Between cardinal 0 and cadinal 1

[Organic]

x is some real number (a singleton)

|{}| = 0

|{x}| = 1


Is there some mathematical branch that define an object, which its cadinality exists between 0 and 1 ?




Organic

 

[HallsofIvy]

No. By definition, finite cardinalities are (literally!) "counting numbers", i.e. positive integers. There are no positive integers between 0 and 1.

If you want to talk about something other

 

[Organic]

Hi HallsofIvy,

Therefore, can we conclude that no x(=real number) element (exclude |{}|) can close the "gap" thet exists between |{}| and any arbitrary close x, by definition ?

 

[HallsofIvy]

No, because that has nothing to do with "cardinality". You seem to be having enormous trouble making the jump from the "discrete" integers (which are well ordered) to the rational numbers (which are not). You seem to be repeatedly asserting that since the rational numbers are not well ordered they can't exist! Perhaps a good introductory analysis course would help.

 

[Organic]

Any x(= some number that can belong to Q or R) is a singleton.

So, I am talking about any arbitrary (rational or irrational) number (exclude |{}|).




Organic

 

[HallsofIvy]

1) You did not understand what I said.

2) I am coming to the conclusion that you do not understand what you are saying.

 

[Organic]

Hi HallsofIvy,

I ask you a very simple question.

Take any arbitrary x (which is some singlton which belongs to Q or R, it means a single rationl or irrationl number) exclude |{}|(=0), and try to close the "gap" ( to reach |{}|=0, not to approach |{}|=0 ).

Please show me how it can be done.


Organic

 

[HallsofIvy]

I can not show you how to do something that makes no sense. As long as you insist upon dealing with integers only, you CAN'T "close the gap" from "1" to "0". It doesn't matter what you call the element in the singleton set- as long as you deal only with finite sets, you are dealing only with integers.

 

[Organic]

Hi HallsofIvy,


2^aleph0 = c means no distinguished objects at all.

So, Dedekind's cut, Epsilon-Delta Definition, Cauchy Sequence
are all make no sense because no one of them reaches the target (or limit or what so ever) but approaches it by arbitrary close infinitely many objects.


More than that, their definitions are based on "infinitely many ..."
which is nothing but a simple contradiction of the continuum concept, that can never be defined by "infinitely many ..." .

Any defined continuum is exactly one and only one continuous object, and only this one and only one object can close the gap between any two different (distinguished from each other) objects.

For me it is simple and clear.

Please show how "infinitely many ..." can ever be a continuum.


Thank you.

(Also infinitely many magnitude's levels of ...2^(2^(2^(2^aleph0))) does not have the power of the continuum, because it is based on "infinitely many ..." definition)


Organic

 

[Hurkyl]

2^aleph0 = c means no distinguished objects at all.

No it doesn't. c means the cardinality of the real numbers.

 

[Jeebus]

In common usage, a cardinal number is a number used in counting (a counting number), such as 1, 2, 3, ...

In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. A set has aleph-0 members if it can be put into a one-to-one correspondence with the finite ordinal numbers. The cardinality of a set is also frequently referred to as the "power" of a set.

 

[quartodeciman]

The idea of the unreachability of continuum from discretion by serial process has a distinguished (if decidedly marginal) history.
The most distinguished exponent IMO was Hermann Klaus Hugo Weyl.

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Weyl.html
bio Hermann Weyl

A taste of Prof. Weyl's criticism of the Dedekind-Cantor procedures might be found in his book:

The Continuum : A Critical Examination of the Foundation of Analysis
Dover; reprint edition (1994)

Here is a more recent review of Weyl's ideas:

http://publish.uwo.ca/~jbell/Hermann Weyl.pdf
University Western Ontario - John L. Bell: Hermann Weyl on Intuition and the Continuum

Here is another prominent exponent:

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Brouwer.html
bio Luitzen Egbertus Jan Brouwer

Weyl discusses the Brouwer procedure in another book:

Philosophy of Mathematics and Natural Science
Atheneum; reprint edition (1949)
-----------

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Wikipedia: fuzzy logic

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Wikipedia: fuzzy sets

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fuzzy FAQs

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-----------

 

[Organic]

Hi,

I was not understood, so I'll write it again:

2^aleph0 is the real numbers cardinality, but this cardinality does not have the power of the continuum because:


Dedekind's cut, Epsilon-Delta Definition, Cauchy Sequence
are all make no sense because no one of them reaches the target (or limit or what so ever) but approaches it by arbitrary close infinitely many objects.

To really reach limit we must have the power of the continuum, and as I wrote above, real numbers magnitude can only approaches limit, and never reaches it.

More than that, the above definitions are based on "infinitely many ..." which is nothing but a simple contradiction of the continuum concept, that can never be defined by something which described by "infinitely many ..." .

Any defined continuum is exactly one and only one continuous object, and only this one and only one object can close the gap between any two different (distinguished from each other) objects.

For me it is simple and clear.

I Also clime that no infinitely many magnitude's levels of ...2^(2^(2^(2^aleph0))) have the power of the continuum, because it is based on "infinitely many ..." definition.

A sharp and cear definition of the one continuous object, which can close the gap between any two different (distinguished from each other) objects, can be found here:

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


Organic

 

[Hurkyl]

Please can someone clearly show how "infinitely many ..." can ever be a one continuum.

Because the continuum is defined to be the real numbers.

Dedekind's cut, Epsilon-Delta Definition, Cauchy Sequence
are all make no sense because no one of them reaches the target (or limit or what so ever) but approaches it by arbitrary close infinitely many objects.

Forget about them then. Their purpose is not to tell you what the real numbers are, but to establish a fact called relative consistency: if the axioms of set theory are consistent, then the axioms of the real numbers are consistent.

(Incidentally, I've only heard "ε-δ definition" applied to limits, I'm not sure what you mean by it in this context...)

(And, BTW, I'm presuming that you are referring to the set theoretic constructions of the real numbers)


The real numbers are defined by these three words: complete ordered field.

To quickly move onto the part relevant to this thread, I'll vastly simplify ordered field and state that it means that +, -, *, /, and < all work like they're "supposed" to.


It's the word complete that has earned the real numbers the nickname "the continuum". (and, sometimes, the term continuum is applied to other complete structures)

Intuitively, all this means is that there are no holes in the real numbers. There are a lot of equivalent ways to say what this means for the real numbers, and IMHO the easiest to understand is:

If I have a collection of "small" real numbers and a collection of "large" real numbers, there is a real number "between" them.

More precisely, if:

S and T are nonempty sets of real numbers
For any s in S and t in T, s < t

then

There is a real number x such that for any s in S and t in T:
s <= x <= t


That is what, mathematically, the continuum concept is. No "holes".


And the topology of the real numbers is equipped with something that "closes the gap" between two distinct numbers; it's called an open interval. The open interval (x, y) is the collection of all real numbers z such that x < z < y, and these are the fundamental objects of study in the topology of the real numbers.

 

[Organic]

Thank you Hyrkyl,

You wrote:

It's the word complete that has earned the real numbers the nickname "the continuum". (and, sometimes, the term continuum is applied to other complete structures)

Intuitively, all this means is that there are no holes in the real numbers.

So the word "continuum" is just a nickname of the word "complete"
and the word "complete" does not based on rigorous proof, but based on some intuition that some set of "infinitely many objects" is complete.

You also wrote:

If I have a collection of "small" real numbers and a collection of "large" real numbers, there is a real number "between" them.

More precisely, if:

S and T are nonempty sets of real numbers
For any s in S and t in T, s < t

then

There is a real number x such that for any s in S and t in T:
s <= x <= t


That is what, mathematically, the continuum concept is. No "holes".

It is the Dedekind's cut definitions, and it is based on an intuition
that x can close the gap between s and t.

There is no rigorous proof here, but an unproved intuition which believes that this x, which is one of "infinitely many objects", simultaneously can reaches (not approaches) s and t, which are different objects.

This x can reaches s and t iff it has the form of Double-simultaneous-connection object.

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 = {.___.} ).

Therefore, x is . XOR .___.

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.


By the above definitions, for the first time in Modern Mathematics, there is a clear and sharp distinction between the Continuum and the Discreteness concepts, not by their Quantitative property, but by their Structural property.

By defining the double-simultaneous-connection as a legitimate member, I redefine the original lexicographical meaning, back to the Continuum concept, and change the perception of Continuum and Discreteness concepts in Modern Mathematics.

__ = connector

1) Any connector can be translated to a single number, which is its length. In this case I think any arithmetic that works between "normal" real numbers, must work here too.

An example: a__b = b-a = c (a single absolute number)

2) Any length is not a localized element by definition, therefore there can be infinitely many indistinguishable connectors.

An example: 2 = __ = __ = __ = ...

Here we need to develop some arithmetic that deals with uncertainty and redundancy + combination with case (1), and we get case (3).

3) Any connector can be located (by locating exactly one of its end-points) to some localized element (A unique real number). I think this state is some mixing of (1) and (2) cases.

An example: 2__4 = 7__9 = -1002__-1000 = ... = 2

So, my theory can work as case (1) or case (3) system


Any object with length=0 is not a connector but a "normal" real number, therefore {(a,a)} is not a connector but a "normal" real number.

By the way, we can think of a length with a negative value,
and by this we mean that we have a mirror image of - and +
to the real line, where zero is the center of it.

An example: -b__-a = -b - -a = -c , a__b = b - a = c





Please someone can show some trouble in my definitions ?


Thank you.




Organic

 

[HallsofIvy]

It is the Dedekind's cut definitions, and it is based on an intuition that x can close the gap between s and t.

On the contrary, the whole point of the Dedekind cut definition of real numbers is that one can then easily prove the "least upper bound" property and thus rigrorously PROVE what you are referring to as "close the gap".

Please someone can show some trouble in my definitions ?

Since you refuse to accept the repeated criticism that they ARE NOT definitions and that they don't make sense even as assertions, I see no point in trying to point out more "trouble".

 

[Hurkyl]

So the word "continuum" is just a nickname of the word "complete"
and the word "complete" does not based on rigorous proof, but based on some intuition that some set of "infinitely many objects" is complete.

The definition of complete is rigorous. My advanced calc text opts to define completeness in terms of Dedekind Cuts as follows:

A Dedekind cut of a field is a pair of nonempty sets A and B whose union is that field, and for a in A and b in B, a <= b.

An element c of the field generates a Dedekind cut (A, B) if A = {x | x <= c) and B = {x | x >= c)

And finally, an ordering on a field is called complete iff every Dedekind cut is generated by an element of the field.


My text has a cool little chart of a bunch of statements that are equivalent to the notion of dedikend completeness, so they could be used instead, if you prefer.

These statements are:

(1) The real numbers are Dedekind complete.

(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.

(3) Least-Upper-Bound property

If you have a collection of real numbers that has an upper bound, then that collection of real numbers has a least upper bound.

(4) Heine-Borel theorem

A subset of the real numbers is compact iff it is closed and bounded.

(5) Nested-interval property

If I have a collection of closed intervals I_n such that for all i > 0, I_i is a subset of I_i-1, then the intersection of all of these intervals is nonempty.

(6) Bolzano-Weierstrass theorem

If I have an infinite collection A of points that has a lower and upper bound, I can find a real number x such that every open interval containing x also contains an infinite number of points in A.

(7) Monotonic sequence property

Any bounded monotonic sequence converges.

(8) Archimedean property + Cauchy sequences converge

The sequence (1/n) converges to 0 as n goes to infinity, and Cauchy sequences converge

It is the Dedekind's cut definitions, and it is based on an intuition
that x can close the gap between s and t.

Actually, my sets don't generally form a Dedekind cut.

And I still have no clue what you mean by "close the gap"; all I'm claiming is that x is between s and t for all s in S and t in T.

There is no rigorous proof here, but an unproved intuition which believes that this x, which is one of "infinitely many objects", simultaneously can reaches (not approaches) s and t, which are different objects.

That's certainly not my intuition because I have no clue what you mean. In any case, as HallsofIvy and I have stated, the definition is rigorous; nothing is left to intuition.



And here's something to think about. You indicate a connection (though you still don't define what it is) between 2 and 4, which you wrote as 2__4. Does that mean that 3__5 does not exist?



And back to the original question in this thread...

Suppose z is a cardinal number. Let S be a set with cardinality z.

Either z = 0 or z > 0.

Suppose z > 0. That means S is not the empty set, so there exists an element x in S.

Let f be the function from {y} to S that maps everything to x.

So I have a one-to-one function from {y}, a set of cardinality 1, into S. This means that 1 <= z.

So I have proven this theorem:

If z is a cardinal number and z is not 0, then z >= 1. In particular, it cannot be the case that 0 < z < 1.

 

[Organic]

No HallsofIvy,



Supremum (is also called the "least upper bound") for a bound is a number which a function, sequence, or set, never exceeds.

Never exceeds and never reaches. it is only approaches and never close the gap, therefore does not have the power of the continuum.

No one can prove the "least upper bound" property, and thus rigrorously PROVE what I am referring to as "close the gap",
if he can't clearly show how he really reaches it.

It is all based on unproved intuition of some mathematicians 100 years ago.



Organic

 

[Organic]

Hi Hurkyl,


You wrote:

You indicate a connection (though you still don't define what it is) between 2 and 4, which you wrote as 2__4. Does that mean that 3__5 does not exist?

If you wrote such a thing it means that I am still not understood.

Example 1:

Please think on finitely long x axis, where its length > 0.

Any point on it is like a y(=0) axis and we get some x,0 location.

You can define any x,0 location but always you have the x exis as a one continuous object between any two arbitrary close x,0 locations along it.

If you define 2_4 then 3_5 can be defined Separately from 2_4 or if not Separately then the result is 2_3_4_5 .

Because my number system is at least a singleton and a connector, we can think in terms of a 2D "space", where there exist two parallel real lines with boolean connective between them.

For example:

__2__4__

__3__5__



Example 2:

Please think on finitely long x axis.

This object is a one object with no points in or on it (exclude two endpoints), and its length is > 0.

It also simultaneously exists in its both endpoints, therefore can be
an object that really close the gap between any two different (distinguished from each other) objects.

No "normal" distinguished real number has this property, which is to simultaneously exists in more than one distinguished value.

You 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) ?

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 = {.___.} ).

Therefore, x is . XOR .___.

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 disjoint subsets A and B of the real numbers.

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



Organic

 

[HallsofIvy]

Never exceeds and never reaches. it is only approaches and never close the gap, therefore does not have the power of the continuum.

No, that's not correct. I'm not at all surprised that you do not know the definition of "least upper bound". I am frustrated though not surprised that you refuse to listen to what you are told.

 

[Organic]

Hi HallsofIvy,

Hurkyl wrote:

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.

I say:

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) ?

Please read my last post (before this post) to hurkyl on this subject.