re:The Axioms of Complementary Set Theory
From: Doron Shadmi (complementarytheory_at_yahoo-dot-com.no-spam.invalid)
Date: 08/03/04
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Date: 3 Aug 2004 14:31:35 -0500
> Doron,
>
> I don't understand your claim that, according to standard
mathematics, [3,4]=[2,5]. I would say that this expression is
undefined in standard mathematics.
>
> If, by the notation [3,4] you mean the interval which includes all
numbers from [3,4], then to compare this to any other interval you
need to define a measure of the interval. For example, one possible
measure would be:
>
> f([x,y]) = |x-y|
>
> By this measure, we would have:
>
> f([3,4]) = |3-4| = 1,
>
> and
>
> f([2,5]) = |2-5| = 3.
>
> We can then validly compare the measures and say
>
> f([2,5]) > f([3,4])
>
> Alternatively, we might require that [x,y] = [p,q] iff x=p and y=q,
in which case it is only possible to test for equality or inequality
of two intervals.
>
> It is not clear to me by what process you say that [3,4] <
[2,5].
>
> Perhaps you can explain.
> [quote:f88ae5a29e]
> I don't understand your claim that, according to standard
mathematics, [3,4]=[2,5].
>
It is very simple:
[3,4] and [2,5] are closed intervals.
By standard Math if these two intervals are related
only to [b:f88ae5a29e]N[/b:f88ae5a29e]
members, then there is nothing between the numbers of each interval
and we get 1-1 and onto map of:
3 <--> 2
4 <--> 5
In short, we get the same cardinality for both closed intervals.
And the reason for this is: In standard Math an interval {._.} is
defined by points {.}, where in my new system I have this axiom:
The axiom of independency:
[b:f88ae5a29e]p[/b:f88ae5a29e] and
[i:f88ae5a29e][b:f88ae5a29e]s[/b:f88ae5a29e][/i:f88ae5a29e] cannot be
defined by each other.
>
> It is not clear to me by what process you say that [3,4] <
[2,5].
>
Since in my system {.} and {._.} are independed, then [3,4] <
[2,5].
Please remember that in both cases we are talking
[b:f88ae5a29e]only[/b:f88ae5a29e] on [b:f88ae5a29e]N[/b:f88ae5a29e]
members.[/quote:f88ae5a29e]
> Doron,
>
> You seem to be using the word "interval" in a strange way. When you
refer to the "interval" [2,5], do you mean the set of integers
{2,3,4,5}, or do you mean all real numbers between 2 and 5,
inclusive?
>
> If you mean the set of integers, then the set has cardinality 4,
which is obviously greater than the cardinality of the set of two
integers {3,4}. If, on the other hand, you mean all real numbers
between 2 and 5, then the cardinality of [2,5] is the same as the
cardinality of [3,4], which would be c, or Aleph1, if you believe the
continuum hypothesis.
>
> Of course, that's standard mathematics, and perhaps what you are
doing is suggesting a different way to determine the cardinality of a
set of real numbers. That's what I'm not sure about.
> [2,5] is {2,5}.
>
> [3,4] is {3,4}.
>
> It means that all we have is the lowest and uppest bounds where
nothing exists between them.
>
> This is the way of how by standard Math we can find 1-1 and onto
between the entire [b:f88ae5a29e]N[/b:f88ae5a29e] members and some
proper subset of [b:f88ae5a29e]N[/b:f88ae5a29e] members
>
> Please read this again:
>
> The main idea behind the integers (unless we choose to change it) is
to look on the number line as if it has a one and only one scale
factor, which its value is 1 and only 1.
>
> In this case any arbitrary interval cannot be but 1 (or -1 if we
take zero's left side).
>
> For example:
>
>
..___-2___-1___0___1___2___3___4___5___6___...
>
>
..___-2___-1___0___1___[color=Red:f88ae5a29e]2___3___4___5[/color:f88ae5a29e]___6___...
>
> [color=Blue:f88ae5a29e]3___4[/color:f88ae5a29e] <
[color=Red:f88ae5a29e]2___3___4___5[/color:f88ae5a29e] -->
[3,4]<[2,5] [b:f88ae5a29e]by the new approach[/b:f88ae5a29e].
>
> By this approach no proper subset of [b:f88ae5a29e]N[/b:f88ae5a29e]
can be put in 1-1 correspondence with the entire
[b:f88ae5a29e]N[/b:f88ae5a29e], for example
[b:f88ae5a29e]N[/b:f88ae5a29e] and its odds:
>
> ...___1___2___3___4___5___6___7___8___9___... (Entire N)
> | | | | |
> ...___1_______3_______5_______7_______9___... ( Entire
Odds)
>
>
> [color=Blue:f88ae5a29e][b:f88ae5a29e]In the standard way the
interval {.__.} is omitted and we
get:[/b:f88ae5a29e][/color:f88ae5a29e][code:1:f88ae5a29e]
> ... 1 2 3 4 5 6 7 8 9 ... (Entire N)
> | | | | | | | | |
> ... 1 3 5 7 9 11 13 15 17 ... ( Entire Odds)
> [/code:1:f88ae5a29e]
> [color=Blue:f88ae5a29e][b:f88ae5a29e]As we can clearly see, standard
math does not find 1-1 map between numbers, but between their
represented notations, and we can clearly see that the standard point
of view does not distinguish between a number and its represented
notation.[/b:f88ae5a29e][/color:f88ae5a29e][b:f88ae5a29e]As we can
clearly see, standard math does not find 1-1 map between numbers, but
between their represented notations, and we can clearly see that the
standard point of view does not distinguish between a number and its
represented
notation.[/b:f88ae5a29e][/color:f88ae5a29e][/quote:f88ae5a29e]
> [quote:f88ae5a29e]
> james, i think what he is trying to show that standard math does not
find a 1-1 map between numbers (as envisioned by him using a number
line concept - that's how i think of it).
>
> i think i can illustrate his definition of one-to-one. imagine a
number line like a ruler, and if you take two rulers together,
one-to-one means that the numbers line up directly to 1,2,3,4,5.
since his definition of enumeration states that a ruler like number
line must line up with 1,2,3,4,5 so he says that there can be no
one-to-one relation between all integers, and all odd integers, since
3 lines up to 3 and not 2 and thus leaves 2 without a member to
enumerate. therefore, if you believe, like he does, that numbers are
more than just symbols but that they have some inner structure, say
this structure is represented by the length of a part of the ruler
around the number, then standard math cannot find one-to-one map
because it compares the symbols 1,2,3,4, and not the actual numbers
themselves (represented here by the length of the ruler around the
symbol 1,2,3,4). then it is clear... i think
>
[b:f88ae5a29e]Dear fallen angel, this is a perfect explanation of my
idea about numbers.[/b:f88ae5a29e]
Let us think about Mandelbrot set
http://aleph0.clarku.edu/~djoyce/julia/julia.html .
The set itself is the black areas, where no information can be found.
This black area is the invariant or the constant side of Mandelbrot
set, but the other side of it is its border area, where the
interesting information is created when Mandelbrot set gradually
disappearing at infinity.
No one of these sides can be ignored if we want to understand what is
a Mandelbrot set.
The same approach has to be used if we want to understand what is
[b:f88ae5a29e]R[/b:f88ae5a29e] collection.
Any [b:f88ae5a29e]R[/b:f88ae5a29e] member is a unique (invariant and
constant) element in the collection, but on the same time each
constant is a scale factor of the entire
[b:f88ae5a29e]R[/b:f88ae5a29e] collection.
It means that the entire [b:f88ae5a29e]R[/b:f88ae5a29e] collection
exists between two opposite states (minus is the mirror -not the
opposite- of plus side).
In one state, when 0 is the scale factor, no
[b:f88ae5a29e]R[/b:f88ae5a29e] member except 0 can be found.
On the other state No [b:f88ae5a29e]R[/b:f88ae5a29e] member can be
found when we reach oo (as clearly can be shown here:
[url]http://www.geocities.com/complementarytheory/RiemannsLimits.pdf[/url]
).
Furthermore, because of this duality of any
[b:f88ae5a29e]R[/b:f88ae5a29e] member, we get a system which is both
absolute (when a single scale is examined) and relative (where the
same place of the real line is examined simultaneously on several
different scales).
Another example:
Pi = the relations between the perimeter and the diameter of a
circle.
Pi is invariant in any arbitrary given scale, but when several scale
levels a simultaneously compared, we can clearly see that each circle
has a different curvature.
If our system is a circle, then if we want to understand what is a
circle, then both its invariant and variant properties cannot be
ignored.
(We also have to be aware to the fact the no circle can be found when
Diameter or Perimeter = 0, or Diameter or Perimeter = oo.
In short, our basic approach is to find the gateways between opposite
properties, and the best way to do it, is by an including-middle
logical reasoning
([url]http://www.geocities.com/complementarytheory/CompLogic.pdf[/url]).
>
> i still don't understand what is the benefit of this concept? it
seems to reduce the abstractness of mathematics. after all, if you
cannot enumerate numbers, what can we do?
>
By my number system, you do not have to reduce any abstract or
non-abstract element to a quantitative model, in order to analyze and
conclude some meaningful and useful things about it.
And the reason is, my number system is both structural/quantitative
information system, where each element of it is examined by the
Symmetry concept, which is the most powerful tool of the language of
Mathematics.
In my system Symmetry degree and information clarity degree have
complementary relations, where each concept simultaneously
preventing/defining the other concept.
The result of these complementary interactions is infinitely many
gateways to infinitely many topologies, where each one of them can be
a building-block for another logical reasoning.
If we ignore the structural property of a given number and look only
on its quantity, then this point of view is not more abstract then
the structural/quantitative point of view, but more trivial then the
structural/quantitative point of view.
We don't have to be happy if there is a way to take two complex
systems like, for example, two persons, and then to say that we have
two objects.
This is a trivial point of view of these persons, and by my number
system we can choose if we want to ignore or not their internal
complexity.
Therefore the result of this kind of view is richer then the standard
quantitative-only point of view, for example:
By this new approach we can build, for example, a totally new
Turing-like machine, that can change forever our abilities to deal
with complexity which is based in simplicity.[/quote:f88ae5a29e]
> Please read this:
>
> I thought about something and I'll be glad to know your opinion.
>
> It goes like this:
>
> When two violins in the same room are tuned with each other, if we
play on one of them we find that the strings of the other violin are
also vibrate.
>
> Now let as say that intuition is our tuned instrument, and if a
person expresses its intuitions by developing a way of thinking, the
people that embrace this way of thinking probably share the same
intuitions.
>
> On the basis of these common intuitions a community can be
established.
>
> Let us say that this community is the first organization that deals
with some part of the human knowledge, so in these early stages this
community has no comparators on this part of the human knowledge.
>
> Quickly this community becomes the most developed organization,
which holds this part of the human knowledge, and other parts of
human civilization look at this organization as the one and only one
possible intuition which standing in the basis of a one and only one
way (school) of thought (and I am not talking about variations, which
are actually different brunches of the same way of thought, or the
same school of thought if you like).
>
> 2500 years are passing and this school of thought survives because
of two main reasons:
>
> [b:f88ae5a29e]1)[/b:f88ae5a29e] This way of thought was fitting to
the needs of the human civilization along these 'slow' (linear)
years.
>
> [b:f88ae5a29e]2)[/b:f88ae5a29e] Any other alternative intuitions (if
they where at all) where put aside because:
>
> [b:f88ae5a29e]a)[/b:f88ae5a29e] They where not useful in their time.
>
> [b:f88ae5a29e]b)[/b:f88ae5a29e] And if they where useful and also a
real alternative to the current school, then the current school used
its power and money to block this alternative intuition by forcing
its educational methods on the public.
>
> [b:f88ae5a29e]We have to understand that intuitions cannot be
learned, but a lot of external power can distort them until they lost
their ability to be the source of a new school of
thought.[/b:f88ae5a29e]
>
>
> The 120 century is the time where our civilization moved from linear
time to a non-linear time.
>
> In this time the power of few holds the destiny of our civilization,
and most of their power is based on the technical abilities that where
developed by this school of thought, that was established 2500 years
ago.
>
> But our technical achievements, which are not balanced by another
ways of thought, are like a government with no opposite.
>
> We have learned that evolution needs diversity; otherwise we quickly
get a dead planet.
>
> The field of evolution in our non-linear time splits to "hardwhere"
and "softwhere" parallel pathes, where the hardwhere side is our
technology and the softwhere side is our morality.
>
> We can clearly see that there is no balance between the levels of
these two paths, and this lack of balance in a non-linear time can
quickly lead us to a dead-end street.
>
> Therefore I think that we have to do the best we can to find the
balance between our morality level and our technical abilities.
>
> The first place that binds both paths is the language of
mathematics.
>
> In my opinion people which learn this powerful language, must first
of all to develop their moral abilities by opening themselves to
another intuitions which are not their intuitions and let them
flourish in their communities.
>
> By this way we develop our tolerance and learn how to live side by
side, and if other intuitions are better then our intuition in this
period of time, we do our best to help them flourish instead of
trying our best to shut them down.
>
> And we have the motivation to do that because we understand that we
are all in the same boat.
>
> My intuitions and ideas about the language of mathematics are
different then the standard school of mathematics.
>
> But in my opinion the most important difference, which I think fits
to our non-linear time (more then the standard school) is that I
include the mathematician cognition's ability to develop Math as a
part of the mathematical research.
>
> By this self-reference attitude I hope to develop the gateway that
can connect between our moral abilities to our technical abilities.
>
> And for that I need your help.
>
> What do you think?
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