Deeper than primes

[epix]

This is batter than only verbalize:

3.14159...[base 10] is the following endless fractal (exactly because on amount of locations can reduce any given line segment between the locations, into a point):

[qimg]http://farm5.static.flickr.com/4086/5105470874_17e5aaa5f3_b.jpg[/qimg]
[qimg]http://farm2.static.flickr.com/1256/5105473512_aa49164260_z.jpg[/qimg]
[qimg]http://farm2.static.flickr.com/1390/5104877745_f8f2627061.jpg[/qimg]
[qimg]http://farm2.static.flickr.com/1086/5105473650_5aaa36b4b5_z.jpg[/qimg]
[qimg]http://farm5.static.flickr.com/4085/5105473724_9ee904f365.jpg[/qimg]
[qimg]http://farm2.static.flickr.com/1093/5104877951_6cb1bafc28.jpg[/qimg]

...

etc ... ad infinitum, where 3.14159...[base 10] do not reaches pi location (marked by the green vertical line).

Doron, 1, 2, 3, 4, ... doesn't reach the "infinity location" either. In the domain of the approximate numerical format, Pi also refers to a series that forms the infinite sequence of digits

3.1415926535897932384626433832795028841971693...

With each digit added, the number is getting more and more precise, but there is really no known point pi in the approximate numerical format on the line that the series is approaching, coz you can't skip the order of integers in the formula that lives in the infinite loop that spits out the irrational numbers that more and more resemble pi. There is no way to jump ahead and see exactly what number is the series approaching, but can't ever reach, as much as there is no way to skip time to see what kind of crap is happening on 12/21/2012. When you see something like this

∞
∑ (50k - 6)/(2^k*(3k nCr k)) = pi
k=1

it just means that the series generates a particular infinite sequence of digits denoted "pi."

Of course, there is an empty space between "precise" and "more precise" or P<MP in case of pi, but that space is constantly being filled with additional points after a certain number of iterations in one of the formulas that generate the sequence of digits known as pi. (Some formulas are faster than others.)

So what is really "the limit" that the 3.14159... is approaching but can't ever reach, so there is always a space between the limit and the approaching value -- something that makes you to believe that you are onto something? I give you a clue: Reconcile the difference between divergence and convergence and find out what is the limit/number of 0, 1, 2, 3, 4, ... that the sequence of naturals is approaching but can't ever reach.

 

[punshhh]

Why didn't you use "unit of length" instead? For example, 1 meter is the base unit of length in the International System of Units.

How long would 1 Drag be?

Well, this is the same as if you asked what is the temperature of 1 meter. The unit of movement 1 Drag got nothing to do with distances, as much as the unit of inhalation 1 drag got nothing to do with volume. When a moving object comes to rest and then starts to move again, then the process is measured in Drags -- it's a measure of occurrence. For example 10 Drags/s means that an object came to rest and again accelerated ten times in one second.

Ok, so two 0 dimensional objects with 1 drag between them are further apart than two 0 dimensional objects with an infinite number of 0 dimensional objects placed side by side between them. Even if 1 drag is infinitely short, it would always be longer.

 

[doronshadmi]

Again please show some location(s) on any of your “ever smaller sub-line segments” that is not and can not be covered by (a) point(s).

Again please show the set of all infinitely many different points that exist along a line segment,without ever smaller sub-line segments between them.

It does matter how many times you repeat your “uncovered” nonsense when you can’t show anything is uncovered

It does matter how many times you repeat your “covered” nonsense when you can’t show anything that is the closest AND different points.

or define your smallest line segment such that it is uncovered (which would give you a desecrate space).

Again you are using a finite reasoning in order to understand the infinite reasoning of the ever smaller.

It is your usual self-contradictory problem Doron, you want an “infinite amount of ever smaller sub-line segments” and no “smallest line segment” (a continuous space) but also want there to be some “uncovered line segment” (a desecrate space).

The uncovered space is exactly the continuum itself as an ever smaller sub-line segment, which is irreducible to a discrete existing thing like a point.

The Man, the collection of all infinitely many different points is not a continuous space, exactly because it is no more than a collection of all different AND smallest elements, where this existence is impossible without the ever smaller actual continuum state between the arbitrary closer smallest (and therefore discrete) collection of points.

You still can't comprehend the the co-existence of all different AND smallest elements AND all sub-line segments, where the all different AND smallest elements not reach the actual continuum of any ever-smaller sub-line segment, along some considered line segment.

 

[doronshadmi]

(Some formulas are faster than others.)

Exactly as some points are closer than others to pi.

The fastest formula does not exist exactly because no ever smaller sub-line segment is reducible to a point along the line segment, which its size= circumference/diameter.

 

[doronshadmi]

Let's take it one step at a time. I'll ask a short question (and I expect a short straight answer). What are the elements of {}?

Do you understand what a range of mapping from no mapping to bijection is?

A = {}

B = {A}

↔ A (no mapping)

A ↔ A (bijection)

is some example of such range.

 

[doronshadmi]

Again I ask: Can we expect to see your bijections between the elements of {} and its power set ...

http://www.internationalskeptics.com/forums/showpost.php?p=7072978&postcount=14985

 

[sympathic]

Do you understand what a range of mapping from no mapping to bijection is?

A = {}

B = {A}

↔ A (no mapping)

A ↔ A (bijection)

is some example of such range.

You see - you are talking to yourself again. I asked a simple question: what are the elements of {}? - can I get a simple answer?

 

[doronshadmi]

No, it is not. Cantor's Theorem is a theorem. Doron has been corrected on this before, but he appears unable to understand the difference between a theorem and its proof.

Nonsense.

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. http://en.wikipedia.org/wiki/Theorem

jsfisher does not distinguish between Conjecture and Theorem, where some example of theorem is called Cantor's theorem.

In the standard proof for Cantor's Theorem a contradiction develops by assuming a bijection exists be all of the members of P(S) and all of the members of S, not just one from each as Doron states.

Indeed cantor assumes that there is a bijection between all of the members of P(S) and all of the members of S.

But then he shows a member of P(S) that any attempt to define mapping between this member and some S member, leads to contradiction (because according to Cantor's construction rules of the considered P(S) member, S is a member AND not a member of the considered P(S) member, which is a logical contradiction, or in other words, there can't be a mapping between this P(S) member any one of the S members) By using this logical contradiction and the fact that |P(S)| is at least = |S| (by the a ↔ {a} trivial mapping) Cantor concludes that the initial assumption (that there is a bijection between all of the members of P(S) and all of the members of S) is a false assumption.

And this may well be the core of wrongness. Doron has convinced himself Cantor's Theorem is a construction method for P(S) using some set of bijections that he even admits do not exist.

And this may well be the core of wrongness. jsfisher simply can't grasp that Cantor's construction method of all P(S) members (without missing any P(S) member), is independent of Cantor's theorem and its logical restrictions.

Doron also continues to be quite proud of himself for is claimed method to build P(S) given P(S).

jsfisher really believes that there is a way to build some non-empty set, without the members of this set.

That pretty much explains what Doron considers achievement.

That pretty much explains jsfisher's "understanding" of this fine subject.

 

[doronshadmi]

You see - you are talking to yourself again. I asked a simple question: what are the elements of {}? - can I get a simple answer?

You see, you ask rhetoric questions exactly because you don't want to look beyond your box.

EDIT:

The range of mapping from no mapping to bijection is a straightforward answer that does not follow after your rhetoric question.

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=7006557&postcount=14647 in order to really understand my answer.

You and your friends here do not understand that in additional to no mapping, one can define any wished degree of mapping between a non-empty set that its members are not subsets, and a non-empty set that its members are subsets.

 

[sympathic]

You see, you ask rhetoric questions exactly because you don't want to look beyond your box.

EDIT:

The range of mapping from no mapping to bijection is a straightforward answer that does not follow after your rhetoric question.

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=7006557&postcount=14647 in order to really understand my answer.

You and your friends here do not understand that in additional to no mapping, one can define any wished degree of mapping between a non-empty set that its members are not subsets, and a non-empty set that its members are subsets.

See - talking to yourself again. You are beyond all hope. Enjoy your mumblings.

 

[doronshadmi]

See - talking to yourself again. You are beyond all hope. Enjoy your mumblings.

Once again.

You and your friends here do not understand that in addition to no mapping, one can define any wished degree of mapping between a non-empty set that all its members are not subsets (for example {a,b,c,d,...}), and a non-empty set that most of its members are subsets (for example {{},{a},{b},{c},{d},... , {a,b,c,d,...}}).

Enjoy your box. It can't let you to understand http://www.internationalskeptics.com/forums/showpost.php?p=7006557&postcount=14647 .

 

[epix]

Ok, so two 0 dimensional objects with 1 drag between them are further apart than two 0 dimensional objects with an infinite number of 0 dimensional objects placed side by side between them. Even if 1 drag is infinitely short, it would always be longer.

No, it's not okay. Read again my reply where it says that

The unit of movement 1 Drag got nothing to do with distances, ...

and so your reference to 1 Drag being infinitely short misses entirely the definition of the measure in question. Also, you wrote "1 drag" and not 1 Drag, which adds to the confusion, coz as I've already mentioned, 1 drag is a unit of inhalation where 1 drag = 1.000028 grasps -- not to be confused with 1 Grasp (upper-case G), which is a unit of knowledge acquired through comprehension. Btw, do you know that the unit of disbelief 1 eyeroll equals pi/6 neckturns?

 

[doronshadmi]

Even if 1 drag is infinitely short, it would always be longer.

Exactly.

 

[doronshadmi]

The unit of movement 1 Drag got nothing to do with distances,

It does not matter, you get at least two distinct elements.

 

[jsfisher]

No, it is not. Cantor's Theorem is a theorem. Doron has been corrected on this before, but he appears unable to understand the difference between a theorem and its proof.

Nonsense.

In what way is it nonsense? I provided a clear example of a proof which you referred to as Cantor's Theorem. Proofs are not theorems and theorems are not proofs, yet you continue to confuse the two.

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. http://en.wikipedia.org/wiki/Theorem

jsfisher does not distinguish between Conjecture and Theorem, where some example of theorem is called Cantor's theorem.

Where did I do that?

In the standard proof for Cantor's Theorem a contradiction develops by assuming a bijection exists be all of the members of P(S) and all of the members of S, not just one from each as Doron states.

I stand by this statement.

Indeed cantor assumes that there is a bijection between all of the members of P(S) and all of the members of S.

Cantor's Theorem does no such thing. Again, you have conflated theorem with proof. The standard proof, on the other hand, does conditionally assume such a bijection does exist.

But then he shows a member of P(S) that any attempt to define mapping between this member and some S member, leads to contradiction

No. The mapping was already defined; it was conditionally presumed to exist. There is no attempt later in the proof to define parts of it. The proof shows there is an element of P(S) that isn't included in the conditionally assumed to exist bijection.

...By using this logical contradiction and the fact that |P(S)| is at least = |S| (by the a ↔ {a} trivial mapping) Cantor concludes that the initial assumption (that there is a bijection between all of the members of P(S) and all of the members of S) is a false assumption.

...and therefore |P(S)|>|S| for all S. The only person objecting to this conclusion is you, Doron.

And this may well be the core of wrongness. Doron has convinced himself Cantor's Theorem is a construction method for P(S) using some set of bijections that he even admits do not exist.

And this may well be the core of wrongness. jsfisher simply can't grasp that Cantor's construction method of all P(S) members (without missing any P(S) member), is independent of Cantor's theorem and its logical restrictions.

There is no such construction method for "all P(S) members", neither in the theorem nor the standard proof.

Doron also continues to be quite proud of himself for is claimed method to build P(S) given P(S).

jsfisher really believes that there is a way to build some non-empty set, without the members of this set.

I have no idea what this non sequitur is supposed to mean, but it doesn't address your continued trivial feat of deriving P(S) from P(S).

 

[epix]

The uncovered space is exactly the continuum itself as an ever smaller sub-line segment, which is irreducible to a discrete existing thing like a point.

This is not true for all cases of reduction. Consider a line segment 5 units long. That means there is a distance 5 between points [0,0] and [5,0] when these points are drawn in the Euclidean space. When you reduce the line segment through subtraction using the entire length, then

[5,0] - [5,0] = [0,0] (as 5-5=0 and 0-0=0)

So the point [0,0] still exists and therefore a line segment is reducible to a single point.

But that is not the case with reduction through division where

[5,0]/[5,0]

leads to division by zero and consequently to an indeterminable result.

You still can't comprehend the the co-existence of all different AND smallest elements AND all sub-line segments, where the all different AND smallest elements not reach the actual continuum of any ever-smaller sub-line segment, along some considered line segment.

You seem to have a hard time to understand the meaning of the expression "fully covered by points." There is nothing else that can divide a line segment but a point. With each single division, new point is created. The "full coverage" means that there is no line segment that a point wouldn't be able to divide in any location along the length of the segment. I've already asked you about this. Do you think that there is a line segment that cannot be further divided, meaning that no point is capable of "covering" that segment? If so, then you need to prove that

r/d = 0 where d≠0 and r>0

The infinite co-existence between line segments and points assures the continuous division of the initial line segment-- it doesn't mean that there are locations on the line segment that a point is not capable of "covering" and therefore dividing.

 

[epix]

Originally Posted by epix
The unit of movement 1 Drag got nothing to do with distances,

It does not matter, you get at least two distinct elements.

If you drag an orange from one corner of the table to the other, do you get at least two oranges to peel? If you drag a banana the same way what do you get? If you press hard enough, then you get zero bananas in exchange for a diagonal smear on the surface of the table.

Your idea that it doesn't matter if it is a Boeing 737 or a pigeon as long as it flies, is the first step for NOT understanding the combined units. For example, the unit of contraction is 1 Squeeze. But this unit is not the same as 1 Drag.

forall Orange:1 Squeeze /dot product/ 1 Drag = JuicyLine

Since a point is a 0-dimensional object, there is nothing to grab it by and drag. That's why 1 Drag = 10 SP (self-propelants).

 

[epix]

Exactly as some points are closer than others to pi.

The fastest formula does not exist exactly because no ever smaller sub-line segment is reducible to a point along the line segment, which its size= circumference/diameter.

You may think that the expansion of the digits of pi goes as

3
3.1
3.14
3.141
3.1415
3.14159

and so on, but that's not so. Here is one of the seemingly faster formulas:

But if you want to compute pi to the precision of just 3.14, you need x = 1256, coz

2*arctan[1256] = 3.14000029724...

Arctangent[x] is given by this infinite series

x¹/1 - x³/3 + x⁵/5 - x⁷/7 + ...

It really takes a considereble amount of time just to get to the precision of pi which can be applied to the known physical micro world.

The limit of pi in this approximate format, which actually shows the digits, cannot be obviously known due to the infinite series, but the limit of 3.14159... is pi -- just these two letters, or you can switch to the exact format and define pi as pi=circumference/diameter. At this moment a line segment cannot be "covered" by a point, which happens to be pi computed to the precision of one trillion digits, coz there is an obstacle in the speed the computers can operate with. But math is not about that; math is about knowing how to get here and there. Your insistence on the idea that if n is covered, then n+1 is not, and when it is, then n+2 isn't covered and so on suggests that you are not the greatest mathematicians who was, is and ever will be, as you laid down your aspiration in your OP. Your formula for eternal youth has a bug in it, Doron.

 

[doronshadmi]

No. The mapping was already defined; it was conditionally presumed to exist. There is no attempt later in the proof to define parts of it. The proof shows there is an element of P(S) that isn't included in the conditionally assumed to exist bijection.

Exactly as I wrote, so?

...and therefore |P(S)|>|S| for all S. The only person objecting to this conclusion is you, Doron.

My proof is independent of Cantor's proof.

There is no such construction method for "all P(S) members", neither in the theorem nor the standard proof.

Again, my proof is independent of Cantor's proof.

I have no idea what this non sequitur is supposed to mean

Yes, I know that you do not understand that in order to define a non-empty set, its members have to be defined.

 

[doronshadmi]

This is not true for all cases of reduction. Consider a line segment 5 units long. That means there is a distance 5 between points [0,0] and [5,0] when these points are drawn in the Euclidean space. When you reduce the line segment through subtraction using the entire length, then

[5,0] - [5,0] = [0,0] (as 5-5=0 and 0-0=0)

So the point [0,0] still exists and therefore a line segment is reducible to a single point.

No, you simply jump from an element that needs at least two different locations in order to be defined ([0,0],[5,0] element, in this case, which is known as line segment), to an element that needs only one location in order to be defined (([0,0] element, in this case, which is known as point).