Closed Intervals with Infinite Endpoints: Explained

[Organic]

No, a XOR b are some arbitrary members of Rseq=[a,...111)XOR(...000,b]

 

[phoenixthoth]

Originally posted by Organic
a XOR b, don't you see?

a XOR b can be defined as Rseq=[a,...111)XOR(...000,b]

Rseq cunstructed by:
http://www.geocities.com/complement...iagonalView.pdf

i'm just not seeing what this has to do with <.


i take it that a is defined to mean [a,...111)? what does ...111 mean because that's ambiguous: aren't ...1111 and ...0111 different versions of ...111? same question for ...000.

if a means [a,...111) then why doesn't b mean [b,...111)?

i take it that [a,...111) is some kind of interval. for that to make sense, you have to explain how a<...111 without using this definition of <. i understand that the XOR between the two intervals is the "disjoint union": the set of all things in the union but not the intersection.

 

[Organic]

Let's make it simpler.

a is some arbitrary member (an infinitely long 01 sequence) of Rseq=[a,...111)

Rseq cunstructed by:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

Therefore a cannot be ...111, therefore Rseq cannot be completed, and therefore no transfinite number can use Rseq as its building-block.

Therefore tranfinite univereses does not exist.

 

[master_coda]

You realize that to prove that something is impossible, you can't just make a single attempt to do it, fail, and then say "therefore it's impossible".

 

[phoenixthoth]

Let's make it simpler.

a is some arbitrary member (an infinitely long 01 sequence) of Rseq=[a,...111)

yeah, the left endpoint. or do you mean mapped into by some kind of natural map rather than "is some member of."

Therefore a cannot be ...111

that goes without saying. in any interval G=(a,b), if x is in G, then x is not b.

therefore Rseq cannot be completed

doesn't b complete my interval G? you may want to go through either the dedekind cut or cauchy sequence of rational numbers construction of real numbers and tell us where the flaws in those proofs are.

, and therefore no transfinite number can use Rseq as its building-block.

well, since ...111 completes the interval [a,...1), the premise above doesn't hold and so this conclusion doesn't follow.

Therefore tranfinite univereses does not exist.

this doesn't follow from the past premise even if it were true. your statement "no transfinite number can use Rseq as its building-block" doesn't mean that there is no building block for transfinite universes, just that Rseq isn't one. this is what the previous poster was saying.

in short, this is not going to be a way to define a<b better than the usual ways, i don't think. please see the dedekind cut or cauchy sequence of rational numbers constructions for good ways to define a<b. in order to be something new, this theory would have to
1. define a new ordering <* such that if x and y are real numbers than x<*y iff x<y
2. be an extension of < so that <* applies to more objects than real numbers.

the question was what is an interval and i think it has been thoroughly explained. certainly not all properties of intervals have been given, but enough have been to get one's feet wet.

edit: i think you may find my treatment of cantor's diagonal argument interesting in the file attached to page 4 of "the search for absolute infinity." in a new subsets axiom, i show how there is no contradiction obtained from cantor's diagonal argument.

 

[Organic]

Rseq is actually both R and N, DON'T YOU SEE THET?

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

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

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

From the other hand Rseq is N ( = 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 |N|*|N|.

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

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

Therefore transfinite universes do not hold.

Again, Rseq is both R AND N.

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(N)| exists iff P(N)=[...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, |N| is a "never ending story", therefore words like 'all' or 'complete' cannot be related to |N|.

 

[master_coda]

No. Your argument is based upon your concept of infinity, not the mathematical one. Whatever contradictions you find in your concept have no bearing on mathematics.

 

[russ_watters]

Originally posted by master_coda
No. Your argument is based upon your concept of infinity, not the mathematical one. Whatever contradictions you find in your concept have no bearing on mathematics.

You're a little new here - have you seen all of the threads Organic started in the general math forum? You are right, but the only thing you'll accomplish by trying to explain it is headaches from banging your head against the wall. Organic is not interested in math - only in making up his own new math as he goes along.

And so far, I haven't even seen his point - math (to me) is a tool for use in science/engineering. I haven't seen where he's said what he wants to do with his new math once he's finished inventing it.

 

[Organic]

No dear master_coda,

My last post clearly shows the problems that existing in Standard Mathematics about the transfinite definition.

Your last response is too weak.

NOW, YOU HAVE TO PROVE THAT MY CLIMES DO NOT HOLD.

You have no other choice, otherwise any response that can't clearly show why my argument does not hold, is meaningless.

 

[master_coda]

Originally posted by Organic
No dear master_coda,

My last post clearly shows the problems that existing in Standard Mathematics about the transfinite definition.

Your last response is too weak.

NOW, YOU HAVE TO PROVE THAT MY CLIMES DO NOT HOLD.

You have no other choice, otherwise any response that can't clearly show why my argument does not hold, is meaningless.

If you don't use standard mathematical definitions, your remarks don't have anything to do with math.

Beside, in math you never have to prove someones claims don't hold. The person making the claim has to prove it does hold. And you haven't provided anything resembling a proof...you yourself admit that you have no skill at formalizing math. What makes you think you can formalize a proof?

 

[master_coda]

Originally posted by russ_watters
You're a little new here - have you seen all of the threads Organic started in the general math forum? You are right, but the only thing you'll accomplish by trying to explain it is headaches from banging your head against the wall. Organic is not interested in math - only in making up his own new math as he goes along.

And so far, I haven't even seen his point - math (to me) is a tool for use in science/engineering. I haven't seen where he's said what he wants to do with his new math once he's finished inventing it.

I'm well aware of this. But I don't really take it serious enough to get frustrated over it.

 

[Organic]

HO NO russ_watters,

My doors are clearly wide opened, and i clearly show the benefits of my new points of view (which are non-Euclidean) on the standard Euclidean point of view (which is based on Boolean Logic or Fuzzy Logic).

Some examples:

1) Here we can see the complementary associations between multiplication and addition.

These complementary associations, deeply changing and enriching the Number's concept.

Also we can see that a*b and b*a are noncommutative, therefore have more interesting information then the standard commutative system.

See for yourself here (please read all of it, thank you):
http://www.geocities.com/complementarytheory/ET.pdf

2) the logic bases of the above can be found here (please read all of it including all links, thank you):
http://www.geocities.com/complementarytheory/AHA.pdf

3) My general point of view on symmetry can be found here, and there we can clearly show how our standard number system is based on some private case of broken symmetry (please read all of it including all links, thank you):
http://www.geocities.com/complementarytheory/GIF.pdf
http://www.geocities.com/complementarytheory/RealModel.pdf
http://www.geocities.com/complementarytheory/LIM.pdf
http://www.geocities.com/complementarytheory/MathLimits.pdf
http://www.geocities.com/complementarytheory/SPI.pdf

4) This non-Euclidean point of view, which is based on Complementary Logic, has much more power to deal with Quantum universe, because its fundamentals are based on complementarity, redundancy, uncertainty and symmetry, which are all connected in one and simple logical system.

Also because of the same associations, which are associated by Complementary Logic, this point of view can lead us to construct and deal with much more complex systems, then Euclidean point of view can do (Because of the limitations of Blooean or Fuzzy logics).

To examine this please see (read it only if you understand Complementary Logic):
http://www.geocities.com/complementarytheory/CATheory.pdf

5) Beyond the traditional "objective" attitude to Math language, this non-Euclidean can lead us to explore new frontiers that cannot be reached by standard approach, for example (read it only if you understand Complementary Logic):
http://www.geocities.com/complementarytheory/CK.pdf
http://www.geocities.com/complementarytheory/count.pdf
http://www.geocities.com/complementarytheory/Moral.pdf

After you read and understand all of it, then and only than, please reply.

Thank you,

Organic

 

[Organic]

Dear master_coda,

Notations have no meaning by themselves, we give them the meaning, and they are only tools that help us to express our ideas.

What i wrote in the post that you refuse to deal with it (by hiding behind technical excuses) is a very weak response.

This post (that you refuse to answer to it) is written in a clear Mathematical way that any Mathematician can clearly understand and can response to it.

Please show me something that I wrote in this post, which is not based on standard Mathematics.

Yours,

Orgainc

 

[phoenixthoth]

Rseq is actually both R and N

so if R=Rseq=N, then R=N. this can't be because not every element of R is an element of N. 1/2, for example, is an element of R but not N.

 

[phoenixthoth]

N is in bijection with Q, the set of rational numbers, so it is "equal" in a sense, but not with R. but at this point, since we're not talking about intervals anymore, perhaps you should start a new thread or continue under "combinations."

the question which was somewhat generous is how do you define a<b? i say generous because a and b were i think just defined to be "infinite sequences." I'm wondering how you can have an infinite sequence without being able to talk about infinity. but, that taken for granted, the question was asked how to define a<b. in the cauchy construction of the real numbers, real numbers a and b are equivalence classes of rational cauchy sequences. there a<b is defined.

i think you wanted to say that your ideas were like [0,oo) but they're not. oo is actually essentially undefined, though it seems like you don't define it either. that's not automatically a problem. it's just a symbol used for convienience. we could use the symbol * and it wouldn't be construed as any infinity. if you look at the definition of what it means for a limit of a sequence to converge, in the limit symbol there is an infinity but in the definition there is no mention whatsoever of infinity; therefore it can remain undefined.

real analysis is not really the proper setting (pun intended) for infinity; it is cantor's set theory and the alephs.

but if this will no longer be about intervals, perhaps you should start a new thread or continue under "combinations" or some other thread you already started. i remember asking you to show where in cantor's proof it is wrong to show that P(N)>N and you never did. in my absolute infinity theory, partly inspired by you, i did show where it would go wrong: it would fail to be a contradiction using the extension of the subsets axiom in ternary logic. if you want to see how to correctly invent a crackpot theory, read my crackpot article. just notice the overall presentation starting with definitions. you for example, defined a double-simultaneous connection as a connection such that... but you never defined connection. you have to break every definition down into something already defined or you're building a whole new undefined concept; so far, the set is virtually the only undefined concept so to add a new undefined concept would take a lot of convincing in the sense that the theory would have to have a lot of merit and power. what can you do with the theory? if it's just to disprove cantor's diagonal argument or show that transfinite sets can't exist, then not only is that incorrect, it won't get off the ground. the only way those theories can be "wrong" is if you change the axioms. but this can only be done in a way that extends them, not the other direction. my modified subsets axioms, i believe, extends the usual subsets axiom.

you've asked us to show the flaws in your claims:
1. no definitions of key terms
2. lack of rigor.

if you claim that there is a bijection from N to R, then you have to specifiy what it is or else demonstrate it exists.

 

[Organic]

Hi phoenixthoth,


I try to explain a system which is multi-dimensional by nature, to persons who insist to translate it two 2 dimensional system.

Another example, I try to explain a colorful system by nature, to persons who insist to translate it two black and white system (Boolean Logic) Or greyscale system (Fuzzy Logic).

It simply can't be done.

What you write simply show me that I did not succeed to explain what is Complementary Logic.

Complementary Logic is first of all a paradigm changing in the question: "what is Mathematics?".

I am not talking here about some technical improvement, but on something that is changing math from its conceptual fundamental level, no less no more.

If you still trying to look at Complementary Logic through the Boolean or Fuzzy Logic eyes, then you cannot understand even one thing in Complementary Logic.

I hardly tried to open your eyes to Complementary Logic, but from your last post I realize that I did not succeed yet.

You still trying to find it under the spotlight of Boolean Logic and Euclidean Mathematics.

So, let me simply tell you that you will not find it there, again because we are dealing here with a paradigm change, no more, no less.


Yours,

Organic

 

[phoenixthoth]

i would guess that category theory is also a paradigm change, to give you an example, an abandoning of sets as the fundamental object. if you abandon logic, even fuzzy logic, then you can no longer use deduction:
(A&(A-->B)) --> B,
because that's a statement in binary logic, yet you still use deduction don't you? hence you're using binary logic to escape binary logic. this can be done with some delicacy. but to be a mathematical theory, there still needs to be definitions of terms (even category theory has definitions), a set as small as possible of undefined terms (preferably empty), and "rigor." one commonality in your articles is a very rapid jump signaled by your use of the word "therefore." this is not automatically a bad thing, but i would say that as it is, it is too rapid. several of your "therefores" are "non-sequitors," which means one of two things:
1. i don't understand the logic
2. the conclusion just don't follow from the premises.

keep trying and i will work with you to parse out the nonrigor and sift through it. but you have to at least try to incorperate my advice when i point to something.

just let me get one thing straight: what is your primary goal in five sentences or fewer? i can't argue with your goals.

now, after your primary goal has been stated, your thesis statement, give me an outline, not intended to prove it, of how you will accomplish your primary goal as briefly as you can.

we will talk about that.

then, and only then in my opinion, should we talk about the details and how to go about doing it. at that point, if and when we get there, give me small spurts of things to consider, rather than a ton of articles. if i want to referee your work, i will ask you a question about the first thing that seems incorrect or unclear; so it wouldn't help to send me pages of math if i will have a question with the second line. i think this is how we should carry on from this point. in other words, I've read your articles already so sending them again and saying "don't you see?" will not help move things foward. for now, just state your primary goal which has to do with noneuclidean mathematics, not to be confused with noneuclidean geometry i realize, and a very basic outline of how you will achieve that primary goal.

let me give you an example of what i want:
in my ternary universal set theory article, my primary goal is to axiomatize the universal set, the set of all sets, into existence.

that's it. that's my primary goal. one sentence.

steps:
1. use ternary logic to
2. extend the subsets and foundation axioms so that i can
3. axiomatize U into existence and
4. remove the usual problem, russell's paradox while
5. showing where the normal theory wouldn't apply with the extended subsets axiom.

this is all i want for now. no details. this will help me grasp what you are trying to do. then, after you have stated your primary goal and basic steps, we will go over the details line by line and work towards a rigorous theory.

 

[Organic]

My goal is to find a theory that can associate between at list two opposites.

And I want to reach that goal by using the simplest possible ways.

By simplest i mean maximum output out of minimum input, including what among them where output has the lowest possible entropy.

Also the theory has to include its developer and the development process as natural parts of it.

 

[phoenixthoth]

ok. thank you.

now i want to really understand what you're trying to say:

My goal is to find a theory that can associate between at list two opposites.

english question: do you mean "My goal is to find a theory that can associate between a list two opposite descriptions or characterisitics?"

by "list," do you mean "set" or not? for example, if C and D are two categories, do you want to apply your goal to the list {C,D} or only when C and D are sets or something else? and by "opposite," do you also mean "complimentary?"

once you answer these questions, also i need to point out that "by using the simplest possible means" is actually a part of the goal, i think. in other words, it's not an outline of the steps you will take to assiciate a list with two opposites. i don't want details yet, just some outline more specific than "by simplest possible means" and more general than all the details.

a side question not of much importance now is this: do you intend to prove that your means are the simplest? i ask because that could be very difficult to do.

also, can you give me an idea as to what kind of two opposites you mean and in what sense they are opposites?

and, finally, the more you give me, the more i or someone might be able to tell you if this, or something similar, has been done before. if something similar has been done, it will help you immensely to become familiar with it and go where it does not.

 

[Organic]

Forgive me about my English, the right one is:

My goal is to find a theory that can associate between at least two opposites.

And I want to reach that goal by using the simplest possible ways.

By simplest i mean maximum output out of minimum input, including what among them where output has the lowest possible entropy.

Also the theory has to include its developer and the development process as natural parts of it.

 

[phoenixthoth]

what kinds of opposites?

opposite numbers?

opposite sets?

opposite elements in a group? (group theory is a fairly general setting for looking at structures with opposites.)

something else?

in category theory, there is a concept of opposite category...

 

[Organic]

Dear phoenixthoth,

Please read all of this, thank you:

My goal is to find a theory that can associate between at least two opposites.

And I want to reach that goal by using the simplest possible ways.

By simplest i mean maximum output out of minimum input, including what among them where output has the lowest possible entropy.

Also the theory has to include its developer and the development process as natural parts of it.


Step 1:
The first thing is to find the most general concept to start with, so we choose information.


Step 2:
Then we choose the limits of any information system, which can be defined as at least to opposites, so we choose No information, Total Information.


Step 3:
we are useing these limits as the contents of two opposite set's types, where the set's idea is a tool that we call it General Information Framework(GIF), which is the model or the platform that we use to explore our ideas.
http://www.geocities.com/complementarytheory/GIF.pdf


Step 4:
Now we look for simplicity by using the symmetry concept as the balance between {} and {__}.

{} and {__} are the unreachable limits of our system, which is a fading transitions between these limits, and only the products of the fading transitions can be explored as meaningful Information. By using the open interval idea the meaningful information exists in ({},{__}).


Step 5:
The first symmetry break is a model of infinitely many empty information cells existing upon infinitely many scales, where cells size expending (aspirating to) {__} an shrinking (aspirating to) {}.

The second symmetry break is to "left-right|right-left" symmetry by fill the empty information cells with the minimum necessary information that can break the symmetry.
http://www.geocities.com/complementarytheory/LIM.pdf

The third symmetry break is the floating point system that splitting the Information cells to two opposite directions, integer and fractional.

By using Riemann's Ball we find the full symmetry between integer an fractional sides.
http://www.geocities.com/complementarytheory/RiemannsBall.pdf

Also By using Riemann's Ball we find the difference between actual infinity and potential infinity.

Also we find that potential infinity can never be completed and this property do not give us any possibility to use the words 'all' or 'complete' when we explore it.
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf


Step 6:
With this knowledge in our hands, we realize that The Number System of Standard Mathematics is some arbitrary broken symmetry between integer and fractional sides, where fractional side is full, but the integer side includes only numbers with finite length.


Step 7:
At this stage we stop continuing are main program to show the problems that we have found in Standard Mathematics from our new point of view.

The problems of Standard Math that we have found:

1) It is not aware that it is based on some arbitrary broken symmetry between its integer side and its fractional side.

2) It does not distinguish between potential infinity and actual infinity, and therefore using words like 'all' and 'complete' related to potential infinity.

The result of this mistake is the transfinite universes, which is nothing but a "full gas in neutral".

3) It is based on very week methods like Boolean Logic (black XOR white system) OR Fuzzy Logic (Grayscale system).

4) Standard Math is based on the quantity concept, therefore a lot of very interesting information are out of its scope.

5) There is no difference between multiplication and addition.

7) There is no general definition to the Number concept.

8) Concepts like redundancy and uncertainty are not fundamental concepts.


Step 8:
We continue our main program to find the logic system that will be the base of our system. The result is what we call Complementary Logic.
http://www.geocities.com/complementarytheory/CompLogic.pdf
http://www.geocities.com/complementarytheory/4BPM.pdf


Step 9:
By using Complementary Logic, we reexamine the concept of The Number and starting to make the first general sketches of Complementary Logic Number System.
http://www.geocities.com/complementarytheory/AHA.pdf
http://www.geocities.com/complementarytheory/Everything.pdf
http://www.geocities.com/complementarytheory/ASPIRATING.pdf
http://www.geocities.com/complementarytheory/ET.pdf
http://www.geocities.com/complementarytheory/CATheory.pdf
In these sketches we can clearly show that Complementary Logic is based on stapes 1 to 5 and fix the problems that have been found in steps 6 and 7.

We also think that Complementary Logic can be very useful in Quantum Mechanics (the micro level) and also it can be used as a very good basis for modals that dealing with static an dynamic complexity (in mid and macro levels).

The reason that we think so, is because Complementary Logic using in a coherent way concepts like Information's clarity-degree, Symmetry-degree, redundancy, uncertainty, Information structure and quantity.


Because Complementary Logic is a "colorful" system, we try to explore its frontiers by checking subjects like "our abilities to count" and more subjects that are connected to our own cybernetic systems.
http://www.geocities.com/complementarytheory/count.pdf
http://www.geocities.com/complementarytheory/RealModel.pdf
http://www.geocities.com/complementarytheory/CK.pdf


Step 10:
We examine the connections between Complementary Logic ,Moral and Art.
http://www.geocities.com/complementarytheory/Moral.pdf
http://www.geocities.com/complementarytheory/O-Harp.pdf


Step 11:
We hope for some help.
http://www.geocities.com/complementarytheory/HelpIsNeeded.pdf



Yours,

Organic

 

[phoenixthoth]

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.

i don't really know much about researching nonmathematical theories but i would imagine the first thing you should do is define information and specify the scope of your theory. is it going to apply to all kinds of information?

you may want to run a search for "information theory."

 

[Organic]

Please don't say that.

Dont you realize that 'Information theory' of today is a MAthematical theory?

Did you read steps 6 and 7?

If you read them then how can you say that my goal is not also going through Math?

 

[phoenixthoth]

i know very little about information theory. i think that generally, general information should be more general than math and though it can be applied to any kind of information, including math, it's not just a math theory. it can use math in it, but if it's about information in general, then it may not be a math theory. either way, you haven't addressed the other point i wrote about which was that in order to talk about information, you have to either say what its definition is or why we should allow it to remain undefined.

 

[Organic]

What is undefined?

 

[phoenixthoth]

if I'm not mistaken, sets and points from geometry are undefined. in other contexts, points are defined.

 

[Organic]

Dear phoenixthoth,



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

Thank you.

Yours,

Orgainc

 

[phoenixthoth]

information, for one thing.

 

[Organic]

Why do I have to define it?

 

[phoenixthoth]

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 &isin; though &isin; 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.

 

[Organic]

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

 

[Hurkyl]

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.

 

[russ_watters]

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!

 

[Organic]

Hi Hurkyl,

Please read my post (with the steps) and also my post about x-model and x, and see for yourself how Information concept is extremely productive concept in my work.

If you don't think so, please tell me why?


I'd like to know what do you have to say about Complementary Logic:

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

http://www.geocities.com/complementarytheory/4BPM.pdf