The Continuum and The Discreteness concepts in Math language

[Doron Shadmi]

Hi,

To check a bijection between two sets, we have to look at the contents of
the two examined sets as if they are discrete elements, otherwise we can't
use the bijection method.

|R| = Continuum, so my question is, how can we check if there is or there is
not a bijection between |Q| and |R|, if |R| = Continuum ?


Yours,

Doron

 

[mathman]

What is |Q| ?

 

[Hurkyl]

There is not. There are two famous "diagonalization" proofs (at least one is by Cantor... I think both are) that combine to prove this... one proves there is a bijection between the rational numbers and the natural numbers, the other proves there is no bijection between the natural numbers and the real numbers.

 

[Doron Shadmi]

To Mathman: |Q| = the cardinal of the rational numbers.


Hi Hurkyl,

But in case of |N| and |R| "diagonalization" proof, we have to look at R members as if they are many infinitely discrete elements.

Otherwise we can't use the bijection method, which needs both compared
sets to be expressed by the discreteness concept.

All we get is |N|=|Q|<|R| .

So, my question is: how |R| = Continuum ?



Yours,

Doron

 

[Hurkyl]

The "continuum" structure of the real numbers is a topological property, not a set property; it is irrelevant to the issue of counting points.

Being a "continuum" means, loosely, that if we consider the ordering relation and distances between real numbers, they form a complete metric space. Loosely speaking, that means there are no "holes", we can find distances, and they're one-dimensional.

We don't need an ordering relation nor distances in order to count objects; the continuum structure is irrelevant to the issue of comparing the sizes of sets.



Incidentally, there is a snazzy way to prove that |N|<|R| using metric spaces. One can prove that the "length" of a countable set of points must be zero. Since the (metric space of) real numbers has "length" greater than zero, |R| cannot be countable, and thus |N|<|R|.

I put "length" in quotes because I'm really talking about the 1-dimensional measure; it's a modest generalization of the idea of "length", and the measure of a set is equal to its length in any case where length can be defined. (the measure is just applicable to a greater variety of sets)

 

[Doron Shadmi]

Hi Hurkyl,

Being a "continuum" means, loosely, that if we consider the ordering relation and distances between real numbers, they form a complete metric space. Loosely speaking, that means there are no "holes", we can find distances, and they're one-dimensional.

So, if |R| = Continuum, how can we find elements in R ?

By no "holes" we mean (as I understand it) that the Continuum
is a one indivisible smooth structure.

Words like "distances", "points" ,"cuts"(dedekind), "elements"
or any form that can be expressed by "infinitely many ..."
is a contradiction of the Continuum concept.

Please tell me what do you think.

Yours,

Doron

 

[Hurkyl]

By no "holes" we mean (as I understand it) that the Continuum is a one indivisible smooth structure.

A slightly more precise way to state this (and the one mathematicians actually use as a defining property of a set of real numbers) is that no matter how you partition the continuum (aka real numbers) into two disjoint sets, one set will contain a boundary point of the other. Mathematicians call this property "connectedness"

For example, if I partition R into (-&infin;, 0] and (0, &infin;), the first set contains a boundary point of the other, namely 0.

One can show that Q is not connected by partitioning it into the following two sets:

A = {x: x² < 2}
B = {x: x² > 2}

A little thought will show that neither of these sets contain a boundary point of the other (because that boundary point would have to be a square root of two), so the rational numbers are not connected.





But the point I was trying to make earlier is that connectness is a topological property; it is only relevant when you're performing operations that preserve topological properties; i.e. continuous operations; your objection is analogous to saying that we can't slice an orange because it destroys the ball-shape of the orange. If we perform operations that don't care about the topology of a space, then we can't expect the topology that space to have much relevance to the operations performed.

 

[Doron Shadmi]

A slightly more precise way to state this (and the one mathematicians actually use as a defining property of a set of real numbers) is that no matter how you partition the continuum (aka real numbers) into two disjoint sets, one set will contain a boundary point of the other. Mathematicians call this property "connectedness"

As I understand it, if you use a word like "Continuum", any concept or event
that breaks it, is no longer belongs to the Continuum concept.

more precisely, there is a XOR ratio between the Continuum concept
and any concept or event that breaks it.

The Continuum concept contains nothing but its own indivisible smooth
unbreakable state.

So (as I see it) any use of words like "contain a boundary point ..."
is a phase transition to the Discreteness concept, and from the point of view of
this concept, the Continuum does not exist (because of the XOR ratio between
those concepts).

your objection is analogous to saying that we can't slice an orange because it destroys the ball-shape of the orange.

I am talking about the Continuum concept itself, so a more precise example of it is:

Let the Continuum be a X(~=0)-axis.

Let a Point be any Y(=0)-axis on the X-axis.

Let XOR be the ratio between any Point to the Continuum.

 

[Hurkyl]

You see wrong. *shrug* (at least from a mathematical point of view)

Mathematicians have no trouble regarding the real numbers as a set as well as a continuum. When you care about geometric properties, you use geometric operations. When you don't care about geometric properties (such as when counting things), you can use things that aren't geometric operations.


I think you just aren't giving math a chance here; you seem to have rejected out-right anything math likes to do to things without having tried to understand if it has captured the essense of the idea.

And if you don't want to do that, try encoding your ideas about the continuum into a precise set of axioms and rules. No offense, but as is, what you have written is gibberish. e.g. what is a ratio between concepts? Things like that need to be made precise.

 

[Doron Shadmi]

Dear Hurkyl,

There is no way to associate between a discrete set {…} and a Continuous set {___} by means of the Quantity concept, without forcing the Continuum concept to be expressed in terms of the Discreteness concept, and what the Common Math does is:

{.<-- . -->.} = Extrapolation over scales = elements with finite magnitude = N,Z.

{.--> . <--.} = Interpolation over scales = elements with finite or infinite magnitude, where those with an infinite magnitude are built on repetitions over scales = Q.

{. --> . <-- .} = Interpolation over scales = elements with infinite magnitude without repetitions over scales = R.

But the infinite { . --> . <-- . } magnitude never reaches the {___} state, and this is an axiomatic fact that no mathematical manipulation (which is based on the quantity concept) can change.

For example, please show me how we can use the bijection method between {...} and {__} ?

We find that |R| > |Q| by using the bijection method, and for this, the strucrute of each compared elemant in both sides
MUST BE {. <-- . --> .} or {. --> . <-- .}, so we are closed under {...} and can't conclude that |R| = {__} = Continuum.

All we can conclude is that the magnitude of the infinitely many elements of |R| is bigger than the magnitude of the infinitely many elements of |Q|.


Here are some of my non-standard basic definitions:

If we use the idea of sets and look at their contents from
a structural point of view, we can find this:

{} = The Emptiness = 0 = Content does not exist.

Let power 0 be the simplest level of existence of some set's content.

{__} = The Continuum = An infinitely long indivisible element = 0^0 = Content exists (from the structural point of view, there are exactly 0 elements in the Continuum, so its base value = 0, But because it exists (unlike the emptiness) its full notation = 0^0).

(You can break an infinitely long continuum infinitely many times, but always you will find an invariant structural state of {.___.} which is a connector between any two break points.)

{...} = The Discreteness = Infinitely many elements = infi^0 = Content exists.

Any transformation from {} to {__} or {...} is based on phase transition, because we have 0(=does not exist) to 1(=exists) transition.

So, from a structural point of view, we have a quantum-like leap.


Now, let us explore the two basic structural types that exist.

0^0 = infi^0 = 1 and we can see that we can't distinguish between
Continuum and Discreteness by their Quantity property.

But by their Structural property {__} ~= {...} .

From the above we can learn that the Structure concept have
more information than the Quantity concept in Math language.

Through this point of view, any element (with finite or infinite magnitude) that is under a definition like "infinitely many ..." can not be but a member of {...}, which is the structure of the Discreteness concept.


Because we can't know the exact value of any R member, we can't use any R member as a common boundary between any two sets.




Please read carefully the overview of my new theory of numbers:


http://www.geocities.com/complementarytheory/CATpage.html


I wil be glade to get your remarks and insights.


Yours,

Doron