Continuous set and continuum hypothesis

[uncool]

Yes. I agree. I try to figure out another definition.

Good. Then, as the real numbers are uncountable, and the real numbers are the union of the rational numbers and the irrational numbers, the irrational numbers are uncountable. But you've asserted that any discrete set is countable. That's a contradiction. As such, given that you've accepted that the real numbers are uncountable, you have a problem. It's good that you see this and are looking for a new definition.

 

This is why you shouldn't simply make assertions, especially if you are not sure of your definition.

[pengkuan]

Good. Then, as the real numbers are uncountable, and the real numbers are the union of the rational numbers and the irrational numbers, the irrational numbers are uncountable. But you've asserted that any discrete set is countable. That's a contradiction. As such, given that you've accepted that the real numbers are uncountable, you have a problem. It's good that you see this and are looking for a new definition.

 

This is why you shouldn't simply make assertions, especially if you are not sure of your definition.

Agree. It is learning course to expose my idea and receive critiques. Before you explain me that discreteness is not as simple, I just explain intuitively. This is why I make confusion. But the confusion will fade with your help and from others, leaving the good substance clear.

 

I must change my terminology. Using "may be " in place of "be"

[pengkuan]

Good. Then, as the real numbers are uncountable, and the real numbers are the union of the rational numbers and the irrational numbers, the irrational numbers are uncountable. But you've asserted that any discrete set is countable. That's a contradiction. As such, given that you've accepted that the real numbers are uncountable, you have a problem. It's good that you see this and are looking for a new definition.

 

This is why you shouldn't simply make assertions, especially if you are not sure of your definition.

 

1. Discreteness of set

 

Discreteness is the property that qualifies sets that are formed by isolated points. Isolation means these points do not touch one another. Take the set {0, 1}. Because there is a void space between 0 and 1, they do not touch each other. This void space is named empty interval, here it is ]0, 1[. Then we put the number 1/2 in the middle to make a set with three elements {0, 1/2, 1}. The new point cut the interval ]0, 1[ into 2 empty intervals, ]0, 1/2[ and

]1/2 , 1[, which surround the point 1/2 . The set {0, 1/2 , 1} is discrete because 1/2 is separated by the 2 empty intervals it created from touching the points 0 and 1.

 

We keep adding new numbers to the above set and obtain: {0, 1/4 ,1/2 , 1},{0, 1/4 ,1/2 ,3/4 , 1}….Each new number splits an interval into two intervals which keep the number isolated. For a set with n elements, there are n-1 intervals. Each number is surrounded by the 2 empty intervals it creates. Like water surrounding isolated islands, the empty intervals surrounding each element make them isolated. So, the resulting set is entirely discrete.

 

Suppose we have an discrete set with i ordered elements: { x₀, x₁ , x₂ ,… , x_i}. The next set having i+1 elements is created by putting an element x in the interval ]x_k, x_k+1[ such that x_k < x <x _k+1. By repeating this process indefinitely, the resulting set will have infinitely many elements. As the elements are added one by one without end, the cardinality of this set is À₀. This set is also discrete because there is an empty interval between any 2 elements. This property defines the discreteness of sets. Natural numbers and rational numbers are infinite discrete sets.

[uncool]

1. Discreteness of set

 

Discreteness is the property that qualifies sets that are formed by isolated points. Isolation means these points do not touch one another. Take the set {0, 1}. Because there is a void space between 0 and 1, they do not touch each other. This void space is named empty interval, here it is ]0, 1[. Then we put the number 1/2 in the middle to make a set with three elements {0, 1/2, 1}. The new point cut the interval ]0, 1[ into 2 empty intervals, ]0, 1/2[ and

]1/2 , 1[, which surround the point 1/2 . The set {0, 1/2 , 1} is discrete because 1/2 is separated by the 2 empty intervals it created from touching the points 0 and 1.

 

We keep adding new numbers to the above set and obtain: {0, 1/4 ,1/2 , 1},{0, 1/4 ,1/2 ,3/4 , 1}….Each new number splits an interval into two intervals which keep the number isolated. For a set with n elements, there are n-1 intervals. Each number is surrounded by the 2 empty intervals it creates. Like water surrounding isolated islands, the empty intervals surrounding each element make them isolated. So, the resulting set is entirely discrete.

 

Suppose we have an discrete set with i ordered elements: { x₀, x₁ , x₂ ,… , x_i}. The next set having i+1 elements is created by putting an element x in the interval ]x_k, x_k+1[ such that x_k < x <x _k+1. By repeating this process indefinitely, the resulting set will have infinitely many elements. As the elements are added one by one without end, the cardinality of this set is À₀. This set is also discrete because there is an empty interval between any 2 elements. This property defines the discreteness of sets. Natural numbers and rational numbers are infinite discrete sets.[/size]

You seem to be defining whether a set is discrete based on how it is "created", not based on what the elements of the set are. But a set doesn't have the "data" of how it was created, only which elements are in it.

 

Is the Cantor set discrete?

[studiot]

Surely all sets are 'discrete' by definition.

 

What is your definition of a member of a set?

[Xerxes]

 

1. Discreteness of set

 

Discreteness is the property that qualifies sets that are formed by isolated points. Isolation means these points do not touch one another. Take the set {0, 1}. Because there is a void space between 0 and 1, they do not touch each other. This void space is named empty interval, here it is ]0, 1[. Then we put the number 1/2 in the middle to make a set with three elements {0, 1/2, 1}. The new point cut the interval ]0, 1[ into 2 empty intervals, ]0, 1/2[ and

]1/2 , 1[,

First a word on notation and terminolgy. The term "interval" is reserved for a segment of the Real Line

i.e all the Real numbers arranged according to their ordering.

 

The notation [math]]0,1[[/math] or equivalently [math](0,1)[/math] refers to a segment - a subset - of the Real line that includes all the Real numberbs rhat lie betweenn 0 and 1 but does not include 0 or 1.

 

Anyway, let's try to work with what you have given us, taken a face value. So you take a 2-element set [math]\{0,1\}[/math] and then remove 0 and 1 to form [math]]0,1[[/math] which is by any argument empty.

 

Now to this empty set you add [math]\frac{1}{2}[/math] and form the 2sets [math]]0,\frac{1}{2}[[/math] and [math]]\frac{1}{2},1[[/math] both of which are empty. But there can be only one empty set.

 

So what have you done? You have effectively iteratively added a number to the empty set and promptly taken it away and then found you are left with the empty set!!

 

I'm not about to write home to mother about that

 

BTW I don't agree that all sets are discrete in the strict sense - they may have a non-empty intersction, neither do I agree that it is improper to define a set in terms of how it is created, butthese are seperate issues, perhaps

[uncool]

Surely all sets are 'discrete' by definition.

 

What is your definition of a member of a set?

I'm not sure what you mean by discrete here; topologically, a discrete set is a set where all of the points make one-point open sets. That clearly isn't true for the real line.

[studiot]

I used the word discrete in inverted commas because I meant using pengkuan's description, which is more akin to connected.

 

It is important to distinguish between the set along with its properties and the members of the set, along with their properties.

 

One required property of all members of all sets is that each member should be distinct (or distinguishable from any other member).

There is no general requirement about their juxtaposition is real or phase space.

Indeed I see difficulties discussing juxtaposition without a metric.

[Xerxes]

 

One required property of all members of all sets is that each member should be distinct (or distinguishable from any other member).

...................

Indeed I see difficulties discussing juxtaposition without a metric.

Interesting point, and one that persuades me that studiot and uncool are correct - point set topology only can answer pengkuan's issues.

 

Given the set of Real numbers [math]\mathbb{R}[/math], how can one be sure that any 2 members are the same or different? Obviously, given the Euclidean metric one can say that if [math]|x-y|=0[/math] then [math]x=y[/math] and not otherwise.

 

But suppose no such metric exists? Let us suppose the Real Line [math]R^1[/math] (as it is commonly understood) and accept that this has what is called the "standard topology" on [math]\mathbb{R}[/math].That is, it is the union of all open intervals (open sets) of the form [math](a,b)[/math] with [math]a<x,y,z,....<b \in (a,b)[/math].

 

Let us define the Hausdorff property of any topological space as follows:

 

If there exist open sets [math]U,\,\,V[/math] containing [math]x,\,\,y[/math] respectively and if these sets are disjoint i.e. [math]U \cap V= \emptyset[/math], however "small" they might be, we may say that [math]x \ne y[/math] (and not otherwise). Clearly in [math]R^1[/math] they are "small".

 

How small? Well, the smallest set that contains [math]x[/math] as an element is the singleton set [math]\{x\}[/math], but this is not an open set in any topology - it is both open and closed which, for reasons I can't be bothered to explain, means they are elements in a non-connected topological space.

 

So what is the point of this ramble? Well, if we agree that [math]R^1[/math] is "continuous", that it is connected and that it has the Hausdorff property, then, for any [math]x,\,\,y \in R^1[/math] it is always possible to find disjoint open sets not singletons that contain them

 

Phew! If you read all of that, bravo indeed

[studiot]

Thanks Xerxes +1, I was relying on you and uncool to do the algebra.

 

Edited December 14, 2015 by studiot

[pengkuan]

You seem to be defining whether a set is discrete based on how it is "created", not based on what the elements of the set are. But a set doesn't have the "data" of how it was created, only which elements are in it.

 

Is the Cantor set discrete?

Can we create all possible discrete set? Can all discrete set be created? If we can prove this two propositions, then there is no difference on how discreteness is defined. But no answer exists. What I can say only is all so created set are discrete.

 

Cantor's set contains continuum because it is constructed from a continuum by leaving 2 continuums in 3. So it is not discrete.

Surely all sets are 'discrete' by definition.

 

What is your definition of a member of a set?

The more advanced in searching in discreteness, the less I'm sure what is discreteness. I have the feeling that the limit of a discrete set is a continuum.

First a word on notation and terminolgy. The term "interval" is reserved for a segment of the Real Line

i.e all the Real numbers arranged according to their ordering.

 

The notation [math]]0,1[[/math] or equivalently [math](0,1)[/math] refers to a segment - a subset - of the Real line that includes all the Real numberbs rhat lie betweenn 0 and 1 but does not include 0 or 1.

 

Anyway, let's try to work with what you have given us, taken a face value. So you take a 2-element set [math]\{0,1\}[/math] and then remove 0 and 1 to form [math]]0,1[[/math] which is by any argument empty.

 

Now to this empty set you add [math]\frac{1}{2}[/math] and form the 2sets [math]]0,\frac{1}{2}[[/math] and [math]]\frac{1}{2},1[[/math] both of which are empty. But there can be only one empty set.

 

So what have you done? You have effectively iteratively added a number to the empty set and promptly taken it away and then found you are left with the empty set!!

 

I'm not about to write home to mother about that

 

BTW I don't agree that all sets are discrete in the strict sense - they may have a non-empty intersction, neither do I agree that it is improper to define a set in terms of how it is created, butthese are seperate issues, perhaps

Yes, I have not the correct word to name a void space that contains nothing but have nonetheless a length.

 

So, you say that there is only one empty set and (1, 1/2 ) and (1/2, 1) are in fact the same empty set. Agree. How then we name the empty space (1, 1/2 ) and (1/2, 1) since they are not sets?

Interesting point, and one that persuades me that studiot and uncool are correct - point set topology only can answer pengkuan's issues.

 

Given the set of Real numbers [math]\mathbb{R}[/math], how can one be sure that any 2 members are the same or different? Obviously, given the Euclidean metric one can say that if [math]|x-y|=0[/math] then [math]x=y[/math] and not otherwise.

 

But suppose no such metric exists? Let us suppose the Real Line [math]R^1[/math] (as it is commonly understood) and accept that this has what is called the "standard topology" on [math]\mathbb{R}[/math].That is, it is the union of all open intervals (open sets) of the form [math](a,b)[/math] with [math]a<x,y,z,....<b \in (a,b)[/math].

 

Let us define the Hausdorff property of any topological space as follows:

 

If there exist open sets [math]U,\,\,V[/math] containing [math]x,\,\,y[/math] respectively and if these sets are disjoint i.e. [math]U \cap V= \emptyset[/math], however "small" they might be, we may say that [math]x \ne y[/math] (and not otherwise). Clearly in [math]R^1[/math] they are "small".

 

How small? Well, the smallest set that contains [math]x[/math] as an element is the singleton set [math]\{x\}[/math], but this is not an open set in any topology - it is both open and closed which, for reasons I can't be bothered to explain, means they are elements in a non-connected topological space.

 

So what is the point of this ramble? Well, if we agree that [math]R^1[/math] is "continuous", that it is connected and that it has the Hausdorff property, then, for any [math]x,\,\,y \in R^1[/math] it is always possible to find disjoint open sets not singletons that contain them

 

Phew! If you read all of that, bravo indeed

Thanks. I have read your text but I can follow.

[studiot]

 

pengkuan

Yes, I have not the correct word to name a void space that contains nothing but have nonetheless a length.

 

nonsense.

 

I'm sorry to be blunt but that is what it is.

 

 

pengkuan

The more advanced in searching in discreteness, the less I'm sure what is discreteness. I have the feeling that the limit of a discrete set is a continuum

 

You say you have not studied set theory yet you are prepared to ignore what others tell you about it.

 

When asked for a definition of ideas and terms you use you cannot supply one.

 

How exactly do extract a distinct and identifiable member of set from a continuum?

 

A while back I suggested you look at connectedness, compactness, completeness and coverings in relation to set theory as they are relevant.

 

How would you define a connected set?

 

This is what I thing you are striving towards.

 

But there are pitfalls.

 

Would you say that the set of all x² - y² < 0 is connected or disconnected? x,y contained in R.

[studiot]

To expand on this

 

 

Quote

 

pengkuan

Yes, I have not the correct word to name a void space that contains nothing but have nonetheless a length.

 

nonsense.

 

I'm sorry to be blunt but that is what it is.

 

One of the axioms of a metric or distance function d(a, b) is

 

For any two points a, b in a set d(a, b) = 0 iff (if and only if) a = b

 

It follows by axiom that there are no points b of zero distance from a in any set.

 

That is why we have the concept of a neighbourhood.

 

A neighbourhood is another set, the set of all points within a given distance from 'a'.

Note the general statement includes zero distance and thus a itself.

 

Several types of points and types of neighbourhood are distinguished.

 

A 'punctured neighbourhood' excludes the zero distance case and thus excludes 'a' itself.

 

Types of neighberhood are used to identify different types of points.

 

Of particular interest are those called limit points, cluster points or accumulation points (these are all names for the same idea).

Points which are not cluster points are called isolated points.

 

Comparing neighberhoods for different points allows us to examine set properties such as connectedness, compactness (or dense) and discreteness.

 

A definition

 

A discrete set has no cluster points so every point in the set is isolated.

Example the integers form a discrete set, but the rationals and the reals do not.

[Carrock]

A definition

 

A discrete set has no cluster points so every point in the set is isolated.

Example the integers form a discrete set, but the rationals and the reals do not.

Small correction maybe.

Surely the rationals must be discrete or they'd be uncountable and could not be mapped to the integers.

[studiot]

Small correction maybe.

Surely the rationals must be discrete or they'd be uncountable and could not be mapped to the integers.

 

 

That rationals are dense in R

 

http://web.mat.bham.ac.uk/R.W.Kaye/seqser/density.html

[Carrock]

This maths is rather beyond me, which is why I stopped posting in these topics.

However, from http://web.mat.bham.ac.uk/R.W.Kaye/seqser/density.html and some other pages, I couldn't find any any indication that the set of integers is not dense in the same sense that the rational numbers is dense.

e.g.

By a similar sort of argument one can prove much more than this: that every real number is the limit of a sequence of rational numbers.

[studiot]

This maths is rather beyond me, which is why I stopped posting in these topics.

However, from http://web.mat.bham.ac.uk/R.W.Kaye/seqser/density.html and some other pages, I couldn't find any any indication that the set of integers is not dense in the same sense that the rational numbers is dense.

e.g.

 

This is why I keep banging on about neighbourhoods.

 

A set is said to be dense (in itself) if every punctured neighberhood of every element in the set intersects the set itself.

 

Alternatively if between any two members of the set we can insert a third the set is dense.

 

So for the rationals p and q the rational (1/2)(p+q) is another rational between p and q.

 

There is not necessarily such an integer available between integers i and j ; 1/2(i + j) , is not necessarily an integer

 

Countability/cardinality is a different concept and the apparent peculiarity you noted is one of the oddities of infinite sets.

Edited December 15, 2015 by studiot

[pengkuan]

What a pity you did not wait for my thoughts in reply to your post16.

 

I have been preparing one of my 'rough guides' this afternoon.

But this takes a measure of time and effort.

Normally a student of this subject would take around 6 months to cover the material, you say you have not studied, and that I am trying to pull together for you in an afternoon.

Many very clever people from many countries spent 150 years or so pulling and pushing, arranging and rearranging this theory into a shape where it is not only the foundation for a great deal of traditional mathematics but provides a springboard for generalisation beyond the numbers.

 

As such the modern theory is in pretty good shape and does not need a rather flaky competitor.

 

 

 

Talking of measure, you are struggling because you are lacking the idea of measure or distance or metric.

Without this your statement

 

 

has no meaning.

 

What is a gap?

 

What is touching?

 

What is adjacent?

 

For your information modern theory recognises three types of set points, not two. The three types are as I listed in post 15.

 

Since you have now introduced continuity and connectedness

the terms Complete, Compact, Continuous, Connected and Covering all have special (very carefully defined) meanings in relation to this subject.

 

Just to be going on with until I complete my rough guide and to show that connected (=no gap) is a complicated concept here is an extract courtesy Buck Advanced Calculus.

 

I have posted my detailed explanation before your 'rough guide' because the administrator have given me a note that warned me not to force people to open my site or pdf. So, I have to obey quickly.

 

Since then , I have seen your opinion on compactness, boundary, adjacent or neighborhood. I have seen that the continuity that I used to qualify real is equivalent to compactness. So, I have added this aspect to my paper. Also, you precise that neighborhood is a notion of measure theory that has not meaning here. So, I have corrected my paper to avoid this notion. In fact, neighborhood is not necessary to qualify compactness.

 

Instead, I have qualified discreteness by the two void spaces around a elements and the contact in real line with non-presence of void space around a element. So, I have avoid the notion of adjacent, neighborhood, contact. Nevertheless discreteness and continuity can be qualified by these properties. Below is the new version of my paper. By correcting according to your objections and other's, I'm zeroing in a definition of discreteness and continuity more acceptable.

 

1. Discreteness of sets

 

Discreteness is the property that qualifies sets that are formed by isolated points. Isolation means these points do not touch one another. Take the set {0, 1}. Because there is a void space between 0 and 1, they do not touch each other. This void space is named (0, 1). Then we put the number ½ in its middle to make a set with three elements {0, ½, 1}. The new point ½ cut the void space (0, 1) into 2 void spaces, (0, ½) and (½, 1). The set {0, ½, 1} is discrete because ½ is separated from the points 0 and 1by the 2 void spaces it created.

 

We keep adding new numbers to the above set and obtain: {0, 1/4, ½, 1},{0, 1/4, ½, 3/4, 1}….Each new number splits a void space into two void spaces which keep the number isolated. For a set with n elements, there are n-1 void spaces. Each number is surrounded by the 2 void spaces it creates. Like water surrounding isolated islands, the void spaces surrounding each element make the elements isolated. So, the resulting set is entirely discrete.

 

Suppose we have an discrete set with i ordered elements: { x₀, x₁ , x₂ ,… , x_i}. The next set having i+1 elements is created by putting an element x in the void space (x_k, x_k+1) such that x_k < x <x _k+1. By repeating this process indefinitely, the resulting set will have infinitely many elements. As the elements are added one by one without end, the cardinality of this set is À₀.

 

This set is also discrete because there is a void space between any 2 elements. In fact, discrete sets do not need to be constructed this way. It is enough that any element is surrounded by void spaces on both sides and the set will be discrete, no matter how the elements are arranged. For example, any rational number is surrounded by void spaces. In spite of the ever shrinking interval between 2 rational numbers that makes them infinitely close, the set of rational numbers is discrete.

 

2. Continuity of sets

 

In the contrary, compact set is continuous, for example the real numbers. The elements of such set touch one another, that is, between any 2 points in the real line there is no void space in which an external point can be inserted. It is equivalent to say that except of boundary points, both side of a point x are immediately filled with other elements of the set, no void space exists there. As x can be any point in the real line, real numbers are in contact with one another and the set of real numbers is entirely continuous. This property defines the continuity of a set.

 

3. Collectively exhaustive and mutually exclusive events

 

When tossing a coin, all possible outcomes are heads or tails. The values heads or tails are said to be collectively exhaustive, that is, there is no other possibility. Also, when heads occurs, tails can't occur and vice versa. These two values are said to be mutually exclusive, that is, the outcome is either heads or tails, no mixed value is allowed, for example half heads and half tails.

 

If the elements of a set have void spaces on both sides, the set is discrete. If the interior elements of a set have no void spaces surrounding them, the set is continuous. The presence or not of the void spaces makes discreteness and continuity of a set the 2 outcomes of a collectively exhaustive and mutually exclusive game, no other possibility exists.

 

4. Continuum hypothesis

 

The cardinality of an infinite discrete set is À₀, like the rational numbers. The cardinality of a continuous set is |ℝ|, like the real numbers. As shown above, a set must be exhaustively and exclusively discrete or continuous. So, its cardinality must be À₀ or |ℝ| but not strictly between À₀ and |ℝ|. In consequence, the continuum hypothesis is true.

 

nonsense.

 

I'm sorry to be blunt but that is what it is.

 

 

You say you have not studied set theory yet you are prepared to ignore what others tell you about it.

 

When asked for a definition of ideas and terms you use you cannot supply one.

 

How exactly do extract a distinct and identifiable member of set from a continuum?

 

A while back I suggested you look at connectedness, compactness, completeness and coverings in relation to set theory as they are relevant.

 

How would you define a connected set?

 

This is what I thing you are striving towards.

 

But there are pitfalls.

 

Would you say that the set of all x² - y² < 0 is connected or disconnected? x,y contained in R.

I think void space alone have more sense than space void space that contains nothing but have nonetheless a length.

 

I will read connectedness, compactness, completeness and coverings in relation to set theory. Let me more time, as I have to answer your messages also.

To expand on this

 

 

One of the axioms of a metric or distance function d(a, b) is

 

For any two points a, b in a set d(a, b) = 0 iff (if and only if) a = b

 

It follows by axiom that there are no points b of zero distance from a in any set.

 

That is why we have the concept of a neighbourhood.

 

A neighbourhood is another set, the set of all points within a given distance from 'a'.

Note the general statement includes zero distance and thus a itself.

 

Several types of points and types of neighbourhood are distinguished.

 

A 'punctured neighbourhood' excludes the zero distance case and thus excludes 'a' itself.

 

Types of neighberhood are used to identify different types of points.

 

Of particular interest are those called limit points, cluster points or accumulation points (these are all names for the same idea).

Points which are not cluster points are called isolated points.

 

Comparing neighberhoods for different points allows us to examine set properties such as connectedness, compactness (or dense) and discreteness.

 

A definition

 

A discrete set has no cluster points so every point in the set is isolated.

Example the integers form a discrete set, but the rationals and the reals do not.

I do not use neighberhoods any more.

Small correction maybe.

Surely the rationals must be discrete or they'd be uncountable and could not be mapped to the integers.

Can Countability be a proof of the discreteness?

 

 

That rationals are dense in R

 

http://web.mat.bham.ac.uk/R.W.Kaye/seqser/density.html

Rational are dense but do not contain their limits that are holes.

[Xerxes]

This is soooo disappointing.

 

Look, nobody here (as far as I know) disputes the truth of the continuum hypothesis, but the contributors to this thread have tried to show you that your "proof" is not valid.

 

Some of us have worked quite hard to explain why, and to give you guidance on fundamental mathematical topics.

 

Your response? Change a word here and there in your original presentation, often leading you even further astray (e.g. the set of Real numbers is NOT compact).

 

I bear no ill will towards you, but if you refuse to learn mathematics either here or elsewhere, then I am done (even though I have enjoyed contributing to this thread).

 

Good luck

[pengkuan]

This is soooo disappointing.

 

Look, nobody here (as far as I know) disputes the truth of the continuum hypothesis, but the contributors to this thread have tried to show you that your "proof" is not valid.

 

Some of us have worked quite hard to explain why, and to give you guidance on fundamental mathematical topics.

 

Your response? Change a word here and there in your original presentation, often leading you even further astray (e.g. the set of Real numbers is NOT compact).

 

I bear no ill will towards you, but if you refuse to learn mathematics either here or elsewhere, then I am done (even though I have enjoyed contributing to this thread).

 

Good luck

 

I'm sorry to disappoint you like this. I have not sufficient mathematical knowledge to write a proof at the standard of mathematics and I cannot swallow the necessary quantity of mathematical knowledge in one day while others learn it in several years.

 

I will let this paper as it is now and move on to something else.

Edited December 15, 2015 by pengkuan

[wtf]

 

Look, nobody here (as far as I know) disputes the truth of the continuum hypothesis

 

 

New here. Amateur math fan.

 

I'll take the other side of that one. I dispute the truth of CH. CH is known to be independent of the standard axioms of set theory (ZFC). The only way to prove CH true or false is to cook up a nonstandard model of the axioms and show that in that model, CH is either true or false. That's what Gödel and Cohen did, respectively.

 

It may be the case that CH is true (or false) about the universe of sets, and that we just haven't found the right axiomatization. Another view is that the matter has no truth or falsity, since sets are just a formal abstractions with ultimately no meaning. Or it may be the case that there's some other resolution.

 

But you can't say that CH is true without supplying further context. By the way, both Gödel and Cohen were of the opinion that CH is false. A heuristic argument for that position is by analogy with finite sets. Between the powerset of a 3 element set and the powerset of a 4 element set are sets of many intermediate cardinalities. Why shouldn't the same be true in the infinite case?

 

Of course that's not even remotely any kind of proof; but it does give some understanding of why people think CH may well be false. The Wiki page for Cohen outlines his argument against CH along the same lines. He feels that the powerset operation is so powerful that it must make a great leap in cardinalities, not a one-unit step. He thought that the cardinality of [math]\mathbb{R}[/math] must be an extremely large Aleph.

Edited December 16, 2015 by wtf

[pengkuan]

 

New here. Amateur math fan.

 

I'll take the other side of that one. I dispute the truth of CH. CH is known to be independent of the standard axioms of set theory (ZFC). The only way to prove CH true or false is to cook up a nonstandard model of the axioms and show that in that model, CH is either true or false. That's what Gödel and Cohen did, respectively.

 

It may be the case that CH is true (or false) about the universe of sets, and that we just haven't found the right axiomatization. Another view is that the matter has no truth or falsity, since sets are just a formal abstractions with ultimately no meaning. Or it may be the case that there's some other resolution.

 

But you can't say that CH is true without supplying further context. By the way, both Gödel and Cohen were of the opinion that CH is false. A heuristic argument for that position is by analogy with finite sets. Between the powerset of a 3 element set and the powerset of a 4 element set are sets of many intermediate cardinalities. Why shouldn't the same be true in the infinite case?

 

Of course that's not even remotely any kind of proof; but it does give some understanding of why people think CH may well be false. The Wiki page for Cohen outlines his argument against CH along the same lines. He feels that the powerset operation is so powerful that it must make a great leap in cardinalities, not a one-unit step. He thought that the cardinality of [math]\mathbb{R}[/math] must be an extremely large Aleph.

Thanks for joining. I think they do not discuss the truth of CH because they they think my proof is not valid not because they are not interested in this topic. Continuum hypothesis is famous because it is the first in the list of Hilbert, but how will it influence mathematics is not clear. One is interested in this topic like a bounty hunter.

 

Can someone explain how mathematics will benefit from continuum hypothesis?

[Keen]

 

I'm sorry to disappoint you like this. I have not sufficient mathematical knowledge to write a proof at the standard of mathematics and I cannot swallow the necessary quantity of mathematical knowledge in one day while others learn it in several years.

 

I will let this paper as it is now and move on to something else.

That would probably be for the best. I don't want you to discourage from mathematics, but if you really wanted to write a serious paper on such a very difficult topic as continuum hypothesis, it would take years of study in mathematics and then even more hard work in mathematical logic.

Either way as far as I know, continuum hypothesis is independent of ZFC axioms, which is a standard framework for modern mathematics, so you can't neither prove nor disprove it: you can only make a theory based on it or its negation. I am sure there are lots of fascinating results in this kind of non standard mathematics, but I am unfortunately not knowledgeable enough in mathematical logic to discuss this topic further.

[pengkuan]

I think that it would be possible to "show a set who's cardinality is between the integers and reals". But it is not done with an equation. It is done with a new description of numbers. If numbers are defined as defined or undefined space and value. And all Reals and Integers exist as defined space and defined value, then there cardinality exists between integers and reals only. All numbers containing a variations of undefined space or undefined value are numbers existing in sets outside of the integers and reals.

Rank of infinite numbers can be arranged because we can count them. Infinity cannot be count, so, we are dealing with something that is not dealable. This is why either CH is true or false does not matter. Infinity is a philosophical idea, which is qualitative. Mathematics deal with quantitative things, that infinity is not.

That would probably be for the best. I don't want you to discourage from mathematics, but if you really wanted to write a serious paper on such a very difficult topic as continuum hypothesis, it would take years of study in mathematics and then even more hard work in mathematical logic.

Either way as far as I know, continuum hypothesis is independent of ZFC axioms, which is a standard framework for modern mathematics, so you can't neither prove nor disprove it: you can only make a theory based on it or its negation. I am sure there are lots of fascinating results in this kind of non standard mathematics, but I am unfortunately not knowledgeable enough in mathematical logic to discuss this topic further.

However, I have learned a lot about mathematics. From my first wrong definition of discreteness that makes irrationals discrete to the present better definition using void space. It is a great leap.

[conway]

pengkuan

 

 

Lol....Quantitize zero. Does it not belong to the real's. It is not really a question of if these things can or can not be counted. It is a matter of how they are defined. If...again...infinite numbers exists on the reals number line....then they belong to that set....despite if we can "count" the cardinality. Defined value and space does not necessarily mean countalbe...as suggested by your "ranks of infinite" as opposed to infinite. If then there is a kind of number that is not defined as (defined space, and defined value)...that is containing undefined space and or value...then here is our "philosophical" separations in the "cardinality" of the reals/integers and "other numbers" Not belonging to that set.