A question on a<b

[Organic]

Let us say that a and b are real numbers, and we have
[a,b] = {x: a <= x <= b}.

Can x be a 1 to 1 correspondence (map) between any two real numbers included in [a,b] ?


Organic

[ahrkron]

You probably need to elaborate a little.

In the expression you show, x is a dummy variable. It stands for anything.

If you want it to be a 1-1 correspondence, then you also have to define a way to compare such a thing with a real number, since you are supposed to look for those x objects that satisfy a <= x <= b (and you already stated that a and b are real numbers).

Also, if you do define a way to decide if a real number is "<=" than a 1-1 correspondence, you should change the notation you use for the set of 1-1corresps, since [a,b] already stands for a set of real numbers.

[Organic]

Hallo ahrkron,

Since x can be a 1 to 1 correspondence between any two real numbers, is the set of all x maps is complete (have the power of the continuum)?


Organic

[ahrkron]

Hi Organic,

You misunderstood my reply.

Your first post is flawed. It does not make sense as stated. You need to define properly the objects and relations you are using. Otherwise, no further comclusions can be drawn from it.

[Organic]

Hi ahrkron,

May be you can help me to write it in a formal way.


The question is:


Theorem: 2^aleph0 < c


Proof:

Let A be the set of all negative real numbers included in (-1,0).

Let B be the set of all positive real numbers included in (0,1).

Let M be the set of maps (1 to 1 and onto) between any two single real numbers of
A and B sets.

Therefore |M| = 2^aleph0.

(0,1) = {x: 0 < x < 1 }, where x is a 1-1 correspondence between any two real numbers included in (0,1), and any x element has no more than 1 real number as a common element with some other x element.

Let T be the set of all x (1-1 correspondence) elements included in (0,1).

Therefore |T| = |M| = 2^aleph0.

B is a totally ordered set, therefore we can find x element between any two different real numbers included in (0,1).

Any x element must be > 0 and cannot include in it any real number.

Therefore 2^aleph0 < c (does not have the power of the continuum).

Q.E.D


A structural model of the above:

set set set
A M B
| | |
| | |
v v v
!__________________!<---- set
!__________________!<---- T members
!__________________!
!__________________! Any point is some real number
!__________________! (A or B members).
!__________________!
!__________________! Any line is some 1-1 correspondence
!__________________! (M or T members).
!__________________!
!__________________!






Am I right ?



Organic

[HallsofIvy]

Doron, is there a reason for not posting this under the name you used before? The mathematician formerly known as "Doron Shadmi"?

The point of the first response by Organic, which you still haven't addressed is that it makes no sense to assert that x is a NUMBER, as you do in say 0<= x<= 1, and then state that x is a 1-1 correspondence: "(0,1) = {x: 0 < x < 1 }, where x is a 1-1 correspondence between any two real numbers included in (0,1)."

x is either a number or a 1-1 correspondence between numbers. It can't be both.

[Organic]

Hi,

x is not a NUMBER but an element that exists between any two different real numbers included in (0,1).



Organic

[HallsofIvy]

So, once again, you are not talking about "normal" mathematics and are not defining your terms. In "normal" mathematics, inequality with numbers is only defined FOR numbers. WHAT KIND of "element" exists between any two numbers but is not a number itself? What in the world do you mean by "element"?

[Organic]

Hi,

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

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

Please do not forget that negative, irrational, complex and transfinite numbers where not "normal" elements, when they first discovered.

I think the important thing is to find if those "non-normal" elements
can be useful for the development of Mathematics Language, by opening new areas for research.

I think if we associate these "non-normal" elements with, so called, "normal" numbers, we shall obtain a new point of view on the abstract concept of a NUMBER.




Organic

[HallsofIvy]

Well, they surely won't be useful if you don't DEFINE them.

[Organic]

Please open the attached pdf file.

[ahrkron]

This is far, far out.
To TD it goes.

[phoenixthoth]

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

what is your PRECISE definition of "<"? so, when you say 0<x, where x is a connector, what precisely does that mean? certainly, this isn't the same < when we say 0<1. it would seem that your < must be an EXTENSION of the binary relation < as defined for real numbers. if it's not an extension, then i would think the whole thing falls apart, for then the two <'s have utterly different meanings even when confined to the concept of real numbers.

let's pretend i don't have adobe. what's a connector? if it's a bijection, what is the domain and range? and how is this bijection inserted into the continum in such a way that it makes sense to say it is between two real numbers?

this is not automatically bizarre because in nonstandard analysis, there are infinitely many hyperreals between two real numbers. each real number has a halo of hyperreals that are infinitesimally close to it.

i'd like to point out that he never said {x &epsilon; R: 0<x<1}, he just said {x: 0<x<1}; that could be a set of hyperreals or what not... clearly, more precision would be nice.

cheers,
phoenix

[Organic]

Hi phoenixthoth,

First, thank you for your reply.

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.


For more detailes please look at:


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


Please look also in this thread:

http://www.physicsforums.com/showthread.php?s=&threadid=6427


I'll be glad to get your remarks and insights.






Yours,

Organic

[russ_watters]

This may help. Its his first post in his other thread. In the attached address you can find A new approach for the definition of a NUMBER... Today I went to the bank and tried to define a new definition of "exchange rate" but for some reason they weren't buying. Hmm, I don't understand why not...

[phoenixthoth]

russ_watters, no, it doesn't help. your comments remind me of those who didn't like the idea of irrational, transcendental, hyperreal, or complex numbers. gauss, as far as in know, invented a new kind of number and they were used in his PhD thesis. i say it doesn't help because it neither proves or disproves the existence of connector elements in such a way that they are naturally embeddable within R.

organic, a few things:

i'm still in need of a more precise definition of what .___. or "connector" means. i know you say it's a 1-1 correspondance (ie, bijection) element between two real numbers. one way to express functions is with notation like this (as we all know): f(x) = stuff. but another way is to represent them as sets of ordered pairs such that no two first-coordinates match (ie, no two different ordered pairs are present with the same first coordinate). so when you say x is a bijection between real numbers, a and b, let's say, that means that x is representable as the following set of ordered pairs (ie, function): { (a, b) }, the singleton containing the ordered pair (a, b). ok, fair enough. x = { (a, b) }.

so far, x is not any more of exotic of a number than the complex number (a,b) or a + b i. but exoticness is irrelevant as long as there is consitency. some people, like david hilbert, define mathematical existance as consistency. (and if physical existance also equals consistency, then, bingo, mathematical existance = physical existence. ha ha ha.)

here's the big question that will improve your theory for as it stands right now, there is a big gap. and here it is.

suppose y is a real number (to say "normal" real number is redundant) and x is a connector, x = { (a, b) }, where a and b are two real numbers. for x to be a connector, this implies a != b. (note that != is standard notation for unequal.)

this is the big question you must answer:

how can you extend the binary relation < so that a statement like

x < y

makes sense?

what do i mean by "make sense?"

well, at first glance, statements like { (1,2) } < 14 don't make sense.

here's how you might fix this. when a = b, then whatever your extended definition of < is should reduce to the definition of < for real numbers for when a = b, x is identified with the real number a. "identified" means that there is a bijection between the set of sets of the form { (a, a) } and the set of real numbers; this bijection is given by
f( { (a, a) } ) = a.
this is what i mean by saying that your definition of < meant for connectors and reals alike should be an extension of what < means for two real numbers.

doing this will, i think, embed the set of connectors within the set of real numbers, which is what i think you want to do.

then, once you have a definition of <, you can then talk about a connector being between two real numbers.

the first big theorem for you, once you have appropriately precise definitions, is an existence theorem. you need to prove the following, which is NOT clear to me:

let a < b be real numbers. there exists a connector x such that
a < x < b. (where < has been precisely defined to apply to connectors and reals.)

do that, and you've actually got your own pet theory on a new kind of number.

to explore a bit further, find the cardinality of the set of all connectors between a and b.

next, find applications!

the main thing about nonstandard analysis, what makes it useful, is that STANDARD analysis results can be obtained by working exclusively with hyperreals. i'm talking about nonstandard analysis for they also talk about there being things between real numbers. godel, well known for his incompletness thoerem, said that nonstandard analysis will be the analysis of the future. well, he's no analyst! go back to the corner, godel, and study your logic!

cheers,
phoenix

[Organic]

Hi phoenixthoth,

First, thank you for your positive attitude.

Well, I have ideas, but I am not good in formal definitions.

So, if you can help me in this point, I'll be thankful to you.

In general I look on Math as a form of comnunication that can be developed through team work between persons, therefore I do not understand the aggression of some persons when I show them my ideas in a wrong "formal" way.

I think this forum will be a better place through positive attitude to each other's ideas.

OK..., more to the point.

I'll write my definitions again, and then I'll try to explain my ideas
to you the best I can (also sorry about my poor English).


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.

Now I'll try to explain the above:

By p<q I mean that there exists some interval > 0 between any two different real numbers, which means that p or q never reach each other by definition (both of them are singletons).

You wrote:

here's the big question that will improve your theory for as it stands right now, there is a big gap. and here it is.

suppose y is a real number (to say "normal" real number is redundant) and x is a connector, x = { (a, b) }, where a and b are two real numbers. for x to be a connector, this implies a != b. (note that != is standard notation for unequal.)

this is the big question you must answer:

how can you extend the binary relation < so that a statement like

x < y

makes sense?

what do i mean by "make sense?"

well, at first glance, statements like { (1,2) } < 14 don't make sense.


I'll try to answer (hope I understand you):

a=p , b=q


Let us write the connctor between a and b as aCb.

So, x = aCb.

By x < y we mean that y is bigger than aCb range (I know that the word "range" or "domain" already have their meaning in Math languauge, so maybe instead of "range" I use "path-place").

If I draw it, it will look like this:

a b y(a D element)
.______. .

If y = bCc ,where c > b then
a b c
.______.____.


If a = b then x is a D element (a "normal" real number).

As you see "<" "=" ">" works for both D and C elements.

D = {.} = a localized element.

C = {.__.} = a non-localized element (Double-simultaneous D places).

By (p, q) = {x : p < x < q} I mean that some C exists between a and b (aCb) where a > p and b < q, therefore x = aCb.

Hope I am understood.

Please tell me what do you think.

Thank you.


Organic

[Anton A. Ermolenko]

Hi Organic.
Originally posted by Organic
Please open the attached pdf file.
from your file:
The most symmetrical and simplest content is Emptiness, which is represented by the empty set notation { } = content does not exist.
On top of this simplicity we can define two opposite types of symmetry contents, {__} and {._. .}.
Huh... really? However, you are not talking about "normal" mathematics. If I get you right, you are talking about symmetry, well, but if a set really has a symmetry - it has topology,
if a set has topology - it has signature, and if it has signature - it must have more than one element (may be uncertain nature)!
Now, just tell us - how do you construct the symmetry on empty set without signature in such a way that it makes sense to say it is a real symmetry? These (and many others) contradictions doesn't allow considering your math's invention...

[Organic]

Hi Anton A. Ermolenko,

Why you are so aggresive, please take Math as some form of communication where persons share each other's ideas.

I'll be glad to get any remarks from you.


More to the point, I connect between simplicity and symmetry concepts.

The more you are simple the more you are symmetric.

The {} content is the most simple, therefore the most symmetric.

Please write to me on any part of my work, where you find contradictions.

You can find my work here:

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

Thank you.



Organic

[phoenixthoth]

if there are any terms in the following post not clear, consult http://www.mathworld.com .

if x = { (a, b) }, which is the bijection with domain {a} and range {b}, is a connector between the real numbers a and b, and y is some other real number, then are you saying that < means the following:

x < y if b < y?

thanks.

cheers,
phoenix

[Anton A. Ermolenko]

Originally posted by Organic
Why you are so aggressive…

Really? Perhaps, you’ve misunderstood...
please take Math as some form of communication where persons share each other's ideas.
Perhaps...
I'll be glad to get any remarks from you.
And I’m glad to see you’re glad...
More to the point, I connect between simplicity and symmetry concepts.

More you simple more you symmetric.

The {} content is the most simple, therefore the most symmetric.
In your case your definition isn’t right. It makes no sense to assert that your symmetry is a real internal symmetry. The space’s symmetry can’t have such definition because this kind of symmetry is internal, not external. Symmetry is not a senseless abstraction, only a simplest variety in the space on non-empty set can be more symmetric depending on simplicity. However, if I get you right, your precisely empty set is not (also can’t be) an element of any definite superset.

Please write to me on any part of my work, where you find contradictions.
Well,
From your file:
Let {___} be Csim (C for Continuum).
Let {._. .} be Dsim (D for Discreteness).
If your set is exactly empty set, then it can’t has continuous or discrete structure because the definition of a continuity or discretization of set by the neighborhood of point, based on relation between elements of this exactly non-empty set, doesn’t allow it.

Thank you.
You’re welcome.

[Hurkyl]

Are the "connectors" you talking about simply intervals?

[phoenixthoth]

"However, if I get you right, your precisely empty set is not (also can’t be) an element of any definite superset."

sure it can. let 0 denote the empty set. {0} is a superset containing 0. in fact, that is the definition of the number 1.

[phoenixthoth]

no, connectors are bijections from a real number to another real number, if i understand.

[Organic]

Hi phoenixthoth,

x < y if b < y?

Yes. and if x = aCb and y = cCd then
x < y iff c > b.


a,b,c,d are all "normal" real numbers (D elements, or singletons), where x and y, in this case, are C elements, or connctors.

So, in general I'm talking about singletons and/or connectors.

By the way, from the above example, x <= y iff b=c,
and there is no such a thing x <= y where c>a and c<b,
because in this case we get 3 connectors aCc, cCb and bCd.




Organic

[phoenixthoth]

supppose we have p < q real numbers.

then p < { (a, 0.5(p+q) ) } < q for all a. this gives an infinite collection of connectors between any two real numbers.

do the set of connectors plus reals form a field? if not, since R is a field, i don't think you could call them numbers. http://mathworld.wolfram.com/Field.html

cheers,
phoenix

[Organic]

phoenixthoth,

Please read my last post to you, beacuse I updated it, sorry.

I hope my updates helps.


Organic

[phoenixthoth]

still wondering if the set of connectors plus reals is a field... that will decide whether or not connectors could be considered numbers, i think.

[Organic]

Hi Hurkyl,

Thank you for reply.


Are the "connectors" you talking about simply intervals?

The connectors, which are C elements, close the interval between any two different "normal" real numbers, which are D elements (singletons).

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

By the way, I hope you have the time to help me in:

http://www.physicsforums.com/showthread.php?s=&threadid=6427

Thank you.


Organic

[Organic]

D + C is a field.

You wrote:

supppose we have p < q real numbers.

then p < { (a, 0.5(p+q) ) } < q for all a. this gives an infinite collection of connectors between any two real numbers.

Let us say that b=0.5(p+q), then between a and b there is an infinite collection of D and C elements, which is a field.


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

[Organic]

Hi Anton A. Ermolenko,


By using the empty set (with the Von Neumann Heirarchy), we can construct the set of all positive integers {0,1,2,3,...}:

0 = { }

1 = {{ }} = {0}

2 = {{ },{{ }}} = {0,1}

3 = {{ },{{ }},{{ },{{ }}}} = {0,1,2}

4 = {{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}}} = {0,1,2,3}

and so on.

So, as you see we can use nothing but the {} to construct numbers, which are not empty.

The common building block -the simplest and invariant(=symmetric) element- is {}.


Organic

[phoenixthoth]

that there are infinitely many of them isn't enough.

how is the arithmetic (addition/subtraction, multiplication/division), of connectors defined?

since these are functions with one element...

for example, what are { (a, b) } + { (c, d) }
and { (a, b) } * { (c, d) }?

if this is to be a field extension of R, then it would have to be the case that
{ (a, a) } + { (b, b) } = { (a+b, a+b) }
and
{ (a, a) } * { (b, b) } = { (ab, ab) }
since the connector { (a, a) } is identified with the real number a.

if it can be shown that if this has ring structure (again, see http://www.mathworld.com for def. of ring), can you show it is an integral domain (ie, no zero divisors)? what is the formula for the multiplicative inverse of a connector { (a, b) }?

a question for the long term is that if the connectors plus reals forms a field, then is it algebraically closed?

cheers,
phoenix

[Organic]

phoenixthoth ,

You wrote:

...since the connector { (a, a) } is identified with the real number a.

Well {(a,a)} can't be but a D element, so there is no connactor (a C element, which exists only between two different Ds) in {(a,a)}.


Please look at the pdf that I sent to your personal email.

Then, please open and read my informal overview in:

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

I have a new point of view on the natural numbers, which are the simplest numbers in Math language.

In general I show that any natural number > 1 is several structural variations of the same quantity, where each structure is some tree-like element, constructed by an AND connective between C and D elements.

So any arithmetic operation must be found among those structures,
and there is no meaning to Cs XOR/OR Ds operations.


Please see if I am understood, and reply for any problem.


Thank you.

(By the way, I opned a thread in your forum at:

http://207.70.190.98/scgi-bin/ikonboard.cgi?;act=ST;f=2;t=48 )




Yours,


Organic

[Anton A. Ermolenko]

"However, if I get you right, your precisely empty set is not (also can’t be) an element of any definite superset."

sure it can.
Hi Phoenixthoth.
I still insist, it can’t.

let 0 denote the empty set. {0} is a superset containing 0. in fact, that is the definition of the number 1.
Let’s see. At first you’ve denoted an empty set as zero (but this denotation is not a number), then you’ve substituted this concept by zero (a number!).
The axiomatic theory of sets needs the concept of an empty set so that the result of any operation with sets is to be a set, too, but the results of all operations with sets form a class; this class is neither a set, nor a superset. In general, a class defines only the properties of some objects (which may be sets), not the objects themselves.
Zero (number) in fact can be an element of a numerical set of an algebraic system. However, it is a number – thus, an element of a numerical set.
But, if we keep on going this way, we’ll change the topic.
The idea I’d like to get across with in my message above is that an empty set can’t possess symmetry as it has no elements.

To Organic:

In general, your ideas aren’t new extension of mathematics, but may have application in the computer science.

[phoenixthoth]

I still insist, it can’t.


Let’s see. At first you’ve denoted an empty set as zero (but this denotation is not a number), then you’ve substituted this concept by zero (a number!).
The axiomatic theory of sets needs the concept of an empty set so that the result of any operation with sets is to be a set, too, but the results of all operations with sets form a class; this class is neither a set, nor a superset. In general, a class defines only the properties of some objects (which may be sets), not the objects themselves.

if you prefer, denote the empty set by e. then {e} is a superset of e containing e.

http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html

here, i'm using the unordered pair axiom (axiom 2) where i'm considering the pair e and e and saying {e, e} is a set. one can then use axiom 1 to show that {e, e} = {e}, so it is a set.





Zero (number) in fact can be an element of a numerical set of an algebraic system. However, it is a number – thus, an element of a numerical set.

in set theory, one defines zero to be the empty set. then it can be shown that all number systems can be built upon this.

But, if we keep on going this way, we’ll change the topic.
The idea I’d like to get across with in my message above is that an empty set can’t possess symmetry as it has no elements.

what is the definition of symmetry, because whatever it is may be vacuously true of the empty set?

cheers,
phoenix

[Anton A. Ermolenko]

By using the empty set (with the Von Neumann Heirarchy), we can construct the set of all positive integers {0,1,2,3,...}:
code:

0 = { }
Huh... really? Let’s see. How do you define an empty set? The axiomatic theory of sets (ATS) defines it as:
("õ"="direct product", "!="="not equal", "Å"="direct addition", "Ú"="or")
A=Æ defined as "B(B !=Æ & AõB=Æ & A+B=B & "C(C=A equivalent to CõA=Æ)).
However,
"B(B !=Æ & AõB=Æ & A+B=B & "C(C=A equivalent to CõA=Æ)) Þ A=Æ Ú A="zero divisor", i.e. it is non-empty set.

i.e. without a non-empty set no one from mathematicians can’t defines an empty set. And this is nature of an empty set (or ATS) – it is necessary to definition of operations with sets.
1 = {{ }} = {0}
What it precisely means? Either an empty set is a subset of «1» (but «1» is not a natural number) or it is an element of «1». The ATS doesn’t let us define 1 (natural number) by non-number. Only a map correspond the elements of different nature to each other. And so on...

[phoenixthoth]

Huh... really? Let’s see. How do you define an empty set? The axiomatic theory of sets (ATS) defines it as:
("õ"="direct product", "<>"="not equal", "+"="direct addition")
AõA=0, B<>0, AõB=0, A+B=0: A=0.
However,
AõA=0, B<>0, AõB=0, A+B=0: A=0 or A="zero divisor", i.e. it is non-empty set.

using axiom 5 in

http://mathworld.wolfram.com/Zermel...nkelAxioms.html

one can define an empty set to be such an x. then you can prove that all empty sets are equal, so it makes sense to give them all one notation.

the word set is undefined.


i.e. without a non-empty set no one from mathematicians can’t defines an empty set. And this is nature of an empty set (or ATS) – it is necessary to definition of operations with sets.

What it precisely means? Either an empty set is a subset of «1» (but «1» is not a natural number) or it is an element of «1». The ATS doesn’t let us define 1 (natural number) by non-number. Only a map correspond the elements of different nature to each other. And so on...
let's look at the first statement and replace "empty set" by the phrase "concept X." i'll also rule out your double negative. it becomes this: without concept X mathematicans can't define concept X. i disagree with this statement. you can certainly define any concept X you like. whether concept X "exists" is another story...

what do you mean when you add (direct sum) and multiply (direct product) sets? is that the same as union and intersection?

the empty set is an element of 1 in set theory.
0 = { }
1 = {0}

n = {n-1}

n+1 = n U {n}

[Anton A. Ermolenko]

if you prefer, denote the empty set by e. then {e} is a superset of e containing e.
Excuse me, is e a subset of {e}, or e an element of {e}?
If e is a subset of {e}, then you’ll not define either {e} or e, because {e}=e, hence, {e, e}={e} imply e either empty set, or zero divisor. Without a non-empty set neither you, nor anybody else will not prove that e is empty set
If e is an element of {e}, then you’ll not prove that e is empty set, because in this case {e, e} always {e, e}.

[Anton A. Ermolenko]

Originally posted by phoenixthoth
using axiom 5 in

one can define an empty set to be such an x. then you can prove that all empty sets are equal, so it makes sense to give them all one notation.

the word set is undefined.

Well, then how do you define zero divisor?

[phoenixthoth]

Originally posted by Anton A. Ermolenko
Excuse me, is e a subset of {e}, or e an element of {e}?
If e is a subset of {e}, then you’ll not define either {e} or e, because {e}=e, hence, {e, e}={e} imply e either empty set, or zero divisor. Without a non-empty set neither you, nor anybody else will not prove that e is empty set
If e is an element of {e}, then you’ll not prove that e is empty set, because in this case {e, e} always {e, e}.

element.

it's not about proving the set is empty. its emptiness is a postulation.

for any x, {x, x} = {x}

[Anton A. Ermolenko]

Originally posted by phoenixthoth
if you prefer,
http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html

Huh... ZF system of axioms? Well, but pair of "axioms" (subsets & foundation) are not precise axioms, only the schemes. I prefer the NBG-system of axioms

[Anton A. Ermolenko]

Originally posted by phoenixthoth
element.

it's not about proving the set is empty. its emptiness is a postulation.

for any x, {x, x} = {x}
0 = { }
1 = {0}

n = {n-1}

n+1 = n U {n}
Huh... well, well, well... I like it. Then following your logic:
0=1=2=3=4=5=6=7=8=9=...="infinity".
You really think so?

[Anton A. Ermolenko]

Originally posted by phoenixthoth
using axiom 5 in

0 = { }
1 = {0}

n = {n-1}

n+1 = n U {n}
In the real
f: f({0})=1
The NBG (not ZF, because ZF hasn't the classes, hence, hasn't a the hierarchies) system of axioms.

[phoenixthoth]

Originally posted by Anton A. Ermolenko
Huh... well, well, well... I like it. Then following your logic:
0=1=2=3=4=5=6=7=8=9=...="infinity".
You really think so?

i don't see how you get 0 = 1 = 2 = ...

0 = { }
1 = {0}

two sets x and y are equal iff (a is an element of x iff a is an element in y).

0 is an element of 1 but 0 is not an element of 0 = { }.

therefore, it is not the case that a is an element of 1 iff a is an element of 0.

therefore, 0 != 1.

[Anton A. Ermolenko]

Originally posted by phoenixthoth
i don't see how you get 0 = 1 = 2 = ...

0 = { }
1 = {0}

two sets x and y are equal iff (a is an element of x iff a is an element in y).

0 is an element of 1 but 0 is not an element of 0 = { }.

therefore, it is not the case that a is an element of 1 iff a is an element of 0.

therefore, 0 != 1.
Exactly 0!=1, because 0! is number of permutations (there is an order!) {01}=1
but if {01}=1, then neither {01,02} nor {02,01} is not equal to {01}, because either a set has the order relations between the elements (even if there is only one element), or hasn't the order relations...
In the real your {0} (0 is an element of {0}) is equal to 0õ{1}, otherwise this {0} is imply that 0 is a subset of {0}, hence, {0}=0...

[phoenixthoth]

!= means "does not equal."

i said, "therefore, 0 != 1." if ! is factorial, then it would be directly adjacent to the zero, which it is not.

if i meant ! as factorial, the statement 0! = 1 is a non-sequitor from my argument.

the statements above the conclusion demonstrated how the pair 0 and 1 don't fit the definition of set equality.

the conclusion was that the two sets are not equal.

i urge you to check out "elements of set theory" by enderton, "set theory" by stoll, or "axiomatic set theory" by suppes for all the details.

[Anton A. Ermolenko]

Originally posted by phoenixthoth
!= means "does not equal."

i said, "therefore, 0 != 1." if ! is factorial, then it would be directly adjacent to the zero, which it is not.

if i meant ! as factorial, the statement 0! = 1 is a non-sequitor from my argument.

the statements above the conclusion demonstrated how the pair 0 and 1 don't fit the definition of set equality.

the conclusion was that the two sets are not equal.

i urge you to check out "elements of set theory" by enderton, "set theory" by stoll, or "axiomatic set theory" by suppes for all the details.
From the ZF system of axioms may be proved (almost) all mathematics. In spite of the fact that the actual number of axiom is equal to infinity (Z5 and ZF9 are not the axioms, only schemes), but there is no these: {0,0}={0}. Empty set (or zero element) defined as the result of these operations:
B != A, A õ A = A, A õ B= A, A + B = B: A=0.
Otherwise, how do you define a zero divisor or a nilpotent device?

[Anton A. Ermolenko]

---

[phoenixthoth]

Originally posted by Anton A. Ermolenko
From the ZF system of axioms may be proved (almost) all mathematics. In spite of the fact that the actual number of axiom is equal to infinity (Z5 and ZF9 are not the axioms, only schemes), but there is no these: {0,0}={0}. Empty set (or zero element) defined as the result of these operations:
B != A, A õ A = A, A õ B= A, A + B = B: A=0.
Otherwise, how do you define a zero divisor or a nilpotent device?

{0,0} = {0} is not an axiom but it can be proven from the axiom of extensionality.

what are the definitions of A and B?

x is a zero divisor if it is nonzero and there is a nonzero y such that x õ y = 0. (source: http://mathworld.wolfram.com/ZeroDivisor.html )

in your equations above, neither B nor A is conclusively a zero divisor because A = 0. in every ring, B õ 0 = 0; that doesn't make B a zero divisor since then, every element would be a zero divisor.

here's how i would define nilponency:
this is pseudo-code:
let N be given. let m be a natural number and x = 1.
N^1 := N. (by, :=, i mean, "is defined to equal")
1. N^(x+1) := N õ N^x. update x so that x = x+1.
2. repeat step 1 until x = m.
when finished, N^m is defined. intuitively, N^m is
N õ N õ ... õ N, where there are m copies of N.

definition: N is nilpotent if N^m = 0 for some m.

[Anton A. Ermolenko]

Originally posted by phoenixthoth
{0,0} = {0} is not an axiom but it can be proven from the axiom of extensionality.

what are the definitions of A and B?


sets

x is a zero divisor if it is nonzero and there is a nonzero y such that x õ y = 0. [/qoute]
I want to see your definition of a zero divisor by ZF system axioms.

[quote]in your equations above, neither B nor A is conclusively a zero divisor because A = 0. in every ring, B õ 0 = 0; that doesn't make B a zero divisor since then, every element would be a zero divisor.
You've misunderstood. I define an empty set (or zero element). In other words, iif B != A, A õ A = A, A õ B= A, A + B = B, then A=0. If B != A, A õ A = A, A õ B= A, A + B != B, then A is a zero divisor. Your postulate {0,0}={0} doesn't allow defining a zero divisor.

[phoenixthoth]

Originally posted by Anton A. Ermolenko
sets


You've misunderstood. I define an empty set (or zero element). In other words, iif B != A, A õ A = A, A õ B= A, A + B = B, then A=0. If B != A, A õ A = A, A õ B= A, A + B != B, then A is a zero divisor. Your postulate {0,0}={0} doesn't allow defining a zero divisor.

zero divisor isn't a term used in set theory. therefore, there's no need to relate it to ZF axioms. zero divisors occur in rings; investigate how rings develop out of set theory. to write down the ZF axioms and then a sequence of statements leading to the definition of zero divisor would take a while.

are A and B allowed to be any two different sets?

is A õ A = A an assumption or a theorem?

is A õ B = A an assumption or a theorem?

is A + B = B an assumption or a theorem?

from the last equation, that B can be "cancelled" is an assumption. cacellation presumes both that there is a zero element and that all elements have inverses. therefore, this is a circular argument.

[Michael F. Dmitriyev]

Greetings,
Using presence of professional mathematicians at a forum, it would be desirable to ask the following question:
Is it present in the mathematics some a theory about self-organizing, development and complication of functions of information system (between elements of some data set)? I keep in a mind our universe as information system.
Thanks.

[Organic]

Hi Michael F. Dmitriyev,


Maybe you will find this book interesting:

http://www.wolframscience.com/



Organic.

[Organic]

Hi phoenixthoth, Hi Anton A. Ermolenko,

The concept of a set is like a "stage" where you can put elements and then find the rules, operations, relations and so on, within and among these elements.

The {} is the "stage" itself and it is not one of the elements "playing" on it.

The "stage" itself must be simpler than any "player" that plays on it, otherwise no player can express its full propery.

The "stage" has no signature at all, therefore it has no content(=emptiness).

It is as if I said that the silence is the base of any sound.

We cannot find any variations in silence, therefore the silence is invariant by its very own nature.

Therefore silence is more symmetric than any sound.

Now, please change silence by emptiness, and some sound by non-emptiness.


Organic

------------------------------------------------------------------------------
To be {true sentence} or to be {false sentence}, that is not the question.


To be(=~{}), or not to be(={}), that is the question.

------------------------------------------------------------------------------

[Organic]

Hi Anton A. Ermolenko,

You wrote:

To Organic:

In general, your ideas aren’t new extension of mathematics, but may have application in the computer science.


Please be more spesific.

Thank you.


Organic

[Anton A. Ermolenko]

Originally posted by phoenixthoth
zero divisor isn't a term used in set theory. therefore, there's no need to relate it to ZF axioms. zero divisors occur in rings; investigate how rings develop out of set theory. to write down the ZF axioms and then a sequence of statements leading to the definition of zero divisor would take a while.

are A and B allowed to be any two different sets?

is A õ A = A an assumption or a theorem?

is A õ B = A an assumption or a theorem?

is A + B = B an assumption or a theorem?

from the last equation, that B can be "cancelled" is an assumption. cacellation presumes both that there is a zero element and that all elements have inverses. therefore, this is a circular argument.
See link below
http://forum.1tv.ru/index.php?act=Attach&type=post&id=249955 (The definition of empty set within the framework of AST)

[Anton A. Ermolenko]

Originally posted by Organic
Hi phoenixthoth, Hi Anton A. Ermolenko,

The concept of a set is like a "stage" where you can put elements and then find the rules, operations, relations and so on, within and among these elements.

The {} is the "stage" itself and it is not one of the elements "playing" on it.

The "stage" itself must be simpler than any "player" that plays on it, otherwise no player can express its full propery.

The "stage" has no signature at all, therefore it has no content(=emptiness).

It is as if I said that the silence is the base of any sound.

We cannot find any variations in silence, therefore the silence is invariant by its very own nature.

Therefore silence is more symmetric than any sound.

Now, please change silence by emptiness, and some sound by non-emptiness.


Organic

------------------------------------------------------------------------------
To be {true sentence} or to be {false sentence}, that is not the question.


To be(=~{}), or not to be(={}), that is the question.

------------------------------------------------------------------------------
Well, just tell me - how do you define an empty set within the framework of AST

[Anton A. Ermolenko]

Originally posted by Organic
Hi Anton A. Ermolenko,

You wrote:


Please be more spesific.

Thank you.


Organic
More specific in what? Why your ideas aren't new extension of mathematics?
Or why your ideas may have application in the computer science?
If first, then I've demonstrate the inconsistency of your definitions and suggestions with AST. There is no new mathematics.
If second, then I think that is wrong way (forum). What kind of physics we research up here?

[phoenixthoth]

Originally posted by Anton A. Ermolenko
Well, just tell me - how do you define an empty set within the framework of AST

i wouldn't know since i wasn't "brought up" with AST; i was brought up on ZF and ZFC with a smattering of the von Neumann/Godel system. maybe a search on http://www.mathworld.com with "AST" will reveal the answer you seek.

in ZF, sets are NOT defined. then it is POSTULATED that there is a "set" with the property that for all x, x is not an alement of this set. the empty set is defined to be a set with this property.

[phoenixthoth]

Originally posted by Anton A. Ermolenko
More specific in what? Why your ideas aren't new extension of mathematics?
Or why your ideas may have application in the computer science?
If first, then I've demonstrate the inconsistency of your definitions and suggestions with AST. There is no new mathematics.
If second, then I think that is wrong way (forum). What kind of physics we research up here?

is there an online reference to AST? i've never heard of it.

the inconsistency of organic's definition with AST is not clear to me nor does it seem relevant because i believe he's using the ZF axioms.

the mathematics is new to me. it may have been already done by someone else, however.

the question about physics is irrelevant because this topic has been moved to the math section.

the connections to computer science aren't clear at all.

[Anton A. Ermolenko]

Originally posted by phoenixthoth
i wouldn't know since i wasn't "brought up" with AST; i was brought up on ZF and ZFC with a smattering of the von Neumann/Godel system. maybe a search on http://www.mathworld.com with "AST" will reveal the answer you seek.

in ZF, sets are NOT defined. then it is POSTULATED that there is a "set" with the property that for all x, x is not an alement of this set. the empty set is defined to be a set with this property.
You've misunderstood again... "AST" (the axiomatic theory of set), in other word, ZF or NBG (von Neumann/Bernays/Godel). Neither ZF, nor NBG hasn't an "axiom of empty set". Can you demonstrate this axiom within the framework of ZF or NBG??? Just tell where it is?

[Anton A. Ermolenko]

Originally posted by phoenixthoth
is there an online reference to AST? i've never heard of it.

the inconsistency of organic's definition with AST is not clear to me nor does it seem relevant because i believe he's using the ZF axioms.

Which of them (axioms)???
the mathematics is new to me. it may have been already done by someone else, however.

the question about physics is irrelevant because this topic has been moved to the math section.
Is the section of General physics>Theory Development math section? I thought, that it is a Physics theory development section... If I've misunderstood, then forgive me

[phoenixthoth]

Originally posted by Anton A. Ermolenko
You've misunderstood again... "AST" (the axiomatic theory of set), in other word, ZF or NBG (von Neumann/Bernays/Godel). Neither ZF, nor NBG hasn't an "axiom of empty set". Can you demonstrate this axiom within the framework of ZF or NBG??? Just tell where it is?

http://mathworld.wolfram.com/AxiomoftheEmptySet.html

note that ! means "not" in this case.

[phoenixthoth]

Originally posted by Anton A. Ermolenko
Which of them (axioms)???

Is the section of General physics>Theory Development math section? I thought, that it is a Physics theory development section... If I've misunderstood, then forgive me

i'm not sure why this topic is under physics theory development.

[Integral]

Originally posted by phoenixthoth
i'm not sure why this topic is under physics theory development.

Would you have it in Math?

[phoenixthoth]

yes. as far as i remember, i found it under math where it had been moved to. i can't see what it's direct relation to physics is.

[Organic]

Hi Integral,

This thread was opened by me under Mthematics > General Math, and it was there until yesterday.

Please check if it can be returned to its original place.


Maybe it is the right time to open Theory Development under Mathematics.

What do you think ?


Thank you,


Organic

[Organic]

Hi Anton A. Ermolenko,

The ZF Axiom of the Empty set:

There is a set A such that, given any set B, B is not a member of A.

(There is a "stage" A with no "players" B)


Please tell me if you find any problem in this axiom.


Thank you.


Organic

[Anton A. Ermolenko]

Originally posted by phoenixthoth
http://mathworld.wolfram.com/AxiomoftheEmptySet.html

note that ! means "not" in this case.
For Zermelo-Frenkel system axioms (AUTHOR's DENOTATIONS, not Ito and somebody else) see link below
http://physics.nad.ru/img/Sets.gif
If you want to understand what exactly author thought, then always use of the author's books or reference

[Anton A. Ermolenko]

Æ

[Organic]

Dear Anton A. Ermolenko,

By [null] do you mean that you accept my point of view
which is:

An analogy: [null] is like an empty sheet of paper with no written thing on it.


Organic

[Anton A. Ermolenko]

Originally posted by Organic
Dear Anton A. Ermolenko,

By [null] do you mean that you accept my point of view
which is:

An analogy: [null] is like an empty sheet of paper with no written thing on it.


Organic
No, I don't.
My post is there only because of a technical problem.

[Organic]

In my opinion, the empty set is the building block of the idea of the set.

Let us call it a zero-order logic object.


A zero-order logic is based on one and only one sentence which is:

"There is x".

In the case of the idea of the set, x is the empty set.


Orgnic

[Anton A. Ermolenko]

Originally posted by Organic
In my opinion, the empty set is the building block of the idea of the set.

Let us call it a zero-order logic object.


Why do you need another axiom? Don't you think that we could do with existing ones?
Originally posted by Organic
A zero-order logic is based on one and only one sentence which is:

"There is x".

In the case of the idea of the set, x is the empty set.

In the case of the idea of the set, x is only an uncertain set, nothing else.

[Hurkyl]

I've been keeping tabs on this thread... but I can't seem to figure out what your point is, Anton, can you enlighten me? [:)]


Some comments on the empty set that haven't been mentioned:

Given the existance of any set S, the ZF axiom of subsets can be used to construct the empty set:

ExAy:(y in x) <=> (y in S and ~(y in S))


It's easy enough to prove that a set constructed this way is unique and doesn't depend on the identity of S. I'll call the empty set 0 because I don't know the code to give me the empty set symbol.

It is easy enough to show that 0, constructed this way, satisfies, for any set T, 0 U T = T, 0 x T = 0, and it is also easy enough to show that 0 is the unique set satisfying these two properties. In fact, in ZF, the four statements:

z = 0
Ay: ~(y in z)
AT: z U T = T
AT: z x T = z

are equivalent; if one holds for a set z, then they all hold.


Why include the axiom of the empty set if you can use the axiom of subsets to construct it? Minimalism. In order to get started, the existance of at least one set must be postulated. Because no matter what set gets postulated one can construct the empty set, the most minimal postulate you can state is that the empty set exists.


Oh, and anyone have an online reference to the axioms of NBG?

[Anton A. Ermolenko]

Originally posted by Hurkyl
I've been keeping tabs on this thread... but I can't seem to figure out what your point is, Anton, can you enlighten me? [:)]


Some comments on the empty set that haven't been mentioned:

Given the existance of any set S, the ZF axiom of subsets can be used to construct the empty set:

ExAy:(y in x) <=> (y in S and ~(y in S))


It's easy enough to prove that a set constructed this way is unique and doesn't depend on the identity of S. I'll call the empty set 0 because I don't know the code to give me the empty set symbol.

It is easy enough to show that 0, constructed this way, satisfies, for any set T, 0 U T = T, 0 x T = 0, and it is also easy enough to show that 0 is the unique set satisfying these two properties. In fact, in ZF, the four statements:
Why include the axiom of the empty set if you can use the axiom of subsets to construct it? Minimalism. In order to get started, the existance of at least one set must be postulated. Because no matter what set gets postulated one can construct the empty set, the most minimal postulate you can state is that the empty set exists.
Of course! Bingo! And I think so, too! Maybe I was not clear enough. My point is that ZF exclude an axiom of the empty set because the empty set concept is a theorem within the framework of ZF, not an axiom. But I asked Organic to give the empty set definition of his own, because the ZF empty set definition doesn't allow to make the deductions he has made.

Originally posted by Hurkyl z = 0
Ay: ~(y in z)
AT: z U T = T
AT: z x T = z

are equivalent; if one holds for a set z, then they all hold.[/B]
Could you specify what you mean? And where are the contradictions with my definitions?

[Hurkyl]

y point is that ZF exclude an axiom of the empty set because the empty set concept is a theorem within the framework of ZF, not an axiom.

That is not the case!

The theorem is that if there exists a set, there exists an empty set. IOW:
(Ex) => (EyAz:!(z is in y))


The catch is that the axiom of the empty set is the only ZFC axiom capable of proving Ex. (Note: you can't even state the axiom of infinity unless the empty set is known to exist)


Let me restate this to emphasize what this means:

Without the axiom of the empty set, ZFC cannot prove the existance of even a single set.



Could you specify what you mean? And where are the contradictions with my definitions?

I am asserting:

Az: [ (z = 0) <=> (Ay: ~(y in z)) <=> (AT: z U T = T) <=> (AT: z x T = z) ]

(where 0 is as I defined it in my previous post)

I'm not sure why you ask about constradictions with your definitions; I think your definition (if I interpret you correctly) is equivalent to the usual definition, but is redundant, complicated, and confusing.

[Anton A. Ermolenko]

Originally posted by Hurkyl
That is not the case!

The theorem is that if there exists a set, there exists an empty set. IOW:
(Ex) => (EyAz:!(z is in y))


The catch is that the axiom of the empty set is the only ZFC axiom capable of proving Ex. (Note: you can't even state the axiom of infinity unless the empty set is known to exist)


Let me restate this to emphasize what this means:

Without the axiom of the empty set, ZFC cannot prove the existance of even a single set.





I am asserting:

Az: [ (z = 0) <=> (Ay: ~(y in z)) <=> (AT: z U T = T) <=> (AT: z x T = z) ]

(where 0 is as I defined it in my previous post)

I'm not sure why you ask about constradictions with your definitions; I think your definition (if I interpret you correctly) is equivalent to the usual definition, but is redundant,
complicated, and confusing.
Once more, which of my definitions (formulas) exactly redundant and complicated? The emty set is only a formula, not an axiom (prove me that my definition of FORMULA of empty set is an axiom; the formulas !x=Æ and z=Æ (in Z7 and Z8) can be substituted with full definition of the empty set... am I worng??? Just prove it!).
http://physics.nad.ru/img/Sets.gif

[Organic]

Anton A. Ermolenko,

Please answer to a simple question.

Who needs who to exist ?

Does the formula need the empty set, or the empty set needs the formula ?

I say: There is a set (a collector) with no content.

I don't need any formula to say that, because it is minimal and clear.

From this point any statement or formula that you give, will be more complicated then what I said, therefore cannot be in the same minimal level.

Therefore, you have no argument.



Organic

[Hurkyl]

Once more, which of my definitions (formulas) exactly redundant and complicated?

Specifically:
z = 0 is defined as
Au(u U z = u & u x z = z & Ev(v = z <=> (v x z = z & v U z = z) & AyAx (y * x = z <=> At(t in z <=> (t not in x & t not in y))))

(where * means set intersection)

is redundant and complicated.

Incidentally, I think it's complicated enough that you didn't even write down your intent correctly; "Ev" should have been "Av".


This formula is equivalent to each of the following statements:

(z = 0) := Au(u U z)
(z = 0) := Au(u x z = z)
(z = 0) := Au(~(u in z))

Your definition is redundant because you only need to have one of these three terms, and then the rest of your definition could be proven as a theorem.


And, as per the observation I made earlier, without the axiom of the empty set, 0 cannot be a formula. You cannot prove 0 exists, so you cannot use it as a constant. The basic formula involving 0 is the relation z = 0.

BTW, your page doesn't specify what Inf(x) is.

[Hurkyl]

I say: There is a set (a collector) with no content.

That is (directly translatable into) a formula:

EzAy: ~(y is in z)

There exists a z such that for all y, y is not in z.

[Anton A. Ermolenko]

Originally posted by Organic
Anton A. Ermolenko,

Please answer to a simple question.

Who needs who to exist ?

Does the formula need the empty set, or the empty set needs the formula ?

I say: There is a set (a collector) with no content.

I don't need any formula to say that, because it is minimal and clear.

From this point any statement or formula that you give, will be more complicated then what I said, therefore cannot be in the same minimal level.

Therefore, you have no argument.

Originally posted by Hurkyl

Specifically:
z = 0 is defined as
Au(u U z = u & u x z = z & Ev(v = z <=> (v x z = z & v U z = z) & AyAx (y * x = z <=> At(t in z <=> (t not in x & t not in y))))

(where * means set intersection)

is redundant and complicated.

Incidentally, I think it's complicated enough that you didn't even write down your intent correctly; "Ev" should have been "Av".


This formula is equivalent to each of the following statements:

(z = 0) := Au(u U z)
(z = 0) := Au(u x z = z)
(z = 0) := Au(~(u in z))

Your definition is redundant because you only need to have one of these three terms, and then the rest of your definition could be proven as a theorem.


And, as per the observation I made earlier, without the axiom of the empty set, 0 cannot be a formula. You cannot prove 0 exists, so you cannot use it as a constant. The basic formula involving 0 is the relation z = 0.

BTW, your page doesn't specify what Inf(x) is.
Well... one and the last time. The set theory operates sets by using operations such as x - direct product of sets, Å - direct addition of sets, U - join of sets, Ï - intersection of sets. The main requirement is that the results of all the above operations with sets are sets as well. Only this requrement makes the set theory use the empty set concept, therefore the empty set can only be defined by non-empty sets and operations. NOTHING ELSE! BECAUSE THE EQUALITY OR UNEQUALITY TO THE EMPTY SET (=Æ OR !=Æ) ARE ONLY FORMULAS. THE ICON Æ WITHOUT "=", "!=", "Î" or "Ï" MAKES NO SENSE. And this is really for the last time.

[Hurkyl]

Or...

0 is a constant in language of formal set theory that represents the empty set.


(incidentally, none of your math symbols appear as math symbols on my computer)


And I will repeat that without an axiom explicity specifying that some set exists, no set can be proven to exist... the existance of at least one set is a crutial part of the proof that (most of) the rest of mathematics is consistent relative to ZFC, so it really serve a practical purpose.

[Organic]

Hi Hurkyl,


That is (directly translatable into) a formula:

There exists a z such that for all y, y is not in z.


By using at least two variables (in this case z and y) we need some formula to describe the relations between them.

No set can be separated from the property of its content, therefore
we have an interesting situation here.

On one hand a collector can exist with no content, but on the other hand its property is depended on the property of its content.

But we also know that the content concept can't exist without a collector.

To define the exact definition of an existing thing z(a collector), is not in the same level of to define the existence of y(a content).

So z can exist with no clear property, but y can't exist at all without z.

Please tell me how Math language deals with these distinguished two levels.

If we say "There is a collector" , do you think that we can come to the conclusion that it has no content, as its minimal existence?



Thank you.



Organic