# Can you help ?

Hi,

I am a bad formalist, but I have a point of view on the continuum concept, which is different from the standard mathematical point of view.

I'll try to explain my idea in a non formal way.

Here is my idea's non formal description:

For me "to close the gap" means that any different and arbitrary close real numbers are simultaneously connected to each other.

No singleton or infinitaly many singletons can do that.

My analogy to connection:

A bridge is the connection between both river's banks.

The bridge must be taken as a one element, otherwise it can't simultaneously connects both river's banks.

When we look at this bridge as if it is a collection of infinitely many objects, the bridge is no longer has the property to simultaneously connect both river's banks.

Now please think of bank1 as p and bank2 as q.

Therefore, Double-simultaneous-connection is the bridge between p and q.

Can you give this idea its exect formal definition ?

My "formal" definitions to this idea:

p and q are real numbers.

If p < q then
[p, q] = {x : p <= x <= q} or
(p, q] = {x : p < x <= q} or
[p, q) = {x : p <= x < q} or
(p, q) = {x : p < x < q} .

A single-simultaneous-connection is any single real number included in p, q ( = D = Discreteness = a localized element = {.} ).

Double-simultaneous-connection is a connection between any two different real numbers included in p, q , where any connection has exactly 1 D as a common element with some other connection ( = C = Continuum = a non-localized element = {._.} ).

Any C is not a "normal" real number but a connector (a 1-1 correspondence element) between any two different "normal" real numbers (D elements).

No single "normal" real number (a D element) has this property, to be a connector between some two different "normal" real numbers (D elements).

Between any two different arbitrary close Ds there is at least one C, and only C has the power of the continuum.

Only C ( a Double-simultaneous-connection object) has the property to connect between two some disjoint subsets A and B of the real numbers.

C is simple and "doing the job" (closes the gap between any disjoint subsets, if needed).

So, this is my point of view on the continuum concept.

Can you help me to address this point of view in formal definitions ?

Thank you.

Organic

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Integral
Staff Emeritus
Gold Member
I doubt if anyone can help you. I have said this before, and I will contiune saying it. You would be better off spending your time attempting to learn about the construction of the real line then wasteting effort looking for the solution to a problem which does not exist. Once you come to understand that there is no gap to bridge you will be well on your way to understanding our number system.

Ok, I'll write the standard point of view on this topic.

Please show me what I missed. I really want to learn from you.

Hurkyl wrote:
2) The real line is connected.
This means that if you partition the real numbers into two nonempty disjoint subsets A and B, there exists a point in one of those sets such that every open interval containing that point contains points in both A and B.

By define disjoint subsets A and B we mean that there are no common real numbers in subsets A and B.

An open interval is a mathematical trick to do the impossible, which is to find
a singleton, surrounded by a mysterious halo (http://mathworld.wolfram.com/OpenBall.html) that the 1D part of it (an open interval) exists in both disjoint subsets.

We have here nothing but a non-elegant way to force the impossible to be possible.

The least upper bound is:

The least upper bound, called the supremum, of a set S, is defined as a quantity M such that no member of the set exceeds M, but if e is any positive quantity, however small, there is a member that exceeds M - e.

The least upper bound of a function, f, is defined as a quantity M such that f(x) <= M for all x in its domain, but if e is any positive quantity, however small, there is an x in the domain such that exceeds M - e.

If I understand the above definitions then they say:

If M - e = k then M - x(or f(x)) < k

Please show me how can we come to the conclusion that x(or f(x)) = M
(closes the gap) ?

Also please show what I missunderstand?

Thank you.

Organic

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HallsofIvy
Homework Helper
The least upper bound, called the supremum, of a set S, is defined as a quantity M such that no member of the set exceeds M, but if e is any positive quantity, however small, there is a member that exceeds M - e.

The least upper bound of a function, f, is defined as a quantity M such that f(x) <= M for all x in its domain, but if e is any positive quantity, however small, there is an x in the domain such that exceeds M - e.

If I understand the above definitions then they say:

If M - e = k then M - x(or f(x)) < k

In your first example, lub of a set, there is no x so I assume you are talking about the second, lub of a function. Since you have asserted f(x)<= M for all x, it has to be f(x) that is compared to M, not x- the two cases are really the same. It's just that in the latter your set is the "image" of f: the set of all values of f. Yes, if M is the supremum (also called least upper bound, abbreviated either lub or sup) then for any &epsilon;>0, there exist an x such that M-&epsilon;< f(x)<= M. If k= M- &epsilon; then that just says
k< f(x)< M. The way you put it "M- f(x)< k" looks like it wouldn't be terribly interesting. Normally the interesting part is when &eps; is very small. That means that k would be relatively large and saying that we can make M- f(x)< k doesn't really tell us anything.
It's true even when k is small. I think what you really want is not "k" but "&epsilon;" itself. Since there always exist x so that M-&epsilon;< f(x), then, adding &epsilon; to both sides and subtractin f(x) from both sides, M- f(x)< &epsilon;
To take a simple example, suppose f(x)= x (0<x<1). Then the image of f (some people would say "the image of (0,1) under f") is just 0<f(x)<1 also and the least upper bound of f is 1. Suppose we take &epsilon; to be 0.001. Then, yes, there exist values of x such that 1- 0.001= 0.999< f(x)<= 1 and so 1- f(x)< 0.001. Defining k to be 1- 0.001= 0.999 and it is fairly obvious that there are values of x such that 1-f(x)< 0.999!

Please show me how can we come to the conclusion that x(or f(x)) = M
(closes the gap) ?
You can't come to that conclusion- it's not, in general, true. If f(x)= x (0<x<1) then the lub, as I said above, is 1 but f(x) is not equal to 1 for any x in 0<x<1. (If it were true, a lot of interesting mathematics would evaporate.) What you may be looking for is the fact that IF f(x) is continuous on a compact set, then not only must f have a supremum but there must be a value of x in that compact set such that f(x) is equal to that supremum. Compact sets have a lot in common with finite sets.

Dear HallsofIvy,

I really want to thank you for your post.

Please tell me if I am wrong, but as much as I know,

1 and 0.9999999999... are two representations of the same number, isn't it ?

for example:

x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1.

By using the word "singleton" I mean that there exist one and only one real number with one and only one value, which is x.

x can have more than one representation (as we can see above) but still x is x.

Let us say that x=0.

Any arbitrary close positive y, which is another unique real number can't be but greater than x.

So, in the case of f(y), f(y) result can be =0 XOR not=0, and it can't be =0 AND not=0.

These are some basics of Boolean logic, which no singleton can break, otherwise we get nonsense.

But my idea on Double-simultaneous-connection between at least two different singletons, defines a new mathematical object, which does not have the property of a singleton, therefore it does not have any respect to the Boolean logic.

I mean, this object can be simultaneously in two different states like > AND =, and it is not a function but an object.

As I understand it, only this kind of object has the power of the continuum, and no infinitely many objects (which must respect the Boolean logic) have this power.

Organic

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HallsofIvy
Homework Helper
I suspected when I responded that I was making a mistake. Your reply to my response has no relationship to either my response or your original post. In particular it has noting to do with the concept of "least upper bound" (and very little to do with the fact that 0.99999...= 1). You seem determined to try to force anything anyone says into your own peculiar view of "connections".

But my idea on Double-simultaneous-connection between at least two different singletons, defines a new mathematical object, which does not have the property of a singleton, therefore it does not have any respect to the Boolean logic.
Then it is not a mathematical object.

I think I understand know why you don't understand my ideas.

For me Boolean logic is only one aspect of mathematics language.

I believe you know the logic of uncertainty that stands in the basics of Quantum Mechanics.

This logic is based on the idea of complementary associations between opposite states.

For example through this kind of logic we can ask questions on things like dead|alive cats or opened|closed doors or >|= objects, localized objects (.) and nonlocalized objects (__), and so on.

My new element belongs to this happy family of objects, and they are mathematical objects as well.

Organic

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HallsofIvy
Homework Helper
Yes, I am aware of "fuzzy logic". Your construct also does not fit into that since you have not supplied the appropriate "truth probabilities".

No, "fuzzy logic" is based on Boolean logic.

0 and 1 are the end conditions of it but you never use both of them as a legitimate result.

This is not the case of the complementary logic, where a legitimate result can be 0 and 1.

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Boolean logic is 0 XOR 1.

Fuzzy logic is fading between 0 XOR 1.

Non-Boolean logic is 0 AND 1.

Organic

HallsofIvy
Homework Helper
Boolean logic is 0 XOR 1.

Fuzzy logic is fading between 0 XOR 1.

Non-Boolean logic is 0 AND 1.

Yes! To quote a famous logician, writing in precisely the same vein you are: "Twas brillig and the slithy toves did gyre and gimble in the wabe. All mimsy were the borogroves and the momes wrath outgrabe!"

HallsofIvy,

Enjoy your Boolean world, because I see that you can't look beyond it.

The real world is based on associations between opposite concepts, where the boolean world is just a tiny part of it.

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Hurkyl
Staff Emeritus
Gold Member
What you write, Organic, doesn't resemble fuzzy logic.

And like it or not, using boolean logic as a foundation (plus real arithmetic), fuzzy logic can be faithfully expressed.

Hi Hurkyl,

That exactly what I mean. Fuzzy logic is based on boolean logic.

I search for noncommutative non-boolean logic.

Can you give me an address of this kind of logical system?

Thank you.

Organic