Limit and Infinity

[Guest Doron Shadmi]

Not correct,

 

Analogies are also used to explain some idea in a different way.

[Sayonara]

Not correct' date='

Analogies are also used to explain some idea in a different way.[/quote']

Gah, you replied while I was editing the post.

 

See additional bits.

[Guest Doron Shadmi]

Here is a list of my axioms, which are related to R:

 

A definition for a point:

A singleton set p that can be defined only by tautology ('='), where p has no internal parts.

 

A definition for an interval (segment):

A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.

 

The axiom of independency:

p and s cannot be defined by each other.

 

The axiom of complementarity:

p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).

 

The axiom of minimal structure:

Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.

 

The axiom of duality(*):

Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.

 

The axiom of completeness:

A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.

 

The Axiom of the unreachable weak limit:

No input can be found by {} which stands for Emptiness.

 

The Axiom of the unreachable strong limit:

No input can be found by {__} which stands for Fullness.

 

The Axiom of potentiality:

p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}.

 

The Axiom of phase transition:

a) There is no Urelement between {} and {.}.

b) There is no Urelement between {.} and {._.}.

c) There is no Urelement between {._.} and {__}.

 

Urelement (http://mathworld.wolfram.com/Urelement.html).

 

 

The axiom of abstract/representation relations:

There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.

 

 

(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.

 

Tautology means x is itself or x=x.

 

Singleton set is http://mathworld.wolfram.com/SingletonSet.html .

 

Multiset is http://mathworld.wolfram.com/Multiset.html .

 

Set is http://mathworld.wolfram.com/Set.html .

 

(By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )

 

More detailes about my work can be found in: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

 

 

 

The Axiom of the paradigm-shift:

 

Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.

 

 

 

Let us stop here to get your remarks.

[Guest Doron Shadmi]

Fist let us write again our last post:

 

Tautology:

x implies x (An example: suppose Paul is not lying. Whoever is not lying, is telling the truth Therefore, Paul is telling the truth) http://en.wikipedia.org/wiki/Tautology.

(tautology is also known as the opposite of a contradiction).

 

(EDIT: instead of the above definition, I change Tautology to: The identity of a thing to itself.

 

(It means that in this framework we do not need 'if, then' proposition in order to define the self existence of some element)

 

 

 

Set:

A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is also ignored.

 

Multiset:

A set-like object in which order is ignored, but multiplicity is explicitly significant.

 

Singleton set:

A set having exactly one element a. A singleton set is denoted by {a} and is the simplest example of a nonempty set.

 

Urelement:(no internal parts)

An urelement contains no elements, belongs to some set, and is not identical with the empty set http://mathworld.wolfram.com/Urelement.html.

 

{.} is both a Singleton set and a Urelement.

 

A definition for a point:

A singleton set p that can be defined only(*) by tautology* ('='), where p has no internal parts.

 

(*) only by tautology means: the minimal possible existence of a non-empty set.

 

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

 

Now let us move to the next step in order to define what is a number in my system.

 

First let us examine a well-known relation between mathematical objects and their relations.

 

=>> is ‘represented by’

 

|{}|=>>0 ; |{{}}|=>>|{0}|=>>1 ; |{{},{{}}}|=>>|{0,{0}}|=>>|{0,1}|=>>2 ;

 

|{{},{{},{{}}}}|=>>|{0,{0,{0}}}|=>>|{0,1,2}|=>>3 ; …

 

A definition for an interval (segment):

A singleton set s that can be defined by tautology* ('=') and ('<' or '>'), where s has no internal parts.

 

(Sign '<' means that we look at the segment from left to the right.

Sign '>' means that we look at the segment from right to the left.

When both '<' , '>' are used then we have a directionless segment.)

 

By the definition of a segment we get {._.}, which is the indivisible singleton set that exists between any two {.}.

Now we have the minimal building-blocks that allows us to define the standard R members.

 

(more detailed explanation of the first two definitions:

 

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

Remark:

In Standard Math we had to write:

 

Point proposition:

If a content of a set is a singleton and a urelement and has no directions, then it is a point.

 

Segment propositon:

If a content of a set is a singleton and a urelement and also has directions, then it is a segment.

 

But since in this framework a Tautology is the identity of a thing to itself,

we do not need an 'if, then' proposition for tautology.

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

 

 

Now, let us examine these first two definitions by using the symmetry concept:

 

1) {.} content is the most symmetrical (the most "tight" on itself) content of a non-empty set.

 

It means that the direction concept does not exist yet and '.' can be defined only by '=' (tautology), which is the identity of '.' to itself.

 

2) {._.} content is the first content that "breaks" the most "tight" symmetry of {.} content, and now in addition to '=' by tautology (which is the identity of '._.' to itself) we have for the first time an existing direction '<' left-right, '>' right-left and also '<>' no-direction, which is different from the most "tight" non-empty element '.'

 

In short, by these two first definitions we get the different non-empty and indivisible contents '.'(a point) or '_'(a segment) .

 

In short, in both definitions (of {.} or {._.}) the conclusion cannot be different from the premise (http://mathworld.wolfram.com/Tautology.html)

 

*A statement for a point:

A point is an indivisible finite content of a non-empty set that has no directions.

 

*A statement for a segment:

A segment is an indivisible finite content of a non-empty set that also has directions. )

 

 

The axiom of independency:

p and s cannot be defined by each other.

 

By the above axiom {.} and {._.} are independed building blocks.

 

The axiom of complementarity:

p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).

 

By the above axiom we define the basic property of the middle domain between {.} and {._.}

 

The axiom of minimal structure:

Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.

 

The above axiom allows us to:

 

1) To define the internal structure of standard R members.

2) To define the internal structures of my new number system.

 

The axiom of duality(*):

Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.

 

The above axiom allows us to construct a collection of R members and also a collection of my new number system.

 

First, let us see how we use my method to construct a collection of R members.

 

 

R members are constructed like this:

 

1) First let us examine how we represent a number by my system:

 

=>> is ‘represented by’

 

a) |{}|=>>0

 

b) There is 1-1 and onto between ‘0’ and the left point of {._.} and we get {‘0’_.}

 

c) |{{}}|=>>|{0}|=>>1

 

e) There is 1-1 and onto between ‘1’ and the right point of {._.} and we get {‘0’_’1’}

 

In short, {.} is the initial place of R collection, which is represented by ‘0’, where {‘0’_.} is the initial place of the second place of R collection, which is represented by ‘1’, and we get our first two must-have building-blocks of R collection.

 

 

2) When we get {‘0’_’1’} we have our two must-have numbers, which are ‘0’ and _’1’.

 

Be aware that ‘0’ is the representation of {.} where ‘1’ is the representation of {._.}.

 

 

3) If we get {.}_AND_{._.}, then and only then we have the minimal must-have information to construct the entire R collection because:

 

a) We have ‘0’ AND _’1’ that give us the to basic scale factors 0 and _1.

 

b) We also have our initial domain _1, which standing in the basis of any arbitrary scale factor that is determined by the ratio between the initial domain _1 and another segment that is smaller or bigger than the initial domain _1 , for example:

0 = .

1 = 0[color=Blue]______1[/color]

2 = 0[color=DarkRed]____________2[/color]  

3 = 0[color=Green]___________________3[/color]

.5 = 0[color=Red]__.5[/color]    

pi = 0[color=Magenta]______________________pi[/color]

 

The negative numbers are the left mirror image of the above numbers.

 

 

There is no division in my number system because both {.} and {._.} are indivisible by definition.

 

In short, any segment is an independent element, that clearly can be shown in the above 2-D representation.

 

If we use a 1-D representation, we get the standard Real-line representation, but then we can understand that division is only an illusion of an overlap of independent elements when they are put on top of each other in a 1-D representation, for example:

0[color=Red]__.5[/color] [color=Blue]__1[/color][color=DarkRed]_____2[/color][color=Green]_____3[/color][color=Magenta]__pi[/color]

 

 

(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.

 

 

Since in my system nothing is divisible, then '/' stands for a ratio between at least any given two (indivisible) numbers.

 

 

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

 

Let us stop here (before we continue to my new number system) to get your remarks.

[Guest Doron Shadmi]

Since I got no remarks so far let us continue to my new number system.

 

As I showed in the previous post, each number which is not 0 is at least a representation of {.}_AND_{._.}.

 

Also each {._.} has 3 basic states which are: '<' for left-right direction, '>' for right-left direction, '<>' for no-direction.

 

Let us write again The axiom of minimal structure:

Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.

 

Let us examie this part: "...where p_AND_s is at least Multiset_AND_Set."

 

We know that the elements of a non-empty "normal" set, which its cardinality > 1,

cannot be identical.

 

But the elements of a multiset, which its cardinality > 1, can be identical.

 

If p_AND_s is at least Multiset_AND_Set, then any given element which its cardinality > 1 has several variations that can be found between '<>' to '<' or '>'.

 

For example, let us represent the variations of cardinals(*) 2,3,4:

 

Let Redundancy be more then one copy of the same value can be found.

 

Let Uncertainty be more than one unique value can be found.

 

Let XOR be #

 

Let a=0,b=1,c=2,d=3 then we get:

   b  b                                        
   #  #                                        
  {a, a,  {a, b}                               
   .  .    .  .                                
   |  |    |  |                                
   |__|_   |__|                                
   |       |                                   

   {x,x}  {{x},x}                              





    c  c  c                                    
    #  #  #                                    
    b  b  b          b  b                      
    #  #  #          #  #                      
   {a, a, a,}       {a, a, c}       {a, b, b}  
    .  .  .          .  .  .         .  .  .   
    |  |  |          |  |  |         |  |  |   
    |  |  |          |__|_ |         |__|_ |   
    |  |  |          |     |         |     |   
    |__|__|_         |_____|         |_____|   
    |                |               |         
    |                |               |         
   {{x,x,x}         {{x,x},x}       {{x},x},x}


               [color=Red][b]Uncertainty[/b][/color]
 <-[b][color=Blue]Redundancy[/color][/b]->^
   d  d  d  d  |
   #  #  #  #  |
   c  c  c  c  |
   #  #  #  #  |
   b  b  b  b  |
   #  #  #  #  |
  {a, a, a, a} V   {a, b, c, d}
   .  .  .  .       .  .  .  .
   |  |  |  |       |  |  |  |
   |  |  |  |       |__|  |  |
   |  |  |  |       |     |  | <--(Standard Math language uses only 
   |  |  |  |       |_____|  |     this no-redundancy_
   |  |  |  |       |        |     no-uncertainty_symmetry)
   |__|__|__|_      |________|
   |                |
   ={x,x,x,x}       ={{{{x},x},x},x}



============>>>

               [color=Red][b]Uncertainty[/b][/color]
 <-[b][color=Blue]Redundancy[/color][/b]->^
   d  d  d  d  |          d  d             d  d
   #  #  #  #  |          #  #             #  #        
   c  c  c  c  |          c  c             c  c
   #  #  #  #  |          #  #             #  #   
   b  b  b  b  |    b  b  b  b             b  b       b  b  b  b
   #  #  #  #  |    #  #  #  #             #  #       #  #  #  #   
  {a, a, a, a} V   {a, a, a, a}     {a, b, a, a}     {a, a, a, a}
   .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
   |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
   |  |  |  |       |__|_ |  |       |__|  |  |       |__|_ |__|_
   |  |  |  |       |     |  |       |     |  |       |     |
   |  |  |  |       |     |  |       |     |  |       |     |
   |  |  |  |       |     |  |       |     |  |       |     |
   |__|__|__|_      |_____|__|_      |_____|__|_      |_____|____
   |                |                |                |
   {x,x,x,x}        {{x,x},x,x}      {{{x},x},x,x}    {{x,x},{x,x}}     

                                     c  c  c
                                     #  #  #      
         b  b                        b  b  b          b  b
         #  #                        #  #  #          #  #         
  {a, b, a, a}     {a, b, a, b}     {a, a, a, d}     {a, a, c, d}
   .  .  .  .       .  .  .  .       .  .  .  .       .  .  .  .
   |  |  |  |       |  |  |  |       |  |  |  |       |  |  |  |
   |__|  |__|_      |__|  |__|       |  |  |  |       |__|_ |  |
   |     |          |     |          |  |  |  |       |     |  |
   |     |          |     |          |__|__|_ |       |_____|  |
   |     |          |     |          |        |       |        |
   |_____|____      |_____|____      |________|       |________|
   |                |                |                |
   {{{x},x},{x,x}} {{{x},x},{{x},x}} {{x,x,x},x}      {{{x,x},x},x}

   a, b, c, d}
   .  .  .  .
   |  |  |  |
   |__|  |  |
   |     |  | <--(Standard Math language uses only this
   |_____|  |     no-redundancy_no-uncertainty_symmetry)
   |        |
   |________|
   |    
   {{{{x},x},x},x}

Also please pay attantion that the last form is the standard R members 0,1,2,3:

 

0 = .

1 = 0[color=Blue]______1[/color]

2 = 0[color=DarkRed]____________2[/color]  

3 = 0[color=Green]___________________3[/color]

And the standrard [b]R[/b] is nothing but of the above 2-D representation 
in a 1-D representation:

0[color=Blue]______1[/color][color=DarkRed]______2[/color][color=Green]______3[/color]

And because no R member is both Multiset_AND_Set, I call it: The "shadow" of my new number system.

 

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

 

(*) Please pay attention that we are not talking about the natural numbers 2,3,4 but the cardinals 2,3,4.

 

It means that our Organic Natural Numbers are actually a general representation of information-trees, where any finite quantity of names of R members can be described by them, for example:

 

Instead of a=0,b=1,c=2,d=3 we can use a=0,b=.5,c=3,d=pi and then we use the same information-trees above.

 

 

I called these general information-trees 'Organic Natural Numbers' because:

 

1) These information-trees of cardinals are always having a structure, which is based on N members.

2) They can be used as natural (not forced) and general representation for any interaction between complementary states, which simultaneously preventing/defining their middle domain.

 

3) Because no R member is divisible by my system, it has its own organic (complete) unique and independent self existence.

[Dave]

I need some time to read through this and deliver you my opinion, since it's all a little hazy atm. Will post again later.

[matt grime]

For other forum users:

 

Doron is a notorious poster on many forums. In physicsforums.com he has posted under the psuedonyms Lama, Organic, www, and many more (each mambership was blocked from posting in the mainstream forums), and all threads he started there were moved to Theory Development, including threads he hijacked.

 

He adopts the terminology of mathematical discourse, and then refuses to use the objects according to their definitions, and generally refuses for a long time to acknowledge this or offer his definitions. In some cases he adamantly insists he is using the terms correctly (his abuse of "tautology" springs to mind when he kept posting a wikipedia definition as if that were his usage, when eventually he admitted he didn't mean it in that sense).

 

He will generally not answer questions directly and will only post more garbage in response. If you wish to keep this forum to mainstream ideas then you probably ought to lock these threads

[Guest Doron Shadmi]

For Other Uesers:

 

Matt Grime is a strange fallow that has an obsession to follow after me over the internet (to say the truth, I invited him 2 years ago to meet me in some forum) to and give advices to other users in other forums, about my threads.

 

I learned a lot during our dialogs, and used what I have learned in order to develop my system, but unfortunately Matt cannot understand the included-middle reasoning because he insists to force an 0_XOR_1 reasoning on it, which cause him not to understand my system.

 

I think that each person can deside by himself what he think on any subject that can be found in this forum or any other place over the internet.

 

So, if to speack more to the point, my axiomatic system, which I call it "Complementary Set Theory" can be found in: http://www.geocities.com/complementarytheory/My-first-axioms.pdf

 

And some example of arithmetics, which is based on this system can be found in: http://www.scienceforums.net/forums/showpost.php?p=76489&postcount=20

[Sayonara]

Matt Grime is a strange fallow that have an obsesion to follow after me over the internet and give advices to other users in other forums, about my threads.

He joined here 5 months before you.

[fuhrerkeebs]

Hahahahahahahahahaha...that must be on the best things I've read in my life...

[Guest Doron Shadmi]

He joined here 5 months before you.

Both of us are participaters of Math forums over the internet, some time I register before him and sometime he register before me, but the one how writes general messages for other users about my work is Matt.

[fuhrerkeebs]

But, back to the topic at hand...Doron Shadmi, if your theories are so correct then why don't you get them published in a reputable mathematics journal? Your previous posts have implied that you have had these ideas for 4 years--possible longer, and if these ideas of your are truely brilliant and correct, the what are you doing still posting on message boards?

[Guest Doron Shadmi]

But' date=' back to the topic at hand...Doron Shadmi, if your theories are so correct then why don't you get them published in a reputable mathematics journal? Your previous posts have implied that you have had these ideas for 4 years--possible longer, and if these ideas of your are truely brilliant and correct, the what are you doing still posting on message boards?

[/quote']

Fist dear fuhrerkeebs, thank you for this question.

 

Since I tried many times to publish my works in many Mathematical journals, I can use my own experience in this case and say:

 

1) Since my ideas cannot be understood without a paradigm-shift in the mind of the reader, there are very low chances that they can be understood by professional Mathematicians from first or second look, and as you know, if some referentor cannot understand your work in the first reading, he will reject it.

 

2) All my works where rejected during the referee process, and I never got a detailed explanation why they where rejected.

 

3) Another reason is that since my ideas cannot fully described by the standard formal common methods, I had no choice but to invent a lot of fundamental non-standard tools in order to explain my non-standard Ideas, which is of course connected to what I wrote in (1).

 

4) You have to understand that what I suggest, (which is an inherent organic connection between our morality side -not in the biblical meaning but as the finest reasoning of “balance with nature” in order to survive our own power- and our technological skills) is so strange for any scholar which is education is based on “self-evident” dichotomy between what he thinks as moral and what he thinks as logical, that there is no chance that my work will be published in any Mathematical Journal, as long as no paradigm-shift first takes place, in the mind of the scholar.

 

5) I have to admit that my work is all the time under a development process, so things become more and more tuned and sharp, and I hope that in the near future, they will be sharp and clear enough, so even professional Mathematicians will be able to accept them.

[DoorNumber1]

I'll admit in advance to being merely a college student without that much mathematical training (I tutor math but only through calc/linear algebra, and I had to teach myself most of calculus because I came from a really really shitty public school) and I've done a bit more number stuff in Comp Sci theory courses but that's it. I don't have a PhD (yet) so I may not be able to put my thoughts into the most precise and bulletproof forms. I'm not even formally studying math; my area is AI because I love studying how people think. So please, bear with me.

 

Your first post stated that the reason you disagree with the current theory of limits is that

No collection of infinitely many elements that can be found in infinitely many different scales, can have any link with some given constant, in such a way that it will be considered as a limit of the discussed collection.

 

In short, Nothing is approaching from the collection to the given constant, as can be clearly seen in my sports car analogy at page 2 of http://www.geocities.com/complementarytheory/ed.pdf

 

Take each separate position of the car, then compare it to zero state and you can clearly see that nothing is approaching to zero state.

 

This fundamental point is one that I disagree with, and it seems as though we understand the word "approaching" in different manners. I read your pdfs and looked at the car example, and it's a great example that explains perfectly well the standard concept of a limit. You say that nothing is approaching the zero state, yet obviously every new point the car is at is closer to the zero state than the last point. That qualifies as approaching zero. Of course, that also qualifies as approaching -1, or any number beyond zero.

 

If you would've extended your car example and drawn even more small cars you would've eventually been making cars so small and so close to zero that they were indistinguishable from it (but would never PASS zero, which is what rules out -1 or anything else). Of course "indistinguishable" in this sense is a matter of screen resolution. But no matter how far you zoom in, you can always repeat that process in which you continue drawing more cars and they will always hit a point at which, eventually, they become indistinguishable from zero. Think of the screen resolution in this case as your epsillon. So how do you, in your reasoning, come to the conclusion that "you can clearly see that nothing is approaching to zero state" and that "Therefore no such constant can be considered as a limit of the above collection"??? Clearly nothing is reaching it, but EVERYTHING is approaching it. That constant is fundamental to the behavior of the series, and to state otherwise requires some serious, heavy duty revelations or an order of magnitude beyond what you have given.

 

Later, drawing on this "conclusion", you state that

It means that if the described collection is A and the limit is B, then the connection between A,B cannot be anything but A_XOR_B.

which I completely disagree with. You're essentially saying that there is no truth beyond the set A that's relevant to A. There is no constant B that holds any relationship to it other than an exclusive or relationship (A or B, but never both A and B... we're using the same definition of xor, right?). You can try, very weakly, to back that up if you were able to prove that the current concept of a limit is incorrect but since you haven't I'm afraid you've failed in this sense as well. You have yet to prove that, for a given set A, there is no number outside of the set that is fundamental to the description of A (as a limit or boundary would be). And if we can describe the behavior of A in terms of some constant B, then again all of your conclusions must be rethought.

 

Then you LATER try to prove that all of our mathematical proofs concerning infinite sets is incorrect based on the first two conclusions that you failed to prove. You attack the ability to use the any/all reasoning here:

Let us examine the universal quantification 'all'.

 

As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements, even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements is an inconsistent idea.

but there's no inconsistency. The inconsistency is how YOU are defining "all" in this sense. You are saying that using the word "all" means that we're assuming everything is inside our domain?... that is completely and utterly false. All is usually describing everything that IS ALREADY inside some defined domain that we give, not somehow pulling everything into it! To give you some credit, I'll assume that you simply worded that incorrectly and you actually had some reasoning behind saying that. Or maybe I just require a paradigm shift or something like that.

 

and further, you say that:

Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.

but this relies COMPLETELY on you having supposedly proven that you can't bind an infinite set by constants (due to the failure of our concept of a limit) which you completely failed to do. You seem to be saying that since you've disproved our concept of a limit and thus taken away our ability to define a boundary to an infinite set, the numbers 0 and 1 are meaningless because to actually touch them our set would have to somehow fundamentally change what it is and undergo a "phase transition" to include them. Dude, do I have to point out how flawed this logic is? If you want, define [0, 1] as the intersection of the sets

 

{1, 0.1, 0.01, 0.001,.... } and

{1 - 1, 1 - 0.1, 1 - 0.01, 1 - 0.001, ....}

k?

 

You later, further defending your theory that we can define no limit, state that

You can show that 1 is really the limit of sequence 0.9,0.99,0.999,0.9999,0.99999,.... , only if you can prove that there is a smooth link (without "leaps") between this sequence and 1, which is not based on {0.9,0.99,0.999,0.9999,0.99999,.... }_XOR_{1} connection.

I happily take the challenge because it's such an easy one, but I refuse to work within the confines of your "smooth link without leaps" framework. I will simply show you that the number 1 is essential to the definition of this series, and thus even though it's not included in the series it's still a fundamental part of it and can be associated with it. First, with whatever theory of numbers you may decide to hold, do you agree that a given number has more than one representation? That 4 is completely equivalent to 2 + 2?

If you do believe this, then I suggest that you simply rewrite the series, instead of being

{0.9, 0.99, 0.999, 0.9999, ...} as

{1 - 0.1, 1 - 0.01, 1 - 0.001, 1 - 0.0001, ...}

which defines the the elements by the difference between them and 1.

 

It's an equivalent set, but in this case it's obvious that 1 is fundamental in the rules that govern the series. Start with 1. Subtract off smaller and smaller and smaller amounts. Where do you end up at? 1. That's the limit of the series. Since the difference between 1 and the current number, the part that's changing, is getting arbitrarily close to 0, the current number must be growing arbitrarily close to 1. Is this so hard to see? Whether we choose to call this fundamental constant 1, a "limit," "propety," or a freaking "dinosaur" is a matter of semantics, but the fact that it's related by something other than an XOR relationship should be obvious.

 

But no, you refuse to see it and seem to claim that

"getting infinitely close" is not reasonable, because nothing can be closer to something when something is some constant and the "closer" element is one of infinitely many elements that can be found in infinitely many different scales.

WHAT? You're claiming that, just because a number in a series might not ever reach the limit of the series that means that other numbers in the series can't grow arbitrarily closer to that limit? Uh, using my definition of "closeness" as the difference between two numbers |A-B| your logic seems absurd. Is it even possible, with whatever your definition is, for one number to be smaller than another? If so, your claim is groundless. If not, you don't have a reasonable definition of "close".

 

When you're unable to produce anything that seems to back up your claims except an unfounded criticism of the current system, you just accuse us all of needing a paradigm shift or, essentially, being squares living and thinking in flatland and waiting for you, oh mighty Sphere, to show us the way to reality.

 

You claim that you bring is a new way to view numbers that will show deeper relationships between them. Fine. Maybe you do have some valid ideas that can add to the field of mathematics. But for goodness sake, just present those in a clear, backed up manner and stop unsuccessfully trying to topple all of current mathematics. Then maybe people would buy your ideas.

[Guest Doron Shadmi]

Dear DoorNumber1,

 

First, Thank you very much for your detailed post about my work.

 

You say that nothing is approaching the zero state' date=' yet obviously every new point the car is at is closer to the zero state than the last point

[/quote']

You have to look at this example from two different points of view.

 

1) If you are a point then nothing is come closer to you simply because no segment (which exists between at least two points) can be defined by a single point.

 

Therefore there cannot be but a phase transition between being a point and being a segment, and this salutation is totally equivalent to the relations between an empty set and a non-empty set.

 

If we speak in terms of empty/non-empty sets, then there is no difference between a non-empty set with cardinality 2^aleph0 and a non-empty set of cardinality 1, when you look at both of them from the empty set point of view.

2) Only After we understand (1) we can clearly understand that on collection of infinitely many elements can be considered as a complete collection, and the impotent insight of it (which is totally different from the standard point of view about infinity) is that no universal quantification (the term ‘all’) can be related to a collection of infinitely many elements.

 

3) Things become closer if and only if we deal with a collection of finitely many elements.

If you would've extended your car example and drawn even more small cars you would've eventually been making cars so small and so close to zero that they were indistinguishable from it

Not correct, if I am looking at the car from zero state, I can distinguish between me (as zero state) and any given car that obviously is not me (zero).

You have yet to prove that, for a given set A, there is no number outside of the set that is fundamental to the description of A (as a limit or boundary would be).

1) Please look at the first part of this post.

 

2) Please tell me what is the deep meaning that stands in the basis of your words “outside of the set”?

The inconsistency is how YOU are defining "all" in this sense

Please look at the first part of this post.

do you agree that a given number has more than one representation?

No I do not agree with this, because in my theory any representation is a unique structural/quantitative information form, and your point of view is only a quantitative point of view on the concept of a Number.

 

If you understand the structural/quantitative approach, then and only then you can say meaningful things about my framework.

 

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

 

In short, you failed at this stage to do the nessacary paradigm-shift in your mind, in order to understan my framework, for example:

 

If you read the axioms of "Complementary Set Theory" ( http://www.geocities.com/complementarytheory/My-first-axioms.pdf ), which are the axioms of my framework, you will see, for example, that my interpretation to Tautology is totally different form the standard interpretation.

 

It means that in my framework the existence of an element or a collection of elements, is totally not depend on anything, which is beyond their own self-existence domain.

 

 

Let me say more:

 

My goal is to fulfill the dream of the great mathematician Gottfried Wilhelm Leibniz ( http://www.andrews.edu/~calkins/math/biograph/bioleib.htm )

 

Actually my number system ( which some arithmetic of it can be found in http://www.scienceforums.net/forums/showpost.php?p=76489&postcount=20 ) is the fulfillment of Leibniz's Monads ( http://www.angelfire.com/md2/timewarp/leibniz.html ), and beyond it.

[DoorNumber1]

1) If you are a point then nothing is come closer to you simply because no segment (which exists between at least two points) can be defined by a single point.

Okay, again I disagree with you right here. Whether it's because I require a "paradigm shift" or whatever, I do. You're basing the truth of that last statement on the idea that no segment can be defined by a single point... who's trying to? That's obvious. I'm saying that you can test the length of any segment between that one point (your reference point) and another. And yes, if those segments are getting smaller, the second point is drawing closer to the first. To deny that is to say that there is no such thing as distance or space.

 

That's what the concept of a limit is based upon, and I'll assume you understand the current concept of a limit at an intuitive level (although maybe not if you disagree with its truth and usefulness so much).

 

3) Things become closer if and only if we deal with a collection of finitely many elements.

 

If you would've extended your car example and drawn even more small cars you would've eventually been making cars so small and so close to zero that they were indistinguishable from it

 

Not correct, if I am looking at the car from zero state, I can distinguish between me (as zero state) and any given car that obviously is not me (zero).

be careful. I try my best not to quote you out of context so don't do it to me. I qualified my definition of "indistinguishable" in the very next sentence in terms of resolution, which itself is directly analogous to the concept of using an arbitrary e. The point it, to any possible degree of accuracy that you choose (a millionth, billionth, trillionth? keep going) you can show that the series, if run long enough, passes your threshold and continues to get closer. I never claimed that they were the same number, just that you can get close enough that they're virtually indistinguishable. The fact that the number 1 can look at the number 0.999999999999999999 and say "you're not quote me" doesn't change the significance of the number 1 to the number 0.999999999999999999.

 

Even if you choose to represent a number in a different manner that allows for a concept of closeness to be defined only on a finite set, the point is that you can always take a finite approximation of an infinite set produced by a series (as long as you want to run it) and apply your concept of "close" to it.

Screw it, let's use the {0.9, 0.99, 0.999, 0.9999, ...} example.

 

You will notice that, as the finite set you choose gets larger and larger (and therefore a better and better approximation to the infinite one) that the numbers in your finite set do grow close to the number one. And since you can't deal with an actual infinity you can at least conclude that since the closer we come to it the closer the numbers pull toward 1, 1 must be the limit of the series that produced the set.

 

Current math can't really deal with infinities, but calculus is based upon the current definition of a limit which itself it tied into infinity. And guess what? Calculus works. It produces answers to real world problems that simple aren't solvable without an ability to deal with those concepts. If our framework is so inherently flawed, then why does it produce answers that work in the real world? After all, the real world IS the final test of truth.

 

I'm not going to keep this up forever, but I genuinely do want to understand where you're coming from with your idea. But before I can, you have to convince me that my reasoning is flawed and you have yet to do that (or it seems, from the previous posts, to convince anybody else). You create tons of terms and repurpose common words and invent new logic systems to create your framework, but have you considered that if it were really true you should be able to explain it in layman terms to anybody? The hallmark of most big breakthroughs is that they're the kind of thing that makes you slap your head and go "Why didn't I see that!?!?" once it's presented to you with all of its supporting facts. And they also have the ability to be explained to arbitrary degrees of precision to convince a really well educated crowd or just convey the concept to the common man. Your ideas seem to lack that quality.

 

Keep in mind that I am impressed by the time and thought you put into your theories and I think that maybe you're on to something. Any criticism I give is meant to be goodhearted and I hope it's interpreted in that manner. I think that sometimes a single person can have insight that can potentially shift or add to the world's understanding, but you also have to ask yourself this question: Is it more likely that I'm wrong or THE REST OF THE ENTIRE WORLD is wrong? I apply that simple question to myself all of the time and even if I stick with my opinions, which I often do, I at least try to give others the benefit of the doubt and look at their reasons for rejecting something that I'm saying. I don't believe that the majority's always right (or even frequently right), but at least pause to reconsider whether you're coming from as solid a position as you think you are.

 

(oh, and usually when I find that I'm wrong it's because of a limited understanding of the system I'm trying to debunk, and when I talk to really experienced professors they can immediately point out the error in my logic and the false assumptions that I'm making. How well do you REALLY understand current mathematics?)

[Guest Doron Shadmi]

Dear DoorNumber1,

 

Please look at my updated version of http://www.geocities.com/complementarytheory/9999.pdf

 

From your last post it is still understood that you did not read my last post in order to understand it.

 

For example: You ignored the equivalent case that exists between a point and a segment, and an empty set and a non-empty set.

 

Also you ignored the fact that in any given segment you can find infinitely many sub-segments and in any one of them you can find another level of infinitely many sub-sub-segments, and so on... and so on ...

 

So, when we speak about infinitely many sub-segments, all we get is the structure of a fractal which has self similarity over infinitely many scales, and from this point of view, nothing become closer to nothing, because we always find the same proportion in any scale that we choose, and this is exactly the deep meaning of self similarity of a fractal to itself on infinitely many scales of it.

 

Mathematics is useful because we choose to stop this diving into this infinitely deep fractal, and come with some finite practical result that can be used in our daily life, and I agree with you that no system of infinitely many elements can be used by us in our daily life.

 

But let us examine the concept of infinitely many elements by this example:

 

Let us say that we have two building-blocks, which are: a point {.} and a segment {._.}.

 

Let us say that {.} is notated by its length, therefore the length of {.} = 0

 

Now there is a question: What is the length of {0_.} or what number is related to the right edge of this segment?

 

The answer is vary simple: Any arbitrary positive R member, which means, that if we have infinitely many segments where each one of them has a different length, we cannot know the value that has to be related to its right side, if we do not determine arbitrarily what segment is our 1 {0_1}.

 

And only then we can determine the right value of each segment, according to this arbitrary 1 {0_1}.

 

It means that any unique segment can be our arbitrary 1 {0_1} and it means that the right edge of any arbitrary segment is related to the entire R members, and each segment gets its unique name only after we choose one of the segments as 1 {0_1}.

 

So any segment is first of all a Urelement with |R| names.

 

If you look at my Organic Natural Numbers you will find the n^n version of it.

 

I hope that you still with me, because now we can understand that {0_1} and {0_.99999....} are clearly two different segments because {0_1} has no fractal properties, where the right side of {0_.9999...} represents a single path along infinitely many scale levels of a fractal.

 

Also I see that you totally ignored my axiomatic system and my arithmetical examples, which are based on it.

 

And also you ignored the connection between my work and the ideas of Gottfried Wilhelm Leibniz.

 

Please read also http://www.scienceforums.net/forums/showpost.php?p=77655&postcount=15

 

Thank you.

[matt grime]

For Other Uesers:

 

Matt Grime is a strange fallow that has an obsession to follow after me over the internet (to say the truth' date=' I invited him 2 years ago to meet me in some forum) to and give advices to other users in other forums, about my threads.

 

I learned a lot during our dialogs, and used what I have learned in order to develop my system, but unfortunately Matt cannot understand the included-middle reasoning because he insists to force an 0_XOR_1 reasoning on it, which cause him not to understand my system.[/url']

 

 

then why do you keep sending me unsolicited emails with you pet theories in them if you do not wish me to comment upon them. the last three are dated the 23,24,25 of august, after you wrote this message.

this a mathematics forum, you do not understand the first thing about mathematics, and I will keep saying so if i ever notice you posting on any forum that i read.

[Dave]

I think that this thread has gone about as far as it can go until we get some kind of decently laid out and progressive set of ideas.

 

Thread's closed unless someone gives me a good reason to re-open it.