Deeper than primes

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HatRack said:
Let {1,2,3} be the set such that if X is 1, 2, or 3 then X is an element of {1,2,3}.
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The first definition defines {1,2,3} by objects that no one of them is {1,2,3},
so X can't be {1,2,3}, and there is no circular reasoning.

HatRack said:
Let {1,2,3} be the set such that if X is not 1, 2, and 3 then X is not an element of {1,2,3}.
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The second definition defines {1,2,3} by using object {1,2,3},
so X can be {1,2,3}, and there is a circular reasoning.

Since the X of the first definition can't be the X of the second definition, then NOT(NOT X) = X is irrelevant, in this case.

HatRack said:
So Doron, we have achieved a milestone in this thread.
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Not even in your dreams.
 
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Doron, it is clear now that you don't even understand basic first-order logic. Maybe you have some problems with that too? Do I sense a need for another paradigm-shift on the horizon? :rolleyes:
 
doronshadmi said:
The first definition defines {1,2,3} by objects that no one of them is {1,2,3},
so X can't be {1,2,3}, and there is no circular reasoning.



The second definition defines {1,2,3} by using object {1,2,3},
so X can be {1,2,3}, and there is a circular reasoning.

Since the X of the first definition can't be the X of the second definition, then NOT(NOT X) = X is irrelevant, in this case.
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Equivalent definitions of the empty set:

Let {} be the set such that if X = X then X is not an element of {}.

Let {} be the set such that if X is not equal to X then X is an element of {}.

The first one you will claim has "circular reasoning" because X can be {}. But, you can't claim that for the second one, because the empty set is indeed equal to itself. And therefore, {} cannot be X.

Thus, the second definition is a valid definition of the empty set even by your standards. Now, by both real logic and by your "logic", you have been proven wrong. :p
 
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HatRack said:
Equivalent definitions of the empty set:

Let {} be the set such that if X = X then X is not an element of {}.

Let {} be the set such that if X is not equal to X then X is an element of {}.

The first one you will claim has "circular reasoning" because X can be {}. But, you can't claim that for the second one, because the empty set is indeed equal to itself. And therefore, {} cannot be X.

Thus, the second definition is a valid definition of the empty set even by your standards. Now, by both real logic and by your "logic", you have been proven wrong. :p
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By your second definition there is no guarantee that elements of the form X = X are not members of {}, because you did not explicitly prevent this option.
 
doronshadmi said:
By your second definition there is no guarantee that elements of the form X = X are not members of {}, because you did not explicitly prevent this option.
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I forgot, you don't understand mathematical definitions. You see Doron, in mathematics, definitions are taken to be "if and only if" statements by convention, even though they are usually written informally as one-way implications. Of course, you wouldn't know this, so I'll state it more explicitly:

Let {} be the set such that X = X IF AND ONLY IF X is not an element of {}.

Let {} be the set such that X is not equal to X IF AND ONLY IF X is an element of {}.
 
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HatRack said:
I forgot, you don't understand mathematical definitions. You see Doron, in mathematics, definitions are taken to be "if and only if" statements by convention, even though they are usually written informally as one-way implications. Of course, you wouldn't know this, so I'll state it more explicitly:

Let {} be the set such that if X = X IF AND ONLY IF X is not an element of {}.

Let {} be the set such that X is not equal to X IF AND ONLY IF X is an element of {}.
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HatRack, you did not understand my claim.

By your second definition the "elements" of {} are of the form X≠X, so there is no guarantee that the elements of the form X=X are not members of {}, because you did not explicitly prevent this option.
 
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doronshadmi said:
laca said:
Wow. Just, wow. So, if I define IPU as horse that is invisible, pink and has exactly one horn on its head then I also implicitly assert that it exists?
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Yes, it exists in an abstract framework.
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Well! We have a new axiom:

Axiom of the Definitional Actualization:
If something is defined, then it exists in Doronetics.

And who said Mathematics had to be hard?
 
jsfisher said:
Well! We have a new axiom:

Axiom of the Definitional Actualization:
If something is defined, then it exists in Doronetics.

And who said Mathematics had to be hard?
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jsfisher you don't have it, IPU horse that is invisible, pink and has exactly one horn on its head, is very very shy, therefore it hides behind jsfisher.
 
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doronshadmi said:
HatRack, you did not understand my claim.

By your second definition the "elements" of {} are of the form X≠X, so there is no guarantee that the elements of the form X=X are not members of {}, because you did not explicitly prevent this option.
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I'm not the one who is misunderstanding. You do not understand the meaning of the statement:

Let {} be the set such that X is not equal to X IF AND ONLY IF X is an element of {}.

I'll word it even more explicitly since you do not know what "if and only if" means:

Let {} be the set such that if X is an element of {} then X ≠ X and if X ≠ X then X is an element of {}.

This, like the three definitions given before it, only more explicitly, rules out the possibility of X = X being in {}, because I clearly have written that "if X is an element of {} then X ≠ X". Moreover, since {} does not satisfy {} ≠ {}, you can't bring up your fallacious "circular reasoning" argument.

I'll even give one more variation:

Let {} be the set of ONLY those elements X where X ≠ X.
 
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HatRack said:
I'll even give one more variation:

Let {} be the set of ONLY those elements X where X ≠ X.
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In other words, all the elements of the form X=X are not covered by your second definition, so this second definition can't prevent them from not being members of {}(and in this case we have circular reasoning).

In other words:

HatRack's second definition is about an existing set of all things that do not exist.

HatRack's first definition is about an existing set of all existing things that are not its members, where one of the existing sets that are not members of that existing set, is the defined set, or in other words, the defined set is used in order to define itself, which is a circular reasoning.
 
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doronshadmi said:
In other words, all the elements of the form X=X are not covered by your second definition, so this second definition can't prevent them from being members of the defined set.
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You do understand that the two 'X's are the same value, so they will always be equal? In the second case, because they are therefore never not equal to each other, they will not appear in the empty set.
 
zooterkin said:
You do understand that the two 'X's are the same value, so they will always be equal? In the second case, because they are therefore never not equal to each other, they will not appear in the empty set.
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X≠X is an "element" that does not exist, and by HatRackw's second definition, HatRackw defines the set of all elements that do not exist.

This definition do not cover the case of the set, which is defined as the set that all existing elements (the elements of the form X=X) that are not its members.
 
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doronshadmi said:
In other words, all the elements of the form X=X are not covered by your second definition, so this second definition can't prevent them from being members of the defined set.
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Let {} be the set of ONLY those elements X where X ≠ X.

This definition certainly does prevent objects X where X = X from being members of {}. I used the word ONLY. An object can ONLY be a member of {} if X ≠ X. This excludes X = X automatically.
 
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HatRack said:
Let {} be the set of ONLY those elements X where X ≠ X.

This definition certainly does prevent objects X where X = X from being members of {}. I used the word ONLY. An object can ONLY be a member of {} if X ≠ X. This excludes X = X automatically.
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Please look at this:

HatRack's second definition is about an existing set of all things that do not exist (X≠X).

HatRack's first definition is about an existing set of all existing things (X=X) that are not its members, where one of the existing sets that are not members of that existing set, is the defined set, or in other words, the defined set is used in order to define itself as a part of its own definition, which is a circular reasoning.

HatRack's first and second definitions are not the same definition, exactly as non-existing thing is not the same as an existing thing.

Furthermore HatRack's second definition is weak and trivial w.r.t the first definition, because it explicitly defines an existing set only by non-existing things.
 
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doronshadmi said:
HatRack's second definition is about an existing set of all things that do not exist (X≠X).
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The "existing set of all things that do not exist" is the same thing as the "existing set of no things that do exist", which is also the same thing as the empty set. Seriously Doron, you're just killing me with laughter at this point.

:dl: :dl: :dl:
 
doronshadmi said:
HatRack's second definition is about an existing set of all things that do not exist (X≠X).
Click to expand...

Let's take a close look at my definition and what it says:

Let {} be the set of ONLY those elements X where X ≠ X.

You claim that my definition is that of an existent set of all things that do not exist. Okay.

But, notice that I also used the word ONLY in my definition. Thus, in addition to containing all things that do not exist, it has the property that it contains ONLY things that do not exist. So, what conclusions can we draw about what elements do exist in the set I am defining? Well, let's have a closer look at these two properties of my set:

(1) All things that do not exist are in my set: Since non-existent things are... well... non-existent, we are lead to the conclusion that this property gives no assertion about what is in the set and what is not.

(2) Only things that do not exist are in my set: This tells us that anything which does exist cannot be in the set. In other words, no existent things can be in the set. In other words, the set does not contain any existent things. In other words, it is the empty set.

Right here is where the defense of my definition ends. I have just shown, in far more detail than is necessary, that it is the definition of the empty set. But, I'll respond to the rest anyway.

doronshadmi said:
HatRack's first definition is about an existing set of all existing things (X=X) that are not its members, where one of the existing sets that are not members of that existing set, is the defined set, or in other words, the defined set is used in order to define itself as a part of its own definition, which is a circular reasoning.
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Let's just ignore the first definition altogether, I'm fine sticking with the second only.

doronshadmi said:
HatRack's first and second definitions are not the same definition, exactly as non-existing thing is not the same as an existing thing.
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First-order logic would disagree with you. But, like I said, I'm fine with ignoring the first definition.

doronshadmi said:
Furthermore HatRack's second definition is weak and trivial w.r.t the first definition, because it explicitly defines an existing set only by non-existing things.
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You can say what you want about it, that doesn't change the fact that it is the definition of the empty set, as I have shown.

doronshadmi said:
You are wrong.
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Oh, great comeback! Doron, over the last 2 pages of this thread, you have been arguing against the single most stupid and trivial concept in all of mathematics. You've come up with increasingly dumber reasons as to why the empty set does not exist, and you've been shown to be wrong every time. You have also embarrassingly displayed your complete lack of ability to grasp basic logic numerous times.

This has nothing to do with mathematics and everything to do with the fact that you just cannot admit to being wrong. You would rather descend into the lowest depths of stupidity and completely make a mockery of whatever intelligence you do have than admit to defeat. Mathematics is the most purest and beautiful field of study we have, and you'll sadly never be able to experience it due to your own unrelenting stubbornness.
 
HatRack said:
First-order logic would disagree with you. But, like I said, I'm fine with ignoring the first definition.
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HatRack said:
You can say what you want about it, that doesn't change the fact that it is the definition of the empty set, as I have shown.
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You have shown the existence of the empty set of a mathematical universe that has one and only one existing thing.

Do you wish to know the name of this one and only one existing thing?
 
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doronshadmi said:
You have shown the existence of the empty set of a mathematical universe that has one and only one existing thing.
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And there it is, Doron finally admits that the empty set exists, at least in its own mathematical universe. Baby steps.

:bigclap
 
Hi Doron, long time no see! (Although I imagine you don't remember me =P)

Some time ago, about one year now, I made a few posts in this thread... something concerning Zeno's paradox if I recall. We had a brief back and forth, and you linked me to a collection of papers you had written.

I started to read them, but I got bogged down and ended up never finishing (and hence, never returning to this thread); however, at the time I had wanted to better understand and therefore better be able to refute your claims.

The little bit that I read led me to believe, however, that a good part of a number of peoples issues with you and your mathematics comes from some fundamental misunderstandings.

I was led to believe, that your non-local mathematics (I think it was called?) differs from standard mathematics through a combination of:

a) Somewhat different axioms
b) Somewhat differing definitions for some mathematical constructs and symbols

That second part, I believe, really leads to some of the arguments that never truly convince the other side of anything. When they say X they mean Y, when you say X you mean Z (where Y and Z are closely related but not quite the same). And so, when you read their post, you use your definition and when they read yours they use theirs and nobody gets anywhere.

If that collection of papers still exists, and still gives an accurate representation of the basics of your mathematics, I would like you to do two things for me:

I) If you could link me to those papers again I would be much obliged

II) If you could recommend to me an order in which to read said papers, that would be absolutely stupendous.

A good part of the reason I never finished reading those papers so long ago is that I would begin to read one, and then find a term with which I was unfamiliar and have to wade through a different paper (presumably a previous paper) to find the definition. It made for slow reading and headaches =(.

Don't get me wrong, I still believe you are mistaken about a number of things... but, the beauty of mathematics lies in the fact that it does not matter what I believe. Things are either true, or false. And I must yield the possibility that I may be mistaken, and all it will take for me to believe so is a clearly worded and logically sound proof of my mistakes. And so, I suppose that is what I seek - to either be shown the error of my ways or to show you yours.

tl;dr: You once had a series of papers that described your mathematical ideas, do you still have them? If so, what order should I read them in so as to understand them the best.
 
Sanelunatic said:
Don't get me wrong, I still believe you are mistaken about a number of things... but, the beauty of mathematics lies in the fact that it does not matter what I believe. Things are either true, or false.
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The problem is that when things show up either true or true, or false or false, then mathematics ends up with a few unsolved problems -- problems that were formulated way back in the nineteenth century.
 
HatRack said:
And there it is, Doron finally admits that the empty set exists, at least in its own mathematical universe. Baby steps.
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Sure HatRack, finally you are right and totally lonely in your singleton-only universe, like anyone who wins the battle but loses the war:



:boxedin:
 
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Sanelunatic said:
Hi Doron, long time no see! (Although I imagine you don't remember me =P)

Some time ago, about one year now, I made a few posts in this thread... something concerning Zeno's paradox if I recall. We had a brief back and forth, and you linked me to a collection of papers you had written.

I started to read them, but I got bogged down and ended up never finishing (and hence, never returning to this thread); however, at the time I had wanted to better understand and therefore better be able to refute your claims.

The little bit that I read led me to believe, however, that a good part of a number of peoples issues with you and your mathematics comes from some fundamental misunderstandings.

I was led to believe, that your non-local mathematics (I think it was called?) differs from standard mathematics through a combination of:

a) Somewhat different axioms
b) Somewhat differing definitions for some mathematical constructs and symbols

That second part, I believe, really leads to some of the arguments that never truly convince the other side of anything. When they say X they mean Y, when you say X you mean Z (where Y and Z are closely related but not quite the same). And so, when you read their post, you use your definition and when they read yours they use theirs and nobody gets anywhere.

If that collection of papers still exists, and still gives an accurate representation of the basics of your mathematics, I would like you to do two things for me:

I) If you could link me to those papers again I would be much obliged

II) If you could recommend to me an order in which to read said papers, that would be absolutely stupendous.

A good part of the reason I never finished reading those papers so long ago is that I would begin to read one, and then find a term with which I was unfamiliar and have to wade through a different paper (presumably a previous paper) to find the definition. It made for slow reading and headaches =(.

Don't get me wrong, I still believe you are mistaken about a number of things... but, the beauty of mathematics lies in the fact that it does not matter what I believe. Things are either true, or false. And I must yield the possibility that I may be mistaken, and all it will take for me to believe so is a clearly worded and logically sound proof of my mistakes. And so, I suppose that is what I seek - to either be shown the error of my ways or to show you yours.

tl;dr: You once had a series of papers that described your mathematical ideas, do you still have them? If so, what order should I read them in so as to understand them the best.
Click to expand...

Please start here:

http://www.internationalskeptics.com/forums/showpost.php?p=6597986&postcount=12671 .


I am open to your your criticism.
 
doronshadmi said:
The Man said:
So tell us Doron does your “empty set” have itself as a member?
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If {} is used to determine the empty state of {} (by {} as not being a member of {}), then a circular reasoning is used, because one can't define X by using X as a part of the definition of X.
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That does not answer the question.

The Man said:
So tell us Doron does your “empty set” have itself as a member?
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Why do you think you need to “determine the empty state of {}” are you not sure you defined your empty set as, well, empty?
 
doronshadmi said:
The answer no, but it used as one of the elements that determine the property of being empty existing thing, by not being a member of the empty set.

Any use (even negative use) of X as a part of the definition of X, is a circular reasoning.
Click to expand...

So as usual your conflict is just with yourself. As you define your empty set as not having itself as a member yet apparently deliberately misinterpret that as "circular reasoning".
 
doronshadmi said:
Sure HatRack, finally you are right and totally lonely in your singleton-only universe
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No Doron. I take the Axiom of the Empty Set, which you now agree is valid, and combine it with the other axioms of ZFC. This gives me the vast and infinite world of mathematics.
 
HatRack said:
No Doron. I take the Axiom of the Empty Set, which you now agree is valid, and combine it with the other axioms of ZFC. This gives me the vast and infinite world of mathematics.
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You can't, because your empty set exists in a universe of one and only one object, which is determined by one and only one axiom.

Combination is impossible in this universe, so you can't combine your axiom to the other ZFC axioms, because there is no room for them in your universe.



:boxedin:
 
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doronshadmi said:
In general, defining the existence of X by using the existence of X as a part of the definition, is a circular reasoning.
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That's not what you said before. Remember you cited your trusty wikipedia. Surely, you remember. There were words like "premise" and "proposition", remember?

So, tell us again (meaning for the first time) what premise in the axiom implicitly or explicitly assumed the proposition to be proved?

And just because I'm a caring individual, here's the axiom in question for your convenient reference:
[latex]$$$\exists x\, \forall y\, \lnot (y \in x)$$$[/latex]​
Now, what was that premise you had in mind?
 
doronshadmi said:
You can't, because your empty set exists in a universe of one and only one object, which is determined by one and only one axiom.

Combination is impossible in this universe, so you can't combine your axiom to the other ZFC axioms, because there is no room for them in your universe.



:boxedin:
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Combination is certainly far from impossible. Let us now introduce the Axiom of Pairing into "my universe", which states given that sets A and B exist the set {A,B} also exists. Now, we've established that the empty set exists in my universe, so now we know that {{},{}} exists. Or, more briefly, {{}}. From this, we deduce the existence of { {}, {{}} }, {{{ }}}, { {}, {{}}, {{{ }}} }, and so on. Although I have not defined infinity at this point, we now have a universe with an infinite number of existent sets from an intuitive point of view.

In fact, I can even define a finite amount of natural numbers at this point.

0 = {}
1 = {0}
2 = {0,1}
3 = {0,1,2}
etc...

Won't be long before I have the Peano Axioms + enough elementary set theory, which leads to the negation of your "central result". Of course, you know this, so you'll make up any excuse as to why I can't combine more ZFC axioms with the Axiom of the Empty Set. Unfortunately for you, you have no good reason as to why the assumed existence of more sets along with the empty set is impossible.
 
HatRack said:
Combination is certainly far from impossible. Let us now introduce the Axiom of Pairing into "my universe", which states given that sets A and B exist the set {A,B} also exists.
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In that case your axiom of the empty set contradicts your axiom of Pairing, because according to your axiom of the empty set there can be one and only one existing and empty object in your mathematical universe.

If more objects are defined in that universe, then your axiom of the empty set is a false determination.

So there is no room in your :boxedin: universe for any axiom that determines something about more than one existing and empty object.

Your entire universe is at most {}, which is an existing object with (traditional) cardinality 0 = |{}|.

{{},{}} is impossible in your :boxedin: universe, because there are no existing objects like {}={} outside of {}, such that "{} is not a members of {}", so {{}} does not exist in your universe.
 
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jsfisher said:
That's not what you said before. Remember you cited your trusty wikipedia. Surely, you remember. There were words like "premise" and "proposition", remember?
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http://en.wikipedia.org/wiki/Circular_reasoning
Circular reasoning is a formal logical fallacy in which the proposition to be proved is assumed implicitly or explicitly in one of the premises.
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This is exactly what I say: If the Premise = Conclusion (X's existence is used as a part of the definition that defines the existence of X) we are using a circular reasoning.
 
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jsfisher said:
At least you finally admit it's a contradiction. You've been denying that for years.
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You are wrong, it is clearly written in http://www.scribd.com/doc/18453171/International-Journal-of-Pure-and-Applied-Mathematics-Volume-49 :

We understand that a new Mathematical framework can be invented if one assumes that Points and Lines are two different independent elements. For instance, we examined the term “belongs to”. When we examine the way a point ‘belongs to’ a line, we can see that the point belongs locally to the line. In this local viewpoint there is an XOR connective between ‘belonging’ and ‘not belonging’ that prevents them from being simultaneously truthful. That is: A point can either belong or not belong to a line. Looking at this relationship from the line’s viewpoint we see that the line simultaneously belongs AND does NOT belong to the point. This can happen only if we see the line as an indivisible element. This might seemingly appear to be a logical contradiction but, after our investigation during which we demonstrated that a lot of our world is non-local, we understand that the contradiction exists when only the local viewpoint is used.
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This is another proof of your sloppy reading of my papers.
 
doronshadmi said:
In that case your axiom of the empty set contradicts your axiom of Pairing, because according to your axiom of the empty set there can be one and only one existing and empty object in your mathematical universe.
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Nope. My axiom does not say anywhere that there can be one and only one existing object in the universe. Try again.

doronshadmi said:
there are no existing objects like {}={} outside of {}, such that "{} is not a members of {}"
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And how do you come to the (false) conclusion that there are no existing objects outside of {}? This has the potential to be even more entertaining than the empty set / "circular reasoning" fiasco you created for yourself.

For reiteration:

Axiom of the Empty Set: The set {} exists, such that {} is the set of ONLY those elements X where X ≠ X.
 
doronshadmi said:
This is exactly what I say: If the Premise = Conclusion (X's existence is used as a part of the definition that defines the existence of X) we are using a circular reasoning.
Click to expand...

Your continual inability to grasp what a definition is and the true meaning of circular reasoning is one for the books.
 
HatRack said:
Nope. My axiom does not say anywhere that there can be one and only one existing object in the universe. Try again.
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If your empty set exits if and only if all of its "members" are ONLY of the form X≠X, then this axiom can't exist in a universe, where X=X AND X is not a member of the defined set.

In this strong universe of many existing things (where X=X AND X is not a member of the defined set) one of the existing things that are not members of defined set, is the defined set, so in this strong universe the premise that such a set exists can't be satisfied, unless the defined set itself is assumed not to be one of the objects that are members of the defined set.

In other words, Premise = Conclusion exactly because X's existence is used as a part of the definition that defines the existence of X, which is a circular reasoning.
 
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doronshadmi said:
If your empty set exits if and only if its members are of the form X≠X,
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Let's just analyse this opening sentence fragment, as there's so much wrongness and lack of comprehension demonstrated in it. Ignoring typos, what do you mean, "are of the form"? The members are not of a particular form, they are identified by comparing them with themselves. If they are not the same as themselves, then they are a member of the empty set. Since everything you examine is the same as itself, that means there are no members of the empty set. Sorry if I'm belabouring the point, but it seems to me that the way you keep referring to things, e.g. by saying "are of the form X≠X", that you are failing to grasp something that is so obvious and intuitive that if you did understand it you simply wouldn't express it like that.

then this axiom can't exist in a universe, where X=X AND X is not a member of the defined set.
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I have no idea what you are trying to say. Yes, X=X, in all cases, which means that any X you examine will not be a member of the set being defined. The empty set is empty.

In this strong universe of many existing things (where X=X AND X is not a member of the defined set) one of the existing things that are not members of defined set, is the defined set, so in this strong universe the premise that such a set exists can't be satisfied, unless the defined set itself is assumed not to be one of the objects that are members of the defined set.
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No assumption is needed. You simply apply the test. Does it equal itself? Yes; then it is not a member of the empty set.
 
doronshadmi said:
If your empty set exits if and only if all of its "members" are ONLY of the form X≠X, then this axiom can't exist in a universe, where X=X AND X is not a member of the defined set.
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The premise of this implication is false, so any conclusion you draw from it has nothing to do with my mathematical universe. The existence of my empty set is due to an axiom, not due to its contents, or lack thereof. Let's examine your premise closer to see exactly why it is nonsense, after a couple of corrections:

The empty set exists if and only if all of its members satisfy X≠X.

This is a biconditional statement, so it is equivalent to:

If the empty set exists, then all of its members satisfy X≠X, AND if all of the empty set's members satisfy X≠X, then the empty set exists.

The second implication is absurd. The empty set exists in my universe because of an axiom, not because its members satisfy a certain property. So Doron, try again.

doronshadmi said:
In this strong universe of many existing things (where X=X AND X is not a member of the defined set) one of the existing things that are not members of defined set, is the defined set, so in this strong universe the premise that such a set exists can't be satisfied, unless the defined set itself is assumed not to be one of the objects that are members of the defined set.
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You need to learn to read for comprehension. But, even more so, as this sentence demonstrates, you need to learn to WRITE for comprehension. And what is a strong universe as opposed to a weak one or a regular one?

doronshadmi said:
In other words, Premise = Conclusion exactly because X's existence is used as a part of the definition that defines the existence of X, which is a circular reasoning.
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Again with your inability to understand what mathematical definitions mean. Say it with me: DEFINITIONS DO NOT ASSERT EXISTENCE.
 
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doronshadmi said:
This is exactly what I say: If the Premise = Conclusion (X's existence is used as a part of the definition that defines the existence of X) we are using a circular reasoning.
Click to expand...

Here is the axiom, again:

[latex]$$$\exists x\, \forall y\, \lnot (y \in x)$$$[/latex]​

Now, what was that premise you had in mind?
 
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