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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.

Please help me to address it in formal definitions, thank you.

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

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} .

Any

No single "normal" real number (a

Between any two different arbitrary close

Only

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

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.

Please help me to address it in formal definitions, thank you.

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

**is not a "normal" real number but a connector (a 1-1 correspondence element) between any two different "normal" real numbers (***C***elements).***D*No single "normal" real number (a

**element) has this property, to be a connector between some two different "normal" real numbers (***D***elements).***D*Between any two different arbitrary close

**s there is at least one***D***,***C***and only***C*has the power of the continuum.Only

**( a***C***Double-simultaneous-connection**object) has the property**to connect between two some disjoint subsets A and B of the real numbers**.**is simple and "doing the job" (closes the gap between any disjoint subsets, if needed).***C*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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