Deeper than primes

[The Man]

Since we are dealing with a collection of distinct points that its cardinality = |R|, then this is your challenge to show how to define a target without missing any point, by using the agreed reasoning.

So please show how it is done.

What “agreed reasoning”? You mean your ridiculous game where you must skip an infinite number of points to select your first “target”?

No, a continuous 1D space is not a collection of distinct 0D elements, as your reasoning claims.

Doron we have been over the definition of a continuous space before. Again stop trying to simply pawn off your own misinterpretations (again perhaps deliberately) as some one else’s “reasoning”.

 

[The Man]

Specially for you:

There is a closed 1D path, such that:

(x>0)/0 is 0 0D elements on it.

(x>0)/1 is 1 0D elements on it.

...

(x>0)/|R| is |R| 0D elements on it.

Here are (x>0)/0 and (x>0)/1 cases (at the top of diagrams 1,2,3):

[qimg]http://farm5.static.flickr.com/4039/4297878664_d74c38b77e_o.jpg[/qimg]

"Specially for you:"

Nonsense

 

[The Man]

The Man,

You asked me to show a missing point along a line segment.

My answer is: the statement "there are distinct 0D elements with no gaps between them", is always false (a contradiction).

You are invaded to show that the statement "there are distinct 0D elements with no gaps between them", is always true (a tautology).

Your “statement” still does not show any location on a line or line segment that can not be covered by points.


Again as it is your claim that “the statement "there are distinct 0D elements with no gaps between them", is always false (a contradiction).” Then show those gaps or that gap that “always” must be there.


You are “invaded” to show the validity of any of your statements yourself, including the one about points not covering a line.

 

[epix]

No, a continuous 1D space is not a collection of distinct 0D elements, as your reasoning claims.

I guess that in OM 1-D space isn't the same as 1-D geometric object. Fortunately, "1-D space" is a non-existent term outside OM -- a term that would ravage the description of methods of computation designed to produce real results that could be actually applied to solve problems in science and technology.

Can you draw that 1-D space without verbalizing the heck out of it?

 

[epix]

No problem, if we are talking about the magnitude of existence of 0D elements on a closed 1D element then:

Here is that all-inspiring term "magnitude of existence" once again.

We know from the experience of everyday life that "magnitude" relates to various scales. The word "magnitude" usually makes a headline when an earthquake of magnitude like 7.3 on the Richter scale hits a populated area. But when "magnitude" is joined by "existence," the marriage between both nouns produces a child called "7.3 magnitude on the existence scale," for example. In other words, can you exist 6.48 times or 2.78 times?

Contemporary English somewhat shies away from the plural form of the word "existence," but if the plural is used, the quantity is limited to whole numbers or held in an indefinite form, as common sense doesn't permit otherwise. Here is an example:

To affirm that there is an "infinite, intelligent Being apart from the universe" is to distinguish it from the universe, and to contend for two existences.

Everything to be thought of must exist in some place and in some relation to other existences, and therefore to speak of one being apart from all else is the annihilation in thought of that one.

And the icing on the cake . . .

For again let me observe that the uncaused existences which could produce the universe, itself infinitely splendid, superb, and intelligent, must -- were it possible -- be still more wonderful an superb than the universe or Nature, which they are said to have produced; and consequently there is greater difficulty in conceiving them self-existent than in conceiving the unbounded universe self-existent.

Who could stitch together ill-fitting pants better than a 19th century atheist?

There is a possibility that Charles Watts exists twice -- in sequence. If you experience frequent dreams about Victorian England, then it could be a real possibility.

Of course, there is other option that limits Charlie Watts' magnitude of existence to 1.

Out of monkeys we all come.

 

[doronshadmi]

"Specially for you:"

Nonsense

It is nonsense for any one how gets things only in terms of collections of elements that have 0 degrees of freedom.

 

[doronshadmi]

Doron we have been over the definition of a continuous space before. Again stop trying to simply pawn off your own misinterpretations (again perhaps deliberately) as some one else’s “reasoning”.

The Man we have been over the definition of a continuous space before, where real continuum is not a collection of elelemts that have 0 degrees of freedom. Again stop trying to simply pawn off your false agreed misinterpretations (again perhaps deliberately) as the right reasoning of the continuum.

 

[doronshadmi]

Your “statement” still does not show any location on a line or line segment that can not be covered by points.

The Man, the statement "there are distinct 0D elements with no gaps between them" is exactly a claim that is derive from the agreed reasoning about R.

This statment is an axiom of the reasoning that you are using, so you are using a reasoning which is based on contradiction.

 

[epix]

The Man, the statement "there are distinct 0D elements with no gaps between them" is exactly a claim that is derive from the agreed reasoning about R.

Listen, all what you see around are 3D objects. But if you look at the table, for example, from above at a precise angle, you may come under the impression that the table is a 2D object of a rectangular shape. Now when you look at a 0D object, you can't see it, as much as you can't see 0 grapefruits. So how do you know if there are any gaps between 0 grapefruits arranged in a line?

>>>> There is a way to find out, though: let G stand for a grapefruit/point, which is a 0D object: point = G0D.

What, what? What's happening? What time is it?

It's exactly 00:00:00 UTC.

You didn't wake me up at the midnight hour to discuss ordered 0-dimensional objects, did you?

Your omniscience works round the clock, doesn't it?

There are two blemishes on my Creation: you and Doron. Get the hell outta my bedroooooom! OOOOOOOUT!!!

See it, Doron? O stands for Objects and Zero as well. That UT stands for Universal Time. That means O also stands for the face of the clock. Time is measured in different units, which are essentially points on the time line, such as years. If there is a gap between years/points, then we will go from 2011 right to 2013 to avoid God's retaliation for the "midnight question." You better be right about those gaps.

It would be great to be able to physically digitize time. That would make the time travel possible - theoretically. If there were gaps between time points, you could use them to rearrange time segments.

 

[doronshadmi]

Doron if it is “defined as” a member of itself then it is "identical to" at least one member of itself, namely itself. For some reason that simple concept still seems to be beyond you.

Since your used reasoning is based on "defined by" it can't deal with the difference between Set and Member.

Fact a: The empty set {} is not identical with its members.

Fact b: The non-empty set {{}} is not identical with its member because it is not identical with {}, which is the member of the non-empty set {{}}.

Fact c: The non-empty set {{{}}} is not identical with its member because it is not identical with {{}}, which is the member of the non-empty set {{{}}}.

Fact ∞: Etc… at infinitum.

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

Fact 'a: The empty set {} is defined by its members.

Fact 'b: The non-empty set {{}} is defined by its members.

Fact 'c: The non-empty set {{{}}} is defined by its members.

Fact '∞: Etc… at infinitum.

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

Fact a AND Fact 'a is a contradiction.

Fact b AND Fact 'b is a contradiction.

Fact c AND Fact 'c is a contradiction.

Fact ∞ AND Fact '∞ is a contradiction … at infinitum.

 

[doronshadmi]

A correction of http://www.internationalskeptics.com/forums/showpost.php?p=6313836&postcount=11490.

Since your used reasoning is based only on "defined by" it can't deal with the difference between Set and Member.

In order to get the difference between "identical to" and "defined by", we reduce our universe to empty and non-empty sets.

Fact a: The empty set {} is not identical to its members.

Fact 'a: The empty set {} is defined by its members.

Fact b: The non-empty set {{}} is not identical to its members because it is not identical to {}, which is the member of the non-empty set {{}}.

Fact 'b: The non-empty set {{}} is defined by its members.

Fact c: The non-empty set {{{}}} is not identical to its members because it is not identical to {{}}, which is the member of the non-empty set {{{}}}.

Fact 'c: The non-empty set {{{}}} is defined by its members.

…

Fact ∞: The non-empty set {…{{}}…} is not identical to its members because it is not identical to …{{}}…, which is the member of the non-empty set {…{{}}…}.

Fact '∞: The non-empty set {…{{}}…} is defined by its members.

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

"Fact a is Fact 'a" is false.

"Fact b is Fact 'b" is false.

"Fact c is Fact 'c" is false.

"Fact ∞ is Fact '∞" is false … ad infinitum, therefore a contradiction.

 

[The Man]

It is nonsense for any one how gets things only in terms of collections of elements that have 0 degrees of freedom.

Once again stop simply trying to posit aspects of your own failed reasoning onto others.

The Man we have been over the definition of a continuous space before, where real continuum is not a collection of elelemts that have 0 degrees of freedom. Again stop trying to simply pawn off your false agreed misinterpretations (again perhaps deliberately) as the right reasoning of the continuum.

Once again stop simply trying to posit aspects of your own failed reasoning onto others.

The Man, the statement "there are distinct 0D elements with no gaps between them" is exactly a claim that is derive from the agreed reasoning about R.

This statment is an axiom of the reasoning that you are using, so you are using a reasoning which is based on contradiction.

Once again stop simply trying to posit aspects of your own failed reasoning onto others.

Since your used reasoning is based on "defined by" it can't deal with the difference between Set and Member.

Once again stop simply trying to posit aspects of your own failed reasoning onto others.

Fact a: The empty set {} is not identical with its members.

As it has no members.

Fact b: The non-empty set {{}} is not identical with its member because it is not identical with {}, which is the member of the non-empty set {{}}.

As that “member” (at least as you seem to indicate) is the empty set and not “The non-empty set {{}}”.

Fact c: The non-empty set {{{}}} is not identical with its member because it is not identical with {{}}, which is the member of the non-empty set {{{}}}.

Fact ∞: Etc… at infinitum.

Still comparing apples and oranges “at infinitum“, again a set that has itself as a member is identical to at least one of it members, namely itself.

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

Fact 'a: The empty set {} is defined by its members.

Once again as the empty set has no members it is specifically defined by its lack of members (which excludes even itself as a member).

Fact 'b: The non-empty set {{}} is defined by its members.

Fact 'c: The non-empty set {{{}}} is defined by its members.

Fact '∞: Etc… at infinitum.

Technically it is the set (specifically the definition of that particular set) that defines what constitutes its members.

A “non-empty set” simply defines the set as not being empty and thus not the empty set. It neither restricts nor requires that set to be, or from being, a member of itself.

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

Fact a AND Fact 'a is a contradiction.

Fact b AND Fact 'b is a contradiction.

Fact c AND Fact 'c is a contradiction.

Fact ∞ AND Fact '∞ is a contradiction … at infinitum.

Doron if you feel your ‘Facts’ are contradictory then you should probably try to assert more self-consistent ‘Facts’. As usual you only seem to perceive the self-contradictory nature of your own assertions when you are simply trying to ascribe them to someone else.

 

[The Man]

A correction of http://www.internationalskeptics.com/forums/showpost.php?p=6313836&postcount=11490.

I doubt it will correct much, if anything.

Since your used reasoning is based only on "defined by" it can't deal with the difference between Set and Member.

Once again stop simply trying to posit aspects of your own failed reasoning onto others.

In order to get the difference between "identical to" and "defined by", we reduce our universe to empty and non-empty sets.

What set is neither "empty" nor "non-empty"? Your 'reduction’ seems rather superfluous.

Fact a: The empty set {} is not identical to its members.

Again as it specifically can not include itself as a member.

Fact 'a: The empty set {} is defined by its members.

Again it is specifically defined by its lack of members, which again excludes itself from being a member of itself.

Fact b: The non-empty set {{}} is not identical to its members because it is not identical to {}, which is the member of the non-empty set {{}}.

Again “The non-empty set {{}} is not identical to its members” because it does not include itself as a member as it is evidently only a set with only the empty set as a member.

Fact 'b: The non-empty set {{}} is defined by its members.

Again it is the set that defines what constitutes it members and you are evidently defining a set that has only the empty set as its member.

Fact c: The non-empty set {{{}}} is not identical to its members because it is not identical to {{}}, which is the member of the non-empty set {{{}}}.

Fact 'c: The non-empty set {{{}}} is defined by its members.

…

Fact ∞: The non-empty set {…{{}}…} is not identical to its members because it is not identical to …{{}}…, which is the member of the non-empty set {…{{}}…}.

Fact '∞: The non-empty set {…{{}}…} is defined by its members

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

"Fact a is Fact 'a" is false.

"Fact b is Fact 'b" is false.

"Fact c is Fact 'c" is false.

"Fact ∞ is Fact '∞" is false … ad infinitum, therefore a contradiction.

So you still assert your ‘Facts’ to be contradictory and “false”. That is still simply your problem. Where exactly is this “correction” that you asserted?

 

[epix]

Specially for you:

For me?

Yes. For You and Your Divine Omniscience. Go figure . . .

I can do it here -- or do you have an issue with My Presence?

Oh, no. Just . . . figure.

Isn't that Doron's handwriting?

Yes. You don't suppose that I would bring to Your Attention some of the Stephen Hawking's trivialities.

Let me see . . . Hmm . . . Oh, yes! That's "Nocturne."

Huh?

Oh, I'm pretty positive. That's "Nocturne."

What's your point? I mean . . . How did Your Omniscience arrive at this non-apparent conclusion?

The shape of the geometric figures is O and there are 3x9=27 of those circles. So you need to solve O___? 27. The solution is Opus 27.

Aha. But there are more Opuses 27. How do you know that Opus 27 is "Nocturne?"

Just look at the circles; how these circumference points are connected. See? LAAA-LA-LA-LA-LAAA--------LAAAA-LA-LA-LA. That's "Nocturne." Ain't that beautiful?

It's vastly illogical.

What do you mean by "illogical?"

I saw a dude play that piece on YouTube, and he played white keys too.

Of course. That's how Chopin wrote it.

Yeah, but the word "nocturne" derives from Latin "nocturnus" that pertains to the nighttime. So that dude on YouTube got no business playing the white keys. Only dark, I mean the black keys are supposed to be played to bring the desired feel of the darkness of the night wrestling with the occasional moonlight . . . What?

I said, come closer to Me, for I shall whisper something to your ear. There have been two major catastrophes in the history of mankind: you and Doron.




Don't give up, Doron. Keep thinking and inventing. We'll get Him one day.

 

[jsfisher]

Oh, my. Still mired in the trivial and the meaningless. So many years invested and not a single result. At this point, Doron is too invested (and too embarrassed) to admit it is all for naught.

 

[doronshadmi]

So you still assert your ‘Facts’ to be contradictory and “false”. That is still simply your problem. Where exactly is this “correction” that you asserted?

EDIT:

You actually assert that the statement "A ∈ A" is the statement "A = A".

By following this reasoning there is no difference between the identity of set A to itself and the identity of the member of set A, to set A.

In this case there is no difference between the concept of Set and the concept of Member of a given set, and since the concept Member is the concept of Set all we get is the statement "A = A" ("Set A is identical to itself").

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

But unlike your reasoning, the statement "A ∈ A" is not the same as the statement "A = A" as follows:

A is a set.

"A = {A}" means that A is a member of itself (the statement "A ∈ A" is the statement "A = {A}").

Let as follow your assertion that statement "A ∈ A" is the statement "A = A":

If Member A is identical to Set {A}, then we get {A} as a member of itself,
such that A = {{A}}, which contradicts the assertion that the member of A is set A.

If Member {A} is identical to Set {{A}}, then we get {{A}} as a member of itself,
such that A = {{{A}}}, which contradicts the assertion that the member of A is set A.

…

If Member ...{{{A}}}... is identical to Set {…{{{A}}}…} , then we get {…{{{A}}}…} as a member of itself, such that A = {{…{{{A}}}…}}, which contradicts the assertion that the member of A is set A.

Etc... ad infinitum ...

In other words, the assertion that Member is identical to Set and vice versa is not satisfied, or in other words, it is always false (a contradiction).

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

On the other hand the statement "A set is defined by its members" is always true (tautology).

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

So The Man, Since by your reasoning "A=A" = "A∈A" = "A={A}", it is based on contradiction.

You simply do not distinguish between "defined by" (tautology) and "identical to" (contradiction) w.r.t Set and Member concepts.

 

[doronshadmi]

Oh, my. Still mired in the trivial and the meaningless. So many years invested and not a single result. At this point, Doron is too invested (and too embarrassed) to admit it is all for naught.

"Oh, my." is the name of your box, and so many years you are closed in your box, until you do not have the reasoning to get things beyond it.

 

[The Man]

You actually assert that the statement "A ∈ A" is the statement "A = A".

Nope, one is a statement asserting “A” as an element of “A” and the other a statement asserting the equality of “A” with itself. Again if “A” is an element of itself then “A” is (by that assertion of it being an element of itself) equal to at least one element of itself. However, even this conditional circumstance still does not make your statements the same. As they, again, assert different things and “A” still equals “A” even when “A” is not an element of itself. Your claim and “direct perception” fails, as usual.

Once again stop simply trying to posit aspects of your own failed reasoning onto others.

By following this reasoning there is no difference between the identity of set A to itself and the identity of the member of set A, to set A.

In this case there is no difference between the concept of Set and the concept of Member of a given set, and since the concept Member is the concept of Set all we get is the statement "A = A" ("Set A is identical to itself").

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

But not like your reasoning, the statement "A ∈ A" is not the same as the statement "A = A" as follows:

A is a set.

"A = {A}" means that A is a member of itself (the statement "A ∈ A" is the statement "A = {A}").

Let as follow your assertion that statement "A ∈ A" is the statement "A = A".


But if Member is identical to Set, then we get {A} as a member of itself,
such that A = {{A}}.

But if Member is identical to Set, then we get {{A}} as a member of itself,
such that A = {{{A}}}.

…

But if Member is identical to Set, then we get …{{{A}}}…. as a member of itself,
such that A = {…{{{A}}}…}.

In other words, the assertion that Member is identical to Set and vice versa is not satisfied, or in other words, it is always false (a contradiction).

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

On the other hand the statement "A set is defined by its members" is always true (tautology).

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

So The Man, Since by your reasoning "A=A" = "A∈A" = "A={A}", it is based on contradiction.

Again Doron sets being members of themselves causes problems and paradoxes, which is again why the class of all sets that do not have themselves as members is considered a proper class and a set can not be a proper subset of itself.


http://en.wikipedia.org/wiki/Class_(set_theory)

Paradoxes
The paradoxes of naive set theory can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper. For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper.

http://en.wikipedia.org/wiki/Proper_subset#proper_subset

Examples
The set {1, 2} is a proper subset of {1, 2, 3}.
Any set is a subset of itself, but not a proper subset.
The empty set, denoted by Ø, is also a subset of any given set X. (This statement is vacuously true.) The empty set is always a proper subset, except of itself.
The set {x: x is a prime number greater than 2000} is a proper subset of {x: x is an odd number greater than 1000}
The set of natural numbers is a proper subset of the set of rational numbers and the set of points in a line segment is a proper subset of the set of points in a line. These are counter-intuitive examples in which both the part and the whole are infinite, and the part has the same number of elements as the whole (see Cardinality of infinite sets).

We have been over this before, however if you simply want to continue asserting the problems and paradoxes of naïve set theory, that is up to you.

 

[doronshadmi]

Again if “A” is an element of itself then “A” is (by that assertion of it being an element of itself) equal to at least one element of itself.

If A has elements that are identical to A, then there is no difference between A and the element of A.

In that case A = element of A, which leads to infinite regression, as follows:

A = {A}, where an element of A is identical to A, so since the element of A is identical to A, then the element of A = {A}, and in this case A = {{A}}, but since the element of A is identical to A, then the element of A = {{A}}, and in this case A = {{{A}}}, etc… etc… ad infinitum.

In other words, there is no identity between A and an element of A, which is a simple fact that the agreed reasoning (that you quote form wiki, which is also accepted by you) can't get.

EDIT:

“A” still equals “A” even when “A” is not an element of itself. Your claim and “direct perception” fails, as usual.

So what?, you are the one that asserts that there is no difference between A's self identity and A's element.

Once again your lack of direct perception, fails you.

We have been over this before, however if you simply want to continue asserting the problems and paradoxes of naïve set theory, that actually do not exist, exactly as proper classes do not exist.

Again Doron sets being members of themselves causes problems and paradoxes,...

Again The Man, since no element of a given set is identical to that set, then there is no problem, because the assertion that en element of A is identical to A, can't be satisfied, exactly because it is based on infinite regression, as clearly shown above.

Let us take this part from wiki ( http://en.wikipedia.org/wiki/Proper_subset#proper_subset ):

If A is a subset of B, but A is not equal to B

From this part we understand that if A is equal to B then we have two names for the same object. But we can't conclude that because a given set has more than one name, then an element of that set is identical to that set, because the two names are the set, where no one of the names is an element of that set.

 

[epix]

"Oh, my." is the name of your box, and so many years you are closed in your box, until you do not have the reasoning to get things beyond it.

If Condition A leads to a contradiction via a paradox

AND

Condition A is conceptually identical to Condition B

THEN

a construction of a 4-D object suffers from a contradiction.


Q: What is the particular paradox/contradiction?

A:


knock...knock...knock...

Ask Doron. I'm busy.

How did You know that I wanted to ask You a question?

I'm like omniscient, aren't I?