Deeper than primes

[Sanelunatic]

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.

 

[epix]

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.

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.

 

[doronshadmi]

And there it is, Doron finally admits that the empty set exists, at least in its own mathematical universe. Baby steps.

Sure HatRack, finally you are right and totally lonely in your singleton-only universe, like anyone who wins the battle but loses the war:

 

[doronshadmi]

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.

Please start here:

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


I am open to your your criticism.

 

[The Man]

So tell us Doron does your “empty set” have itself as a member?

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.

That does not answer the question.

So tell us Doron does your “empty set” have itself as a member?

Why do you think you need to “determine the empty state of {}” are you not sure you defined your empty set as, well, empty?

 

[The Man]

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.

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".

 

[HatRack]

Sure HatRack, finally you are right and totally lonely in your singleton-only universe

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.

 

[doronshadmi]

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.

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.

 

[doronshadmi]

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".

You are not updated, please start at least from http://www.internationalskeptics.com/forums/showpost.php?p=6602376&postcount=12742 .


In general, defining the existence of X by using the existence of X as a part of the definition, is a circular reasoning.

 

[jsfisher]

In general, defining the existence of X by using the existence of X as a part of the definition, is a circular reasoning.

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?

 

[HatRack]

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.

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.

 

[doronshadmi]

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.

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 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 universe, because there are no existing objects like {}={} outside of {}, such that "{} is not a members of {}", so {{}} does not exist in your universe.

 

[doronshadmi]

That's not what you said before. Remember you cited your trusty wikipedia. Surely, you remember. There were words like "premise" and "proposition", remember?

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.

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.

 

[doronshadmi]

At least you finally admit it's a contradiction. You've been denying that for years.

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.

This is another proof of your sloppy reading of my papers.

 

[HatRack]

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.

Nope. My axiom does not say anywhere that there can be one and only one existing object in the universe. Try again.

there are no existing objects like {}={} outside of {}, such that "{} is not a members of {}"

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.

 

[HatRack]

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.

Your continual inability to grasp what a definition is and the true meaning of circular reasoning is one for the books.

 

[doronshadmi]

Nope. My axiom does not say anywhere that there can be one and only one existing object in the universe. Try again.

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.

 

[zooterkin]

If your empty set exits if and only if its members are of the form X≠X,

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.

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.

No assumption is needed. You simply apply the test. Does it equal itself? Yes; then it is not a member of the empty set.

 

[HatRack]

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.

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.

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.

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?

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.

Again with your inability to understand what mathematical definitions mean. Say it with me: DEFINITIONS DO NOT ASSERT EXISTENCE.

 

[jsfisher]

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.

Here is the axiom, again:

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

Now, what was that premise you had in mind?