Deeper than primes

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(Predecessor is what is less than a considered thing.

Successor is what is more than a considered thing.)

The axiom of minima:
Emptiness is that has no predecessor.

The axiom of maxima:
Fullness is that has no successor.

(Only Emptiness does not have a predecessor in the absolute sense.

Only Fullness does not have a successor in the absolute sense.

The next axioms are at the level of the existence of collections, which is > Emptiness AND < Fullness, where > or < are the order of exitence w.r.t Emptiness or Fullness.)

The axiom of existence:
Any existing thing has a predecessor.

(y and x are place holders for an intermediate state of existence between Emptiness and Fullness.)

The axiom of infinite collection:
If x exists then y>x exists.

(I do not assume numbers, I explicitly use numbers as measurements of the intermediate levels of existence between Emptiness and Fullness.

Simultaneity does not need any further definition in order to clearly be understood, exactly as step-by-step (the opposite of Simultaneity) does not need any further definition in order to clearly be understood.)

The axiom of Locality:
y, such that x = 1 to ∞ and ((y≥0)<x), is simultaneously at most at one location w.r.t all x.

The axiom of Non-Locality:
y, such that x = 0 to ∞ and x < y, is simultaneously at least at two locations w.r.t all x.
 
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HatRack said:
Before you attempt to correct your axioms or alternatively try to tell me that I'm wrong, let's take a different approach. It's already been explained to you that ZFC is strong enough to deduce the negation of your central result known as the "Axiom of the Continuum". ZFC states some very obvious and intuitive things about the notion of a collection. Tell us, in detail, which ZFC axioms are wrong and why.
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I will do it tommorow.
 
doronshadmi said:
(Predecessor is what is less than a considered thing.

Successor is what is more than a considered thing.)
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So, doronetics only works for well ordered collections. How limiting.

The axiom of minima:
Emptiness is that has no predecessor.

The axiom of maxima:
Fullness is that has no successor.
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Neither of these are axioms.

(Only Emptiness does not have a predecessor in the absolute sense.

Only Fullness does not have a successor in the absolute sense.

The next axioms are at the level of the existence of collections, which is > Emptiness AND < Fullness, where > or < are the order of exitence w.r.t Emptiness or Fullness.)
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No, they are not. You'd need to add extra words to the pseudo-axioms to create those sorts of restrictions.

The axiom of existence:
Any existing thing has a predecessor.
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As I stated before, this establishes that emptiness is not an existing thing.

Oh, and look, the banned universal quantification again.

(y and x are place holders for an intermediate state of existence between Emptiness and Fullness.)

The axiom of infinite collection:
If x exists then y>x exists.
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Great, but will you ever bother to assert that at least one thing exists? So far all you have done is explicit deny the existence of emptiness, and by this pseudo-axiom, you have now denied the existence of fullness, too.

(I do not assume numbers, I explicitly use numbers as measurements of the intermediate levels of existence between Emptiness and Fullness.
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You cannot have it both ways, Doron. If they aren't already part of Doronetics before you introduce these pseudo-axioms, then you cannot use them in these pseudo-axioms.

Since you now deny numbers are assumed, and since you didn't bother to define location and location with respect to something, the remainder of your post is therefore gibberish.
 
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Doron, there are multiple problems with your axioms still. This list is certainly not comprehensive.

  1. The first two "axioms" are not axioms at all, they are definitions.
  2. You never assert the existence of anything, leaving your theory rather empty.
  3. Your third axiom denies the existence of something you went out of your way to define.
  4. The biggest problems lie with your last two axioms. If you wish to use numbers and infinity, you must assume the Peano Axioms + elementary set theory, which makes your theory inconsistent right off the bat. Otherwise, you must introduce numbers axiomatically.

I'm beginning to think you don't understand what mathematical rigor means. Okay that's a lie actually, I've thought that well before I even created a forum account here.

And jsfisher made a great point. You are using universal quantification in one of your axioms, the very thing you said was invalid several pages back to try and claim that 1/2 + 1/4 + 1/8 + ... = 1 was wrong. You and epix need to sit down together and read an elementary calculus text, at the very least, before criticizing the foundations of mathematics.
 
doronshadmi said:
In other words, you can't get Emptiness, Fullness, and the intermediate existence of collections between Emptiness and Fullness.
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No, doron. You define emptiness and fullness but in the next two "axioms" you assert that they do not exist (taking into account your definition of successor). You call that useful in what way?
 
doronshadmi said:
How poor understanding about order (at its most general aspect) you have.
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Forgive me for having a much clearer concept of what predecessor and successor generally mean than you. How is giving a new meaning for an otherwise generally accepted term useful? Why don't you just use the accepted terms?
 
doronshadmi said:
0 is a predecessor for pi and pi is a successor for 0.
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No, 0 is less than pi and pi is greater than 0. Those are the terms you're looking for. Successor and predecessor have other meanings. Since the meaning you're trying to convey has a generally accepted term for it, I suggest you stick to it to avoid further confusion.
 
doronshadmi said:
Predecessor is what is less than a considered thing.

Successor is what is more than a considered thing.

Only Emptiness does not have a predecessor in the absolute sense.

Only Fullness does not have a successor in the absolute sense.

Your notion is still closed under the concept of Collection, which is an intermediate state of existence between Emptiness and Fullness.
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WHAT?! Where has HatRack talked about anything but the general purpose of axioms? What does that have to do with collections? Do you even read people's posts before replying or just glance at it and decide on a stock gibberish reply?
 
doronshadmi said:
y and x are place holders for an intermediate state of existence between Emptiness and Fullness.
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How does something exist between two nonexistent things? Your axioms forbid the existence of both emptiness and fullness. Methinks you need to revise your "axioms".
 
laca said:
How does something exist between two nonexistent things? Your axioms forbid the existence of both emptiness and fullness.
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Emptiness is exactly total non-existence, and Fullness is exactly total existence.

Emptiness is strictly below the concept of Collection and Fullness is strictly above the concept of Collection. Since you get concepts only at the level of the concept of Collection, you can't comprehend Emptiness or Fullness.

jsfisher said:
As I stated before, this establishes that emptiness is not an existing thing.
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It is trivially true that Emptiness is not an existing thing.

jsfisher said:
Oh, and look, the banned universal quantification again.
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Worng, the collection of all existing things above Emptiness AND below Fullness, holds.


jsfisher said:
So far all you have done is explicit deny the existence of emptiness
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I don't have to, Emptiness is total non-existence, and Fullness is total existence.

Collection (finite or infinite (countable or uncountable)) exists relatively to these totalities.

This is a novel approach about the abstract foundation of the Mathematical Science, which your "closed under Collections" approach simply can't comprehend.

You do not do the needed paradigm-shift beyond the concept of Collection, and as long as this paradigm-shift is not done, you can't get Organic Mathematics and its novel reasoning of the concept of Infinity, Collection, Number, Function, Logic, Analysis, Geometry, Operation, Continuum, Discreteness, Order, Cardinality, Distinction or any other fundamental concept of the Mathematical Science.

jsfisher said:
You'd need to add extra words to the pseudo-axioms to create those sorts of restrictions.
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It is done for one purpose at this stage, to give you the needed notions in order to help you to do the paradigm-shift beyond the Collection-only notion.

After this paradigm-shift is done, we do not need the extra words.

By analogy, these extra words are like scaffolds that are removed after the paradigm-shift is done.
 
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jsfisher said:
So, doronetics only works for well ordered collections.
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Not well-ordering is, for example, the open interval (0, 1) that does not contain a least element.

(0,1) exists also by OM, such that the least existing thing that is totally local (a point), and the least existing thing that is totally non-local (a line), are excluded.
 
doronshadmi said:
It is done for one purpose at this stage, to give you the needed notions in order to help you to do the paradigm-shift beyond the Collection-only notion.

After this paradigm-shift is done, we do not need the extra words.

By analogy, these extra words are like scaffolds that are removed after the paradigm-shift is done.
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What you're asking is to believe you without any question. That's not going to happen, sorry. You will have to do the work. Maybe then you'll realize how deluded you are.
 
doronshadmi said:
Emptiness is exactly total non-existence, and Fullness is exactly total existence.

Emptiness is strictly below the concept of Collection and Fullness is strictly above the concept of Collection. Since you get concepts only at the level of the concept of Collection, you can't comprehend Emptiness or Fullness.
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There's only one person having problems distinguishing between the existence of emptiness and what it means.

You first defined 'emptiness', then precluded it from existing. You also have a problem with "existing things". If emptiness does not exist (it does not have a predecessor, so it is not an existing thing), then how does the first thing exist, since it too has no predecessor?
 
laca said:
What you're asking is to believe you without any question.
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Not at all, on the contrary, I challenge you to get things beyond the concept of Collection.

Until know you are using the concept of Collection in order to understand Emptiness and Fullness, but these concepts are the totalities below and above that concept of Collection, where the concept of Collection exists relatively to these totalities.

In order to get the concept of Collection from the level of these totalities, a paradigm-shift of the concept of Existence has to be done in your mind.

Without it, there can't be any meaningful communication between us.

This paradigm-shift has nothing to do with the concept of Belief.
 
doronshadmi said:
Not well-ordering is, for example, the open interval (0, 1) that does not contain a least element.

(0,1) exists also by OM, such that the least existing thing that is totally local (a point), and the least existing thing that is totally non-local (a line), are excluded.
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Well, whether the set of real numbers in the open interval (0,1) is well-ordered would depend on the order relation, now wouldn't it? Are you assuming a particular order relation? (Of course you are. You have many, many hidden assumptions.)

And I see now you assume the real numbers, not just the integers, in doronetics., too

Let's see: You require the real numbers as background to your little pseudo-axiom set without any basis. You need universal quantification, but forbid its use by others. You offer simple word-substitution definitions (which aren't definitions in any constructive sense) while claiming they are axioms. And you require an unspecified ordering relation.

Just out of interest, Doron, how do you propose to order the points in a plane? Or the points on a circle? More hidden assumptions, I assume.
 
zooterkin said:
There's only one person having problems distinguishing between the existence of emptiness and what it means.

You first defined 'emptiness', then precluded it from existing. You also have a problem with "existing things". If emptiness does not exist (it does not have a predecessor, so it is not an existing thing), then how does the first thing exist, since it too has no predecessor?
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The first existing thing has a predecessor, which is the concept of Emptiness.

Only Emptiness does not have a concept that can be used as its predecessor, otherwise it not the concept of Emptiness.

Please try to upgrade your abstraction in order to get that.
 
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jsfisher said:
And you require an unspecified ordering relation.
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At this fundamental level, all is needed is the essentials which enable order, which are exactly < or >. Without them no order (well or non-well) is defined.

You simply force a further resolution at this fundamental level, but then you are not at this fundamental level of the concept of Order anymore.
 
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Elf Grinder 3000 said:
What is the difference between an axiom and a definition?
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A definition gives a name to something. It saves us from having to describe the something every time we refer to it; we can use the defined name instead. For example, rather than having to say repeatedly "a polygon with four coplanar equal sides and four equal internal angles", we can just say "square".

An axiom, on the other hand is a proposition, a statement presumed to be true without proof. Here are a couple of axioms borrowed from some formulations of ZFC: An empty set exists. If x and y are sets, then there exists a set which contains x and y as elements.

Compare the axiom, "an empty set exists," to the definition, "an empty set is a set with no members."
 
doronshadmi said:
At this fundamental level, all is needed is the essentials which enable order, which are exactly < or >. Without them no order (well or non-well) is defined.
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Wow! So, in Doronetics, symbols are all powerful. The sheer might of those seemingly simple angle brackets "enable order", as you say.

You simply force a further resolution at this fundamental level, but then you are not at this fundamental level of the concept of Order anymore.
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No, Doron. The order relation is an important detail. Without it, you cannot establish your "fundamental level."


You really have a great many hidden assumptions. Shame on you.
 
Let us simplify the following axioms:

The axiom of Locality:
There exist y and x, such that x is simultaneously at most at one location w.r.t y.


The axiom of Non-Locality:
There exist y and x, such that y is simultaneously at least at more than one location w.r.t x.

So, order or numbers are not used anymore.
 
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doronshadmi said:
Let us simplify the following axioms:

The axiom of Locality:
There exist y and x, such that x is simultaneously at most at one location w.r.t y.


The axiom of Non-Locality:
There exist y and x, such that y is simultaneously at least at more than one location w.r.t x.

So, order or numbers are not used anymore.
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Good for you!! These definitions (and they are definitions, not axioms) don't do what you think they do, but nonetheless, good for you!! Now, can you do something with that predecessor/successor usage?

Also, you haven't yet defined what you mean by being at a "location with respect to" something.
 
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jsfisher said:
... they are definitions ...
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...definitions for local and non-local, that is. Unfortunately, in this newly structured gibberish, it is no longer clear which thing gets the local (non-local) property. Before it was Y. Now, who knows.

...Or maybe you really, really did mean for these to be axioms of the existential variety. That is to say, a pair for somethings exist having the stated (that's being polite) characteristics.

Is that what you meant?
 
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doronshadmi said:
Not at all, on the contrary, I challenge you to get things beyond the concept of Collection.

Until know you are using the concept of Collection in order to understand Emptiness and Fullness, but these concepts are the totalities below and above that concept of Collection, where the concept of Collection exists relatively to these totalities.

In order to get the concept of Collection from the level of these totalities, a paradigm-shift of the concept of Existence has to be done in your mind.

Without it, there can't be any meaningful communication between us.

This paradigm-shift has nothing to do with the concept of Belief.
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Bollocks. Even if there was some sense in what you're advocating (and there really isn't), I don't see why the paradigm shift is needed. Why is the currently accepted framework inadequate for developing new theorems to enhance the tools used when describing reality? Do you have an answer for this? While the lack of an explanation does not in itself mean the work is not useful, it would certainly help us understand what is driving you.
 
doronshadmi said:
The first existing thing has a predecessor, which is the concept of Emptiness.

Only Emptiness does not have a concept that can be used as its predecessor, otherwise it not the concept of Emptiness.

Please try to upgrade your abstraction in order to get that.
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Nope, Doron. Since you're building an axiomatic framework, it is your responsibility and yours alone to specify everything. And by everything I mean everything. Nothing can be left out. Everything has to be spelled out as clearly as day. This is very important.
 
jsfisher said:
Good for you!! These definitions (and they are definitions, not axioms)
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I disagree with you, they are "self-evident propositions requiring no formal demonstration to prove its truth, but
received and assented to as soon as mentioned.”

It all depends on your ability to get their self-evident truth, and you can't do that if you are closed under the concept of Collection.

http://www.ocf.berkeley.edu/~easwaran/papers/axioms.pdf

One initially plausible story about the role of foundational axioms is
that they are intuitively obvious statements that we can use to establish
our theorems with epistemic certainty. Feferman quotes the Oxford
English Dictionary defining an axiom in mathematics as “A self-evident
proposition requiring no formal demonstration to prove its truth, but
received and assented to as soon as mentioned.”
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http://en.wikipedia.org/wiki/Axiom
In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.
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http://mathworld.wolfram.com/Axiom.html
An axiom is a proposition regarded as self-evidently true without proof.
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http://www.cut-the-knot.org/WhatIs/WhatIsAxiom.shtml
axiom (noun), axiomatic (adjective): via Latin, from Greek axioma, "that which is thought fitting; decision; self-evident principle." The Indo-European root is ag- "to drive, to lead." A subsidiary Greek meaning, "to weigh," led to axioma, literally "something weighty." In mathematical terms, axioms are concepts felt weighty enough that you can base a logical system on them.
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laca said:
While the lack of an explanation does not in itself mean the work is not useful, it would certainly help us understand what is driving you.
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The answer goes like this:

I wish to share with you my reasoning about the concept of Complexity, and how it is related to Ethics and Logic.

It is well known that one of the most powerful tools that our civilization uses is The Mathematical Science.

One of the main reasons of the efficiency of this science is the universal principles that stand at its foundations.

Because of these universal principles our civilization achieved its current technology, but the motivations and use of these technologies are not based on universal principles.

In my opinion non-universal principles that are fragmented to different cultures, religions, nations etc… + technology that is derived from universal principles, is a very dangerous cocktail that may lead us to self-made destruction.

In my opinion one of the ways to reduce the chance of self-made destruction is to define a universal framework that may be used as a common base ground for both Ethics and Logical reasoning.

For the past 30 years I am trying to develop such a framework, and this goal is definitely beyond the abilities of a single person.

Anyway, I wish to share with you some of my last results (and please forgive me about my English (my language is Hebrew)) which draw some sketches of this universal framework.

I call this framework Organic Mathematics, or OM.

OM ( http://www.scribd.com/doc/17039028/OMDP ) demonstrates Direct Perception as the common foundation of both Intuition and Logical reasoning. Furthermore, Direct Perception is actually the base ground of any mantel activity, whether it is expressed by senses, emotions, or logical reasoning.

Direct Perception is actually the silent presence of any mantel activity, which enables to bridge our ethical aspects with our logical\technological aspects under a one framework.

The luck of Direct Perception as the base ground of a powerful language like the mathematical science, can easily lead us to manipulate deeper forces of Nature, which are not balanced by universal ethical principles (universal ethical principles must not be limited to any particular religion, culture or civilization).

In my opinion if our species will not learn very soon how to develop the universal bridge between Ethics and Logics under a one comprehensive framework, we shell not survive further manipulations of Nature's forces.

Please look at:

Mathematics As a Tool For Survival:
http://www.scribd.com/doc/16547236/EEM

and http://www.scribd.com/doc/16669828/EtikaE
for clearer representation of my argument (and again, sorry about my English).

More comprehensive papers abut this subject are:

Zeno's Achilles\Tortoise Race and Reconsiderations of Some Mathematical Paradigms
http://www.scribd.com/doc/21967511/TOC-NEW2

Organic Mathematics (A Non-formal Introduction):
http://www.scribd.com/doc/16542245/OMPT

ORGANIC MATHEMATICS, Proposing a way to solve Hilbert's 6th Problem:
http://www.scribd.com/doc/18453171/IJPAMOM [1]

[1] Moshe Klein, Doron Shadmi : Organic Mathematics, International Journal of Pure and Applied Mathematics, volume 49 No. 3 2008, 329-340
 
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doronshadmi said:
I disagree with you, they are "self-evident propositions requiring no formal demonstration to prove its truth, but
received and assented to as soon as mentioned.”
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Ok, axioms they can (almost) be. The gibberish factor is omni-present, but we can accept your two axioms. They establish the existence of some pair of things with some property involving locations and another pair of things (not necessarily different from the first pair) with some other property involving locations.

So, what is it you mean by "location with respect to"?
 
jsfisher said:
Ok, axioms they can (almost) be. The gibberish factor is omni-present, but we can accept your two axioms. They establish the existence of some pair of things with some property involving locations and another pair of things (not necessarily different from the first pair) with some other property involving locations.

So, what is it you mean by "location with respect to"?
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Let us re-write these axioms:

The axiom of Locality:
There exist y and x, such that x is at the location of y.


The axiom of Non-Locality:
There exist y and x, such that y is at AND not at the location of x.
 
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doronshadmi said:
jsfisher said:
Ok, axioms they can (almost) be. The gibberish factor is omni-present, but we can accept your two axioms. They establish the existence of some pair of things with some property involving locations and another pair of things (not necessarily different from the first pair) with some other property involving locations.

So, what is it you mean by "location with respect to"?
Click to expand...

Let us re-write these axioms:

The axiom of Locality:
There exist y and x, such that x is simultaneously at most at one location with respect to y.


The axiom of Non-Locality:
There exist y and x, such that y is simultaneously at AND not at the location of x.
Click to expand...

And just as I said, the first axiom stipulates the existence of two things with certain relative properties, and the second axiom stipulates the existence of two things with certain relative properties.

Now, what is it you mean by "location with respect to"?
 
(Predecessor is what is less than a considered thing.

Successor is what is more than a considered thing.)

The axiom of minima:
Emptiness is that has no predecessor.

The axiom of maxima:
Fullness is that has no successor.

(Only Emptiness does not have a predecessor in the absolute sense.

Only Fullness does not have a successor in the absolute sense.

The next axioms are at the level of the existence of collections, which is > Emptiness AND < Fullness, where > or < are the order of exitence w.r.t Emptiness or Fullness.)

The axiom of existence:
Any existing thing has a predecessor.

(y and x are place holders for an intermediate state of existence between Emptiness and Fullness.)

The axiom of infinite collection:
If x exists then y>x exists.

The axiom of Locality:
There exist y and x, such that x is at the domain of y.

The axiom of Non-Locality:
There exist y and x, such that y is at AND not at the domain of x.
 
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Ok, so what is it you mean by "at the location of"?

Are you now assume that the only things that exist must have locations (multiple locations in at least one case)? More and more hidden assumptions, Doron.
 
You have also introduced a contradiction. We knew it wouldn't take long. Or, perhaps, you are assuming a different formal logic: one that allows "A and not A" as a true proposition?

More, and more, and more hidden assumptions.
 
jsfisher said:
Ok, so what is it you mean by "at the location of"?

Are you now assume that the only things that exist must have locations (multiple locations in at least one case)? More and more hidden assumptions, Doron.
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A more general one:

The axiom of Locality:
There exist y and x, such that x is at the domain of y.

jsfisher said:
Or, perhaps, you are assuming a different formal logic: one that allows "A and not A" as a true proposition?
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Yes, this is the property of Non-locality, it is a contradiction only from the local point of view.

The axiom of Non-Locality:
There exist y and x, such that y is at AND not at the domain of x.
 
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doronshadmi said:
The axiom of Non-Locality:
There exist y and x, such that y is at AND not at the domain of x.
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Umm, okay. So, y is "at the domain of x" and "not at the domain of x". The situation is worse than I thought. This is a violation of first-order logic. No longer are you just crapping all over traditional mathematics, you are now violating the basics of the basics.

Doronetics is quite a strange place indeed, where logic and reason do not matter, and the direct perception of Doron Shadmi reigns supreme.
 
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