A question on a<b

[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

 

[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?