objects. How does continuity arise out of discreteness when points make line? The idea of uncountability solves this problem.

Uncountability is not sufficient. Suppose I take the real line and delete the point at 0. The resulting set is not connected and it is not what you would call "continuous," but it's uncountable. In fact the word "continuous" is wrong here because continuity applies to functions and not sets. However if you mean "no holes," plenty of uncountable sets have holes.

A more striking example is the Cantor set, which is an uncountable set of measure zero. It's full of holes.

ps -- I see you referenced the Cantor ternary set. So if you know this example, why does your exposition not deal with it? In other words you already know a striking counterexample to your idea.

pps -- A few more idle thoughts. Bottom line you are confusing cardinal, order, and topological properties with each other.

Remark: The construction of geometrically continuous line proves that the controversial infinitesimal number really exist, otherwise, continuity cannot arise.

That's just not true. The standard construction of the real numbers within set theory shows that we do not need infinitesimals. There are no infinitesimals in the real numbers and they are not needed in math. It's true that there are nonstandard models containing infinitesimals but they don't add anything to the discussion and do not provide any more deductive power; so they are a distraction in these types of discussions.

One-dimensional geometrically continuous line is constructed only with one-dimensional objects.

Of course that's not true. A 1-D line is made up of 0-D points. It is true that it is a philosophical mystery. But it's not a mathematical mystery!

A better way to think of it is that a line is the path of a point through space. If you had a 0-D point moving through the plane, it would trace out a 1-D path. This was Newton's point of view.

How did Georg Cantor link uncountability to continuity? In fact, he constructed the continuum ℝ in two steps: 1) ℝ is uncountable; 2) Uncountability of ℝ creates continuity for the real line.

You haven't defined "continuity" of a point set. Until you do, your argument is not valid. Do you mean dense? Perhaps you mean complete, in the sense that every Cauchy sequence converges.

In order to have an argument you have to say exactly what you mean by a continuous set. It would make it easier to understand what you're trying to say.

The rationals have a dense linear order, but they're not complete because some Cauchy sequences don't converge.

However, does the limit of an and bn really exist? A limit is a real number which must be fully determined, that is, all the digits from 1st to th are fixed,

I have no idea what that means. Every real number has a decimal representation that is "fully determined" in the sense that all of its digits are "fixed." What do you mean determined? Every real number has a decimal expansion (or two). What does that mean to you?

In addition to this flaw which is explained in «On Cantor's first proof of uncountability», Georg Cantors later proofs, the power-set argument and the diagonal argument, contain also flaws, which are explained in «On the uncountability of the power set of ℕ» and «Hidden assumption of the diagonal argument». So, all 3 proofs that Georg Cantor provided fail and uncountability possibly does not exist.

To the extent that you're trying to understand the nature of the mathematical continuum, that is a noble persuit.

To the extent that you're here to deny Cantor's results, that's generally not a productive topic. Nobody doubts Cantor's results.

About the second step Georg Cantor did nothing but simply claim that ℝ is a continuum; probably he assumed that uncountability really created continuity. But it is shown above that uncountability is not related to continuity. So, uncountability has lost its utility and becomes useless except for itself.

He wasn't making any such claims at all. We agree that uncountability is not related to continuity, if by continuity you mean completeness. The Cantor set and for that matter the reals minus a point are uncountable but not complete.

Continuum hypothesis

The continuum hypothesis states that there is no set whose cardinality is strictly between that of the integers which is a discrete set and the real numbers which is a continuum. The idea behind this hypothesis is that there cannot be set that is discrete and continuous at the same time.

The real numbers with the discrete topology are a discrete set. With the usual topology they're a continuum. You're confusuing cardinality, order properties, and topological properties.

Anyway, ℝ is not a continuum and the continuum hypothesis makes no longer sense.

It makes perfect sense. CH asks which Aleph is the cardinality of the reals. It has nothing to do with the topology on the reals. For example if we give the real numbers the discrete topology, then they are a discrete set. Yet their cardinality doesn't change, and it's sensible to ask what that cardinality is.

Edited by wtf

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As I think I've told you before: countability and uncountability are concepts belonging to set theory; "continuity" of the real line (really connectedness) is a concept that belongs to point-set topology, which builds on top of set theory, but has some extra structure.

The major theorem you are groping towards is this one: Any connected normal (T4) space with more than 1 point is uncountable. Note that this is a one-way theorem - you can't reason from uncountability of a set to anything set-theoretic about the set itself.

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I looked at your paper. You say:

"So, I propose the following definition of continuity:

A line is continuous between 2 points C and D if the space between them is zero."

The problem is that the distance between any two real numbers is zero if and only if the two numbers are the same. This follows from the definition of the distance between real numbers, which is just the absolute value of their difference.

You keep thinking real numbers are like bowling balls lined up in a row; and this false visualization is leading you into mathematical errors.

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Some questions.

The integers are countable (by definition) but they do not possess the property that between every pair of integers there exists another integer.

For example there is no integer between 2 and 3.

So if you line up the integers you can reasonably state there are 'holes' in the line.

The rational numbers are made by ratios of the integers and do posses the property that between every pair of rationals there exists another rational.

In fact this means that between every pair of rationals there is an unending sequence of rationals.

So you must prove the assertion there are holes in the rational lineup.

By holes in the rational lineup I mean the holes left by irrational numbers that are not in the rational lineup.

Uncountability is not sufficient. Suppose I take the real line and delete the point at 0. The resulting set is not connected and it is not what you would call "continuous," but it's uncountable. In fact the word "continuous" is wrong here because continuity applies to functions and not sets. However if you mean "no holes," plenty of uncountable sets have holes.

A more striking example is the Cantor set, which is an uncountable set of measure zero. It's full of holes.

https://en.wikipedia.org/wiki/Cantor_set

ps -- I see you referenced the Cantor ternary set. So if you know this example, why does your exposition not deal with it? In other words you already know a striking counterexample to your idea.

pps -- A few more idle thoughts. Bottom line you are confusing cardinal, order, and topological properties with each other.

By holes in the line of rational number, I mean the points that irrational numbers occupy.

In fact, I do not think that uncountable set is continuous, rather the contrary, uncountable set is not continuous, even the real line. However, I write in the introduction the idea of continuity of the mainstream idea because I cannot make people agree with my point from the beginning. I say what they agree with, then I deduce the point of none line and none continuity.

I have said about Cantor’s ternary set in the end of my paper.

The rest of my paper explains why real line is not a line and why it is not continuous.

That's just not true. The standard construction of the real numbers within set theory shows that we do not need infinitesimals. There are no infinitesimals in the real numbers and they are not needed in math. It's true that there are nonstandard models containing infinitesimals but they don't add anything to the discussion and do not provide any more deductive power; so they are a distraction in these types of discussions.

Of course that's not true. A 1-D line is made up of 0-D points. It is true that it is a philosophical mystery. But it's not a mathematical mystery!

A better way to think of it is that a line is the path of a point through space. If you had a 0-D point moving through the plane, it would trace out a 1-D path. This was Newton's point of view.

Yes that the standard construction of the real numbers does not need infinitesimals. But the real line is not continuous. This is why I create the definition of continuity, which needs infinitesimal length because real numbers have zero length and zero length does not allow continuity.

If you draw a line that is the path of a point through space, then between points there is not continuity. See the electron that passes from one point to the following, it will not be able to forego the point in between.

You haven't defined "continuity" of a point set. Until you do, your argument is not valid. Do you mean dense? Perhaps you mean complete, in the sense that every Cauchy sequence converges.

In order to have an argument you have to say exactly what you mean by a continuous set. It would make it easier to understand what you're trying to say.

The rationals have a dense linear order, but they're not complete because some Cauchy sequences don't converge.

I do not say "continuity" of a point set, but continuity of a line. In the section 3 “So, I propose the following definition of continuity: A line is continuous between 2 points C and D if the space between them is zero. Equivalently, the line is continuous between C and D if a moving point can go from C to D without crossing other point else than C and D. If all points of a line satisfy this condition, then the line is everywhere continuous.

Line is not made by point, is not a set, but a geometric form. Line can be continuous , not set.

I have no idea what that means. Every real number has a decimal representation that is "fully determined" in the sense that all of its digits are "fixed." What do you mean determined? Every real number has a decimal expansion (or two). What does that mean to you?

I mean that we cannot fix all the digits of the limit because its last digit cannot be found. Then the limit is not a real number. This is why this limit does not exist and Cantor's claim that there is forcefully a limit is wrong.

To the extent that you're trying to understand the nature of the mathematical continuum, that is a noble persuit.

To the extent that you're here to deny Cantor's results, that's generally not a productive topic. Nobody doubts Cantor's results.

He wasn't making any such claims at all. We agree that uncountability is not related to continuity, if by continuity you mean completeness. The Cantor set and for that matter the reals minus a point are uncountable but not complete.

Cantor said Real numbers are a continuum, I take this as a claim that ℝ is continuous.

The real numbers with the discrete topology are a discrete set. With the usual topology they're a continuum. You're confusuing cardinality, order properties, and topological properties.

It makes perfect sense. CH asks which Aleph is the cardinality of the reals. It has nothing to do with the topology on the reals. For example if we give the real numbers the discrete topology, then they are a discrete set. Yet their cardinality doesn't change, and it's sensible to ask what that cardinality is.

I'm not discussing topology, I do not know it. But if uncountability is not true, then cardinality is not true either.

I looked at your paper. You say:

"So, I propose the following definition of continuity:

A line is continuous between 2 points C and D if the space between them is zero."

The problem is that the distance between any two real numbers is zero if and only if the two numbers are the same. This follows from the definition of the distance between real numbers, which is just the absolute value of their difference.

In fact, I define a line not as a string of real numbers, so the points of the line cannot be real numbers. This is why I call the "points" blocks and which have infinitesimal length.

Please think the real line as a line that is fully filled, not only by real numbers that are discrete points.

As I think I've told you before: countability and uncountability are concepts belonging to set theory; "continuity" of the real line (really connectedness) is a concept that belongs to point-set topology, which builds on top of set theory, but has some extra structure.

The major theorem you are groping towards is this one: Any connected normal (T4) space with more than 1 point is uncountable. Note that this is a one-way theorem - you can't reason from uncountability of a set to anything set-theoretic about the set itself.

uncountability is a property of set. But continuity is that of geometric line, which is not built by points, that is , by set. I think the notion of continuity that I have explained is not in the present mathematics.

Edited by pengkuan

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If you draw a line that is the path of a point through space, then between points there is not continuity. See the electron that passes from one point to the following, it will not be able to forego the point in between.

That might be the heart of your misunderstanding. Physics $\neq$ math. Euclidean space is a continuum modeled by the real numbers. It's not known whether physical space is like that or not. You are right that electrons and photons don't flow into each other like the points on the real line. That's because the real line is not (as far as anyone knows) an accurate model of the real world.

It's been generally understood that math describes logically consistent worlds, and not necessarily the physically true world, since the discovery of non-Euclidean geometry in the 1840's.

Edited by wtf

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That might be the heart of your misunderstanding. Physics $\neq$ math. Euclidean space is a continuum modeled by the real numbers. It's not known whether physical space is like that or not. You are right that electrons and photons don't flow into each other like the points on the real line. That's because the real line is not (as far as anyone knows) an accurate model of the real world.

It's been generally understood that math describes logically consistent worlds, and not necessarily the physically true world, since the discovery of non-Euclidean geometry in the 1840's.

You are right to point out that the real line is not an accurate model of the real world and non-Euclidean geometry is a good example.

One of the role of mathematics is to provide accurate model of the real world to physics. If there is a physical phenomenon that does not have a accurate mathematical model for representation and prediction, then someone will surely invent one. Like non-Euclidean geometry which represents spacetime for Einstein, a type of continuity different from Euclidean space and real line must be invented to represent continuous line such as electrical circuit where electrons do not jump from point to point, or trajectory of the Earth that does not jump in space. Because real line is discrete, a moving point must jump in order to move.

Other examples of geometrical continuity such as iron chain made of links or pearl necklace exist These are continuous lines made of discrete objects and surely possess interesting properties. The interesting thing in mathematics is not to say whether a idea conforms today's mathematical theories or not, but if the idea can be the base of new things that will built upon the new idea. Construction of new is the big thing. New things are always nonsensical for old mathematics, as non-Euclidean geometry for Euclidean geometry or root -1 for real number theory.

Great mathematicians are those who construct new mathematical theory, those who check mathematical error are good professors.

What I do here is to propose new ideas. But I'm unable to construct mathematical building.

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What I do here is to propose new ideas. But I'm unable to construct mathematical building.

Why not take a few months and study the mathematical formalism of real analysis? The subjects you are thinking about have been studied for thousands of years. What is the nature of the mathematical continuum? And how does it relate to the ultimate nature of the real world?

Standard mathematics, the modern theory of the real numbers, is humanity's best answer yet (at least to the first question, the nature of the mathematical continuum). It's probably not the final answer.

If you seek to overthrow the conventional knowledge, shouldn't you take the time to learn the conventional knowledge first?

Newton, whose work created the modern scientific world, got his start by fully mastering the work of the ancients.

Edited by wtf

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Why not take a few months and study the mathematical formalism of real analysis? The subjects you are thinking about have been studied for thousands of years. What is the nature of the mathematical continuum? And how does it relate to the ultimate nature of the real world?

Standard mathematics, the modern theory of the real numbers, is humanity's best answer yet (at least to the first question, the nature of the mathematical continuum). It's probably not the final answer.

If you seek to overthrow the conventional knowledge, shouldn't you take the time to learn the conventional knowledge first?

Newton, whose work created the modern scientific world, got his start by fully mastering the work of the ancients.

This is surely the best thing to do. I agree.

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I could not really understand a detail why don't you make it a bit easier to audiences,otherwise I am own highly demanding sometimes to write longly.

For instance check such things and apply.

like this

continuoum

f is a function

∀ ɛ > 0 , δ , I x- x0 I < δ , I f(x) - f(x0) I < ɛ

simply if any functşon is continuous ,then

x → x0 , f(x) → f(x0)

discrete

probably there exist several descriptions for this one

when d is a metric. and on a set/space d(x,y)= 1 ,then this is discrete.

Countability

if there exist a function ,f is one by one and

f G → N

G is countable

Cantor theorem.

if any function is continuous i a closed interval,then that continuoum is regular.

Edited by blue89

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I could not really understand a detail why don't you make it a bit easier to audiences,otherwise I am own highly demanding sometimes to write longly.

For instance check such things and apply.

like this

continuoum

f is a function

∀ ɛ > 0 , δ , I x- x0 I < δ , I f(x) - f(x0) I < ɛ

simply if any functşon is continuous ,then

x → x0 , f(x) → f(x0)

discrete

probably there exist several descriptions for this one

when d is a metric. and on a set/space d(x,y)= 1 ,then this is discrete.

Countability

if there exist a function ,f is one by one and

f G → N

G is countable

Cantor theorem.

if any function is continuous i a closed interval,then that continuoum is regular.

The main idea I want to show in my paper is the difference of characteristic between a one dimensional line and a 0 dimensional point. As the point's length is zero, adding points in the dimension of the line is equivalent to adding 0's, which gives zero length at the end. So, a line is not made by points and Cantor's effort to make line with uncountable points is not valid.

But in a discussion in an other forum, I learned that using continuity to qualify line is not appropriate for explaining my idea. In fact, continuity is a property of string of points and I cannot use it to show the difference of characteristic between line and point. I will try using another method another day.

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The main idea I want to show in my paper is the difference of characteristic between a one dimensional line and a 0 dimensional point. As the point's length is zero, adding points in the dimension of the line is equivalent to adding 0's, which gives zero length at the end. So, a line is not made by points and Cantor's effort to make line with uncountable points is not valid.

But in a discussion in an other forum, I learned that using continuity to qualify line is not appropriate for explaining my idea. In fact, continuity is a property of string of points and I cannot use it to show the difference of characteristic between line and point. I will try using another method another day.

Don't the characteristics of points and lines depend on your definitions of them?

I see you are following the ancient Greeks in this respect.

Euclid Definition 1 : A point is that which hath no part.

Euclid Definition 2 : A line is a breadthless length.

The only trouble with this (which Euclid never answered satisfactorily) is describing what happens when two lines intersect.

That is answering the question do they have anything in common?

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Studiot, two lines intersect in a point. That was true for Euclid and it's true in the modern Euclidean plane. Can you explain what you are trying to say?

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Studiot, two lines intersect in a point. That was true for Euclid and it's true in the modern Euclidean plane. Can you explain what you are trying to say?

First let me say thank for for asking this question without assuming something is wrong. +1

I was responding to the claim in the quote by pengkuan that lines are not made up of points.

So I ask you to look carefully at the region indicated as it has no length in any direction so cannot qualify to be classed as a line.

Yet it appears to be part of two lines.

So is a segment of a line not a line?

And if not what is it?

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Studiot you know very well that the intersection of two lines is a point. This is true in ancient Euclidean geometry and in modern analytic geometry. So I ask again, what are you talking about?

Edited by wtf

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Studiot you know very well that the intersection of two lines is a point. This is true in ancient Euclidean geometry and in modern analytic geometry. So I ask again, what are you talking about?

I'm sorry, which exact postulate, definition, axiom or common notion of Euclid states that?

The only one that actually mentions meeting is postulate 5, and it does not say they meet at a point.

Edited by studiot

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In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (Euclid's original axioms contained various flaws which have been corrected by modern mathematicians),[10] a line is stated to have certain properties which relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point.

I do not have sufficient knowledge of Euclid's original work to know if this is something he states or prove. I'd be very surprised to know that Euclid thought two lines can meet in something other than a point. Have you got a reference? I genuinely don't understand why you are pursuing such a seemingly wrong idea. Maybe you know something I don't, in which case I'd be grateful for a reference.

Edited by wtf

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Don't the characteristics of points and lines depend on your definitions of them?

I see you are following the ancient Greeks in this respect.

Euclid Definition 1 : A point is that which hath no part.

Euclid Definition 2 : A line is a breadthless length.

The only trouble with this (which Euclid never answered satisfactorily) is describing what happens when two lines intersect.

That is answering the question do they have anything in common?

Thanks for asking this question that makes things clearer.

Yes, they have a point in common. This means that line includes points. But line is not made of points, which was not explained by me previously. Let me try with an example.

Take a square of sides s and 2 ribbons of width s. When 2 ribbons intersect, the intersection is a square. So, I will say that a ribbon is "made" of squares because when the squares are put one next to another, we will have a ribbon.

Let us shrink the value of s to zero. The square becomes a point that has neither width nor length. The ribbon becomes a line with no width. In this case, the intersection of 2 lines has neither width nor length, it is a point. But if we put the point one next to another, we will not get a line, because zero length added to zero length gives zero length. This is what I mean by "line is not made of points." But line does include points.

Edited by pengkuan

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https://en.wikipedia.org/wiki/Line_(geometry)

I do not have sufficient knowledge of Euclid's original work to know if this is something he states or prove. I'd be very surprised to know that Euclid thought two lines can meet in something other than a point. Have you got a reference? I genuinely don't understand why you are pursuing such a seemingly wrong idea. Maybe you know something I don't, in which case I'd be grateful for a reference.

Euclid Book I Postulate 5

${\rm K}\alpha \iota {\rm{ }}\varepsilon \alpha \upsilon {\rm{ }}\varepsilon \iota \varsigma {\rm{ }}\varepsilon \nu \theta \varepsilon \iota \alpha {\rm{ }}\varepsilon \mu \pi \iota \pi \tau o\mu \sigma \alpha {\rm{ }}\tau \alpha \varsigma {\rm{ }}\varepsilon \upsilon \tau o\varsigma {\rm{ }}\kappa \alpha \iota {\rm{ }}\varepsilon \pi \iota {\rm{ }}\tau \alpha {\rm{ }}\alpha \upsilon \tau \alpha {\rm{ }}\mu \varepsilon \eta {\rm{ }}\gamma \omega \upsilon \iota \alpha \varsigma {\rm{ }}\delta \upsilon o{\rm{ }}o\rho \omega \upsilon$
${\rm{ }}\varepsilon \lambda \alpha \sigma \sigma o\upsilon \alpha \varsigma {\rm{ }}\pi o\eta {\rm{ }}\varepsilon \kappa \beta \alpha \lambda \lambda o\mu \varepsilon \upsilon \alpha \varsigma {\rm{ }}\tau \alpha \varsigma {\rm{ }}\delta \upsilon o{\rm{ }}\varepsilon \upsilon \theta \varepsilon \iota \alpha \varsigma {\rm{ }}\varepsilon \pi {\rm{ }}\alpha \pi \varepsilon \iota \rho o\upsilon {\rm{ }}\sigma \upsilon \mu \pi \iota \pi \tau \varepsilon \iota \upsilon$
${\rm{ }}\varepsilon \phi {\rm{ }}\alpha {\rm{ }}\mu \varepsilon \rho \eta {\rm{ }}\varepsilon \iota \sigma \iota \upsilon \alpha \iota \tau \omega \upsilon {\rm{ }}\delta \nu \upsilon {\rm{ }}o\rho \theta \omega \nu {\rm{ }}\varepsilon \lambda \alpha \sigma \sigma o\nu \varepsilon \varsigma$

Drat it's lost all the spaces

Which translated (Sir Thomas Heath 1908) reads

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

Edited by studiot

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That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

I agree this leaves unanswered the question of whether Euclid regards the intersection of two lines as a point. But I'm still baffled as to why you (a) think this will be helpful to pengkuan, who's confused about the nature of the real line; or (b) is relevant as a response to my questions.

If you tell me you're a scholar of Euclid in the original Greek language then perhaps you can elaborate on your claim that Euclid thinks the intersection of two lines is something other than a point.

Edited by wtf

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I agree this leaves unanswered the question of whether Euclid regards the intersection of two lines as a point. But I'm still baffled as to why you (a) think this will be helpful to pengkuan, who's confused about the nature of the real line; or (b) is relevant as a response to my questions.

If you tell me you're a scholar of Euclid in the original Greek language then perhaps you can elaborate on your claim that Euclid thinks the intersection of two lines is something other than a point.

Two separate issues.

Firstly pengkuan has confirmed that I have more or less correctly deduced the thrust of his argument (post 19)

I did not expect his reply so quickly, I will come back to that.

Secondly pengkuan has correctly identified the continuity question, which was unsolved by Euclid and remained so although discussed for nearly two thousand years before Dedekind in 1872 "Stetigkeit und irrationale Zahlen", but is stuck at the same issue Euclid was.

translated the Dedekind postulate of continuity reads.

"If all points of a straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, there exists one and only one point that produces this division of all the points into two classes, this division of a straight line into two parts.

Daedalus, can you help with the Greek?

I'm sorry I lost all the spacing of the words when pasting in.

Computer typology is not all it's cracked up to be.

Edited by studiot

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Studiot, We agree that pengkuan is confused about the nature of the continuum. The philosophical issues have not been solved. But the mathematical issues are fully solved by modern real analysis based on modern set theory.

In my opinion, it is not helpful to penkuan to encourage his misunderstandings; when in fact there is a modern mathematical theory that resolves them. Mathematically at least. Not, I agree, philosophically.

It is my understanding that pengkuan is asking about the mathematical and not the philosophical nature of the continuum. So there is no need to pretend that non-parallel straight lines in the Euclidean plane intersect in a point. We can use analytic geometry to solve the system of two linear equations. What Euclid might or might not have thought or said two thousand years ago is quite beside the point. In my opinion of course.

Secondly, I simply cannot imagine that Euclid had in mind anything other than a point as the intersection of two lines that are not parallel. For you to imply otherwise is either sophistry or ignorance OR I'm simply missing your point completely! So I'm not being pejorative, simply baffled at your claim that Euclid does not think the intersection of lines is a point.

Edited by wtf

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Thanks for asking this question that makes things clearer.

Yes, they have a point in common. This means that line includes points. But line is not made of points, which was not explained by me previously. Let me try with an example.

Take a square of sides s and 2 ribbons of width s. When 2 ribbons intersect, the intersection is a square. So, I will say that a ribbon is "made" of squares because when the squares are put one next to another, we will have a ribbon.

Let us shrink the value of s to zero. The square becomes a point that has neither width nor length. The ribbon becomes a line with no width. In this case, the intersection of 2 lines has neither width nor length, it is a point. But if we put the point one next to another, we will not get a line, because zero length added to zero length gives zero length. This is what I mean by "line is not made of points." But line does include points.

That last paragraph is exactly Euclid's view as also outlined in my sketch.

He ignored the problem that you ask about, although Heath in his extensive discussion of Greek geometry within his translation of Euclid made it clear that the Greeks knew of it.

The resolution came, as I also said, from Dedekind.

You proposal is a synthesis.

That is it tries to build up something substantial with objects of zero length.

A way to do that did not arrive until Dirac.

Dedekind's solution was analysis.

That is to break down a pre-existing line into particles i.e. points.

Formally he showed that there is always exactly one and only one particle or point at the joint of the partition of the divided line.

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Let us shrink the value of s to zero. The square becomes a point that has neither width nor length. The ribbon becomes a line with no width. In this case, the intersection of 2 lines has neither width nor length, it is a point. But if we put the point one next to another, we will not get a line, because zero length added to zero length gives zero length. This is what I mean by "line is not made of points." But line does include points.

Your ribbon analogy is causing you to have the wrong mental picture of the continuum. You are thinking of points jammed up against each other one after another, like a string of bowling balls.

But between any two points on the line there is a third. There is no notion of "next to" on the line. Between any two real numbers $x$ and $y$, there's a third real number $z$ strictly between them; that is, $x < z < y$. For example we can take $z = \frac{x + y}{2}$.

The linear continuum is modeled by the real numbers. There is a third point strictly between any two given distinct points. The idea of "little squares" side by side simply fails. It's the wrong mental picture.

Suppose we take a line segment and zoom in on it. It still looks the same. Zoom in more. It looks exactly the same. The continuum is self-similar at every zoom level.

It's not a string of bowling balls. It's more like a bowling ball travelling through a big vat of molasses or maple syrup. It's a continuous line.

It is made up of points. But it is not a "row" of points. There's not a first a and a next. You are confused on this I think.

As far as how zero-dimensional points make up a one-dimensional line, that's a philosophical mystery. In math we just ignore it. We allow countable additivity but not uncountable additivity. In other words if you have a countably infinite collection of sets, you can add their lengths. But with uncountability, all bets are off.

You are correct to notice that the question of how points make up a line is a mystery. But you are wrong to let yourself get stuck on that question.

Edited by wtf