Closed Intervals with Infinite Endpoints: Explained

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The discussion centers on the nature of closed intervals in real analysis, particularly those with infinite endpoints, such as (-∞, b] and [a, ∞). Participants argue that these intervals are closed because they contain all their limit points, as any convergent sequence within these intervals converges to a point also within the interval. There is a debate about the definitions of closed sets and intervals, with some arguing that the inclusion of infinite endpoints complicates this classification. The conversation also touches on philosophical implications of infinity in mathematics and the limitations of traditional mathematical logic. Ultimately, the discussion highlights the complexities of defining closed intervals and the broader implications of infinity in mathematical theory.
  • #61
What is undefined?
 
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  • #62
if I'm not mistaken, sets and points from geometry are undefined. in other contexts, points are defined.
 
  • #63
Dear phoenixthoth,



In all of what i wrote, please show me what is undefined and must be defined?

Thank you.

Yours,

Orgainc
 
  • #64
information, for one thing.
 
  • #65
Why do I have to define it?
 
  • #66
it is listed in step 1 as the object of study.

you may not have to define it but there should be several examples of things that are information and things that aren't.

set theory is a theory about an undefined concept but one can say something like, here are four "widgets" and here are seven ways to build new "widgets" from old ones, but i won't tell you what a "widget" is. that's set theory, at least. maybe you can follow a parallel structure in information theory.

if this is to be a mathematical theory, i think you'd have to decide on a set of constants like ∈ though ∈ doesn't have to be one of them.

what would be nice is if you could fit information theory into an existing theory so you get to use all of its power. information itself seems to be more general than even logic and in fact logic would be a subject of study in information, as would illogic. they're both information. it is ok to use logic to study logic or to use logic to study illogic if you do it delicately.

the examples I'm keeping in the back of my mind is how the definition should include the following information:
1. information about what a set contains
2. information about how I'm emotionally feeling
3. information of a poem
4. computerized information
5. information kernels, ie, truly abstract information
6. nonverbal and nonwritten information
7. the relationship between information and truth (eg true information)

so i think that if this is going to be about information in general, it should capture all kinds of information. if information is undefined in terms of standard math words, it will take a lot of "motivation" for anyone to know it. in other words, what will be the major theorems? give at most one for now without proof.
 
  • #67
Dear phoenixthoth,



The major theorem is very simple:

No model of x is x itself, that's all.

To any development of x there is some meaning only in the gap between x-model and x.

Now, x can be Information, Mathematics, and so on.

Shortly speaking, x has two basic forms: x-model, x.

The problem of any research is not to forget the above during the research.

Now let us call x-model potential x, and let us call x actual x.

Modern Math language forgot this and the result is the transfinite universes.

Another importent reason to this result is:

Modern-Math Number-Systems are based on some arbitrary broken symmetry.

To see it, please look again at:

http://www.geocities.com/complementarytheory/LIM.pdf



Modern Math in general does not distinguish between x-model and x.
Therefore it becomes a closed and circular system.

Take for example your comment about Math:
Since this is a theory about information in general, it is, strictly speaking, not a mathematical theory but a theory more general than a mathematical theory.
My response to this is:

There is no such a thing "mathematical theory" because any theory can be only
x-model, and no x-model is x.


Conclusion: Any x-model is an open system that can be changed.

Please read again both of them:

http://www.geocities.com/complementarytheory/GIF.pdf

http://www.geocities.com/complementarytheory/RealModel.pdf



Yours,

Organic
 
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  • #68
Why do I have to define it?

So you can apply logic.


At the very least, you have to enumerate the basic facts about things that allow us to start proving theorems.

For example, ZFC doesn't even try to say what a set is, but it rigorously lists the operations we're allowed to do on sets (e.g. make a pair set, make a power set, make a sumset, make a subset), thus allowing to prove theorems, et cetera.

Euclidean Geometry doesn't try to say what point, line, between, incident, or congruent is, but it precisely lists some facts about them (e.g. for any two distinct points there is a unique line incident with both), thus allowing us to rigorously prove theorems from these basic facts.
 
  • #69
Originally posted by master_coda
I'm well aware of this. But I don't really take it serious enough to get frustrated over it.
Well, you're a glutton for punishment with bonus points for tenacity. Good luck!
 
  • #71
ok, organic. you've stated your goals and your steps and your main theorem. so far, so good. this would be quite an undertaking.

as for your main theorem, what is x? is it an information "system", a set of information, etc.?

you might find this idea interesting. the claim is that the universe contains almost no information:
http://www.hep.upenn.edu/~max/nihilo.html

i already think that (almost) no model of x is x. I'm just taking for example a model for gravity and gravity. what about metamathematics (model theory, set theory, logic)? it is a model for mathematics and it is (part of) mathematics.
 
  • #72
Any Model is on x and never the x.

For example: To eat the cake is the x, but to speak on eating the cake is a model of eating the cake (a x-model).

In mathematics "Eeting the cake" = "Actual infinity".

Shortly speaking, no theory can deal with Actual infinity, but can use a model of it, which is potential infinity.

Also, the main player on this stage is first of all the symmetry concept.
 
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  • #73
i think there can be a finite model of absolute infinity. no one is saying that the model for infinity is infinity i don't think.

now, before you launch into a discussion on that, you still haven't really integrated our feedback into what you're doing, which makes us giving feedback less purposeful. and the feedback was this: define information or leave it undefined but describe examples of it and gives ways to contruct new information from old information as how it is done in geometry and set theory.
 
  • #75
i did a text search for the word "information" and didn't see a definition or a scheme of what is or is not information. therefore, the reader does not really know what your theory applies to. we all have a kind of intuitive sense of what information is but in order to be considered a mathemtical theory, or a philosophy of mathematics theory, information has to be defined or at least how to get new information from old information (with a collection of what some information is) has to be done. otherwise, it will be impossible to prove any statement about information in a rigorous way.

maybe I'm just missing it. if you did this, please point out the specific page number and line number where you define or give an inductive definition of information. i see the word information used on the pages you listed but nothing resembling a definition. a definition or inductive definition is necessary in order to prove anything about information in a mathematical theory. you may want to start with the dictionary definition of information and try to turn that into a mathematical definition. however, the words used in your definition must also be defined or inductively defined. you may also want to look at information theory and see how they define it.
http://en.wikipedia.org/wiki/Information_theory
http://en.wikipedia.org/wiki/Information

by "inductively defined," i mean that if you don't define information, or words used in your definition of information, you should give a few examples of information and then give a list of ways to construct new information from old information. if you use other words, like entropy or information clarity degree, or symmetry, then the same applies to those words because they have either no common definition or a definition that depends on context.

why do you have to define (inductively or not) words?
1. in order to prove something about those words
2. if not, you run the risk of "abusing language" such as but not limited to changing the definition or implied meaning of words in mid article or even mid sentence.
 
  • #76
My Definitions are given by the structures themselves.

The words and the sentences around them just giving an extra explanations
to what is already given by structures.

Shortly speaking, my definitions are "structure oriented".

If you understand this, then look again at http://www.geocities.com/complementarytheory/CATheory.pdf starting from page 7 until the end, thank you.

The beauty in my theory is: the structures are the definitions and the examples.
 
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  • #77
A definition for information:

A product of a mutual influence between, at least, two different things.
 
  • #78
Originally posted by Organic
My Definitions are given by the structures themselves.

The words and sentences around them just giving an extra explanations
to what is already given by structures.

Shortly speaking, my definitions are "structured oriented".

If you understand this, then look again at pages 7 and this time go until the end, thank you.

The beauty in my theory is: the structures are the definitions and the examples.

it sounds like you're trying to get away with defining in your article information by context. in other words, the term is meant to be "defined" by the words around it. this isn't a rigorous definition. the one you just gave is closer to an actual definition.

is the sentence, "yes" an example of information?

how about the formula (x->(y<-> ? it's not a "well formed formula" but would it be considered "information" because it is about five different things? btw, what is a thing? a set? a letter? a symbol?

under your definition, the sentence "yes" is not information whereas "(x->(y<->" is information. i just want to clarify what information is. there is no such thing as a wrong definition; only good and useful definitions or bad and useless definitions.

i can see how "x is x" is information because it is about two differnent things: x and is.

seems that there are at least two kinds of information: sensical information and nonsensical information. there would probably also be degrees in between which suggests an application of a fuzzy approach. there could be a "sense indicator" S so that if x is information then S is a map from the collection of all information to [0,1] such that S(x) is in [0,1] and S(x)=0 means that x is devoid of any sense (perhaps this is total entropy) and S(x)=1 means x is totally sensical (perhaps this is total negative entropy). then you can develop some conditions on what kinds of S's are actual sense indicators because something that makes sense from one perspective may not make sense from another perspective.

the sensical indicator would have nothing to do with the truth of the information, it would just measure how "grammatically correct" the information is.

seems like there should be a definition of "more information" and "less information." a kind of relation between different information resembling subsets and supersets.

my main point here is to just say that defining a word by context simply won't do in a rigorous theory. however, your recent definition is much better. in order to be a mathematical theory, you should define what kinds of things you're considering. a thing is perhaps as general as you can get and goes way beyond math (unless mathematical existence is physical existence, that is).

you'll need a definition of "product" and "mutual influence" where those words don't depend on the definition of "information."

i want to reiterate that i already believe your main theorem without any work: no model of a system is the system, so you may not have to go through all the trouble you're going through.

however, from that, you conclude radical claims about transfinite objects. firstly, it is totally unclear how that follows from "no model of x is x" and secondly, in order to really convince anyone that 150+ years of set theory is wrong, you have to show where the error is. these are short and long term goals, respectively. for now, please just speak about my questions for clarification of what information is and is not, what kinds of "things" you're talking about, what "product" means, and what "mutually influence" means. my main question about "mutually influence" is that i don't see how saying, "the force of gravity acting on two masses M and m is given by the function F(M,m)," shows an influence between the force of gravity and the formula itself. the influence you must be talking about is on some kind of linguistic level because the formula does not influence what the formula refers to (emperically speaking).

to shorten this down for you, maybe just talk about the following things in your next post:
1. is "(x->(y<->" information? (there is no right or wrong answer here)
2. what is "product"
3. what "things" are you talking about (the word "anything" should be used delicately here)
4. what is "mutually influence"

thanks
 
  • #79
"One picture = 1000 words"

Don't you see how rigorous are my structures?
you'll need a definition of "product" and "mutual influence" where those words don't depend on the definition of "information."

I already gave an example for this in page 8 of http://www.geocities.com/complementarytheory/CATheory.pdf
 
  • #80
...the formula does not influence what the formula refers to ...
In any formula is x-model, the infuence is by x.

SASs can link between x-model and x.

Maybe this is the most fundamental SASs property.
 
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  • #81
About the transfinite numbers, I already gave you my answer.

I'll write it again, but this time i use |Q| instead of |N|.

Rseq is actually both R and Q.

( http://www.geocities.com/complementarytheory/NewDiagonalView.pdf }

The way Rseq is constructed is equivalent to both |Q| and 2^|Q| (or |P(Q)|).

This is the reason why we get this result (2^aleph0>=aleph0)={}

Form one hand Rseq is |P(Q)|( =[...000,...111) ).

From the other hand Rseq is |Q| ( = The length of each given sequence ).

Please tell me why it is so hard for you to understand the above?

Let us say it again:

Cantor's diagonal fails because he deals with the wrong input, which is |Q|*|Q|.

By the way Rseq is constructed, for the first time since Cantor we deal with the right input, which is |P(Q)|*|Q|.

By doing this we find that (2^aleph0>=aleph0)={}.

Therefore transfinite universes do not hold.

Again, Rseq is both R AND Q.

More then that:

If Rseq is [...000,...111] then it means that Cantor's diagonal input (which is ...000) does not exist.

Therefore no input --> no output --> no any information to establish the transfinite universes.

More then thet:

|P(Q)| exists iff P(Q)=[...000,...111)

Therefore there is no such a thing like all (or complete) infinitely
many objects.

And when there is no such a thing, transfinite universes do not hold.

Again, |Q| is a "never ending story", therefore words like 'all' or 'complete' cannot be related to |Q|.
 
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  • #82
one thing that had me confused is that when you meant page z, i mistook that for page z of the document and not the page with that z listed at the bottom. so i will go through it again.

Cantor's diagonal fails because he deals with the wrong input, which is |Q|*|Q|.
actually, cantor's diagonal argument doesn't use |Q|*|Q| as input. it uses any set. so |P(Q*Q)|>|Q*Q|, for example.
 
  • #83
Thank you for your correction, but it does not have any influence on my argument that (2^aleph0 >= aleph0) = {}.
 
  • #84
Another interesting thing is the hierarchy of dependency of R in Q, and Q in N.

Please look at this example: http://www.geocities.com/complementarytheory/UPPs.pdf

These Unique Periodic Patterns are prime-like patterns,where any irrational number uses as its building-blocks.

This example perfectly fits my argument about the power of existence that can be found in the second part of this paper (please start from screen 5 of acrobat viewer): http://www.geocities.com/complementarytheory/NewDiagonalView.pdf
 
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  • #85
More general.

Because any mathematical system is only an x-model (therefore an open system) it cannot talk about proofs, because they’re always can be changed (or even replaced) during paradigm’s changes.

When we have a paradigm’s change, a lot of old paradigm's results can become irrelevant.

Therefore, in my opinion, Math language has to use the words 'Current Result' (CuRe) instead of 'proof'.

Please look at this nice article: http://faculty.juniata.edu/esch/neatstuff/truth.html


By using strong words like 'proof', there is (in my opinion) a danger that we become scholastic and closed systems.

And closed systems find their death by entropy.
 
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  • #87
Originally posted by Organic
When we find a 1-1 map between some point x to some R number, then if x in R then for any x in R, we can find some x0 < x OR some x < x2.

Therefore x0 OR x2 are always unreachable for any given x.

Let x0 be -oo(= inifinitely many objects < x).

Let x2 be oo(= inifinitely many objects > x).

No given x can reach x0 or x2.

Therefore x0 OR x2 must be the unreachable limits of any R number.

(x0,x] OR [x,x2), therefore [a,oo) OR (-oo,b] cannot be but half closed intervals.

Therefore the set of all R numbers (where R has a form of infinitely many objects) does not exist.


Shortly speaking, infinitely many objects cannot be related with the word all.

Fore clearer picture please look at:

http://www.geocities.com/complementarytheory/SPI.pdf

What does it mean to find a 1-1 map between some x and some R number? Any function mapping a single element set to R is necessarily injective, so it is either a redundant statement or it has some more meaning you aren't explaining.

next there is this x0 and x1 are not reachable by x. How does one 'reach' a number from another?

then x0 becomes -oo which then becomes a set of numbers. Do you still not understand why this needs rewriting? How can A number be -oo and be A set of infinitely many elements?

ANd in what way does it imply the Reals don't exist? Do you understand their construction as limits of cauchy sequences?

You've still not explained your private definition of the word all that means it is not realistic to talk about the set of all real numbers. give me a real number not in the set R.

matt
 
  • #88
Dear matt,

You take some of my posts in the beginning of this thread, but you first have to read what happened since this post, because maybe your questions have been answered in the next posts.

Please check it, thank you.
 
  • #89
Originally posted by Organic
Dear matt,

You take some of my posts in the beginning of this thread, but you first have to read what happened since this post, because maybe your questions have been answered in the next posts.

Please check it, thank you.

Yes, some of the questions were asked previously. None of your answers have helped illuminate the issue though.

In particular you still insist that by taking the finite strings of 01s and 'completing using the infinity axiom of induction' that you get a set in bijection with N via binary expansions of integers AND a set that contains all strings of 01's. This is patently wrong.

1. Only strings with a finite number of non-zero entres will be mapped to an integer.

2. There are strings with infinitely many non-zero entries.

3. The list has been proven by you to be 'not complete'

4. Induction doesn't allow us to do what you did, inparticular the existence of an inductive set doesn't allow us to construct a set of uncountable cardinality. It just asssures us that an infinite set will exist in our model. It is countable. Then we construct more infinite sets that are of strictly greater cardinality, but not by induction. You can do transfinite induction if you so wish, but I think we've had enough without abusing the axiom of choice as well.

5. Moreover you don't even define the inductive process that tells you how to add in the next successor element.

6. Repeating myself, but, you've demonstrated no countable, enumerable, listable, whatever, set of strings of 01's contains all of them. That is sufficient, albeit that your proof could do with a lot of tidying up (the infinite case does not follow from the finite case by induction!). But you then say the list you've made is all of them anyway. Can you really not see that that is a contradiction in itself? The error is not in maths, but in your assertion you've got a complete list.

Spurious example:

1 is the largest integer. Suppose N is the largest integer, N>1 obiviously, therefore N.N>N

is a contradiction unless N =1. Do you see where that went wrong or have i given you more ammunition?

You make an unsubstantiated (indeed incorrect) claim, deduce a contradiction, but conclude it was something else that was incorrect!

Matt
 

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