1=0.999... infinities and box of chocolates..Phliosophy of Math...

[hansda]

But we are not talking about a point as defined by math.

What is the definition of point in Physics?

We are talking about reducing a ball so that it is the smallest it can be and yet still be greater than zero.

You can reduce the radius upto a Planck's distance which is non-zero.

We may be attempting to explore how that definition may need to be changed.

What is your new definition?

We are not talking about the limitations of a mathematical system. We are talking about reducing a ball so that it is the smallest it can be yet still greater than zero.

That is with Planck's radius.

"If we don't have the tools to deal with the situation then I suggest we invent some"

What is your invention?

Isaac Newton developed/evolved calculus for just that reason. The infinitesimal was developed for just that reason. It appears that even these tools are insufficient.

I think calculus is sufficient to deal with infinitesimals.

The paradox can be expressed simply as:
If a ball has a diameter if 1 Planck Unit what is it's volume?

Planck's distance is finite and not infinitesimal. So, where is the paradox?

If a Planck unit has positive dimension than what makes a plank unit the dimension it is?

Planck's units are already well defined. They are always positive and definite.

What is less than 1 Planck?
Other than zero...

That is infinitesimal but not possible in Physics.

Zero has both 3 dimensions and zero dimensions simultaneously. It is both immaterial and material simultaneously.

Zero is a number and not an object. How it(zero) can have a dimension? Does the number 1(one) have any dimension?

The proof of this is obvious both in Physics and Math. IMO

What is your proof? It may not be correct.

 

[rpenner]

A Planck length, $$\sqrt{\frac{\hbar G}{c^3}}$$, is not the smallest length of the universe. It is merely the scale where our imperfect understanding of quantum mechanics collides with our imperfect understanding of gravitation, and represents a lower limit on the applicability of our imperfect understanding of the universe beyond which we would certainly need a better theory of quantum gravity to talk about the universe with precision.

To assert that x is the smallest physical distance that exists is not consistent with the definition of a sphere of radius x, because if a sphere exists with radius x, then points on the sphere are closer to each other than x or it is not a sphere.

 

[Quantum Quack]

A Planck length, $$\sqrt{\frac{\hbar G}{c^3}}$$, is not the smallest length of the universe. It is merely the scale where our imperfect understanding of quantum mechanics collides with our imperfect understanding of gravitation, and represents a lower limit on the applicability of our imperfect understanding of the universe beyond which we would certainly need a better theory of quantum gravity to talk about the universe with precision.

To assert that x is the smallest physical distance that exists is not consistent with the definition of a sphere of radius x, because if a sphere exists with radius x, then points on the sphere are closer to each other than x or it is not a sphere.

Of course, and so the paradox created by reducing a ball or sphere in this manner is highlighted.

The issue that existed for Zeno of Elea, 490.BC (Achilles & the Tortoise ), still exists today whether we wish to "put it aside " with the use of limits or other arbitrary lengths such as a Planck length etc or not.

I might add that at the time according to some references Zeno was really attempting to reconcile the existing beliefs in "many gods" with the controversial possibility of a monotheistic belief system and used his paradox as a way of convince-ment that all paths lead to a one God reality [ infinity ] as many gods implies "finite limitations" [ x/2 ] placed upon the divine.

Our problem really though, is the "vanishing point" where mass ( matter or substance ) changes from substance to nothingness and from nothing-ness [zero] to substance.
The problem can be expressed in a slightly different way by presuming zero (nothing-ness) and creating 3 dimensions. Asking at what point does zero dimensionality lose it's lacking of dimension and gain some positive dimension?

For us humans we tend to look from substance, reducing to zero. It is the way our minds are wired generally as it is illogical to consider ex-nihilo. So we tend to work from something and not from nothing.
To use a material ball [ finite substance ] and reduce it infinitely [ yet retain it's finite-ness] is the easiest way of looking at this paradox of infinite vs finite. IMO
Ultimately it goes a long way in exploring the reality of inertia, universal constancy of gravitational attraction and then on to "life" and freedom of animation [movement] as well. IMO

 

[Motor Daddy]

c=299,792,458 meters/second

How many meters does light travel in .999... seconds?

An audio/visual alarm stop watch comes in handy for these types of discussions. The alarm is set to go off at exactly t=1 second.

I send a light pulse and the stop watch starts simultaneously. At t=.999... seconds the alarm has not gone off yet.

Question: How far did the light travel in .999... seconds?

 

[hansda]

c=299,792,458 meters/second

How many meters does light travel in .999... seconds?

An audio/visual alarm stop watch comes in handy for these types of discussions. The alarm is set to go off at exactly t=1 second.

I send a light pulse and the stop watch starts simultaneously. At t=.999... seconds the alarm has not gone off yet.

Question: How far did the light travel in .999... seconds?

1 comes after 0.999... only. 1 can not come jumping or bypassing 0.999... ; in that case there will be a discontinuity in the continuum, which is not allowed.

So, it will travel the same distance.

 

[Quantum Quack]

1 comes after 0.999... only. 1 can not come jumping or bypassing 0.999... ; in that case there will be a discontinuity in the continuum, which is not allowed.

So, it will travel the same distance.

Took a while but I would be a bit worried by that statement "1 comes after 0.999... " for if 1=0.999... it (1) does not come after 0.999...
Perhaps a semantic and overly pragmatic point but when debating the value of 0.999.. as being 1 then it certainly can not "come after" 0.999...

Also it is worth noting that 0.999... is 0.999... OF 1 in other words [examples] 9/10th's is 9/10th's of 1, 99/100th's is 99/100th's of one etc etc...
0.999.... can not be a factional value of 1. It must equal 1

 

[hansda]

Took a while but I would be a bit worried by that statement "1 comes after 0.999... " for if 1=0.999... it (1) does not come after 0.999...
Perhaps a semantic and overly pragmatic point but when debating the value of 0.999.. as being 1 then it certainly can not "come after" 0.999...

Also it is worth noting that 0.999... is 0.999... OF 1 in other words [examples] 9/10th's is 9/10th's of 1, 99/100th's is 99/100th's of one etc etc...
0.999.... can not be a factional value of 1. It must equal 1

Consider a straight line. Consider any single point on this line. Denote this point as mathematical zero(0).

Consider another point at the right hand side of this zero point. Denote this point as mathematical one(1).

Distance between '0' and '1' will be finite distance but there can be infinite number of points between '0' and '1'.There also will be infinite numbers between '0' and '1'. So, every point on this segment will correspond to a number and the same point can not correspond to two numbers.

Now consider two consecutive points in this line segment between '0' and '1'. Consecutive means there can not be any third point between these two consecutive points. What will be the distance between these two consecutive points?

(Option-a) If the distance is finite, there will be infinite points between the two consecutive points. This is not possible by definition of consecutive points. So, this option is ruled out.

(Option-b) So, the distance between two consecutive points has to be infinitesimal so that no third point can stay in between these two points. This is the right option.

So, if one of the two consecutive points is '1'; What will be the other point between '0' and '1'?

Isn't it 0.999... .

 

[rpenner]

The above post assumes there can be a "smallest positive number" and that this allows one to have consecutive points in geometry. However consecutive points are not a feature of either Euclidean geometry or set theoretical constructions equivalent to Euclidean geometry. Thus both the novel geometry and novel definition of number would have to be developed before a case for the proposition being sound could be developed.

If 1/2 is a number and ϵ is the "smallest positive number" described by hansda above, then is 0 ≤ (1/2) × ϵ ≤ ϵ < 1/2 < 1 true? Then is 0 < (1/2) × ϵ < ϵ false so that ϵ really is the smallest positive number? In that case, either 0 = (1/2) × ϵ or ϵ = (1/2) × ϵ so that 2 × ( (1/2) × ϵ ) = (1/2) × ϵ + (1/2) × ϵ ≠ ϵ. Does this not seem like a very strange property for a number to have?
Also ϵ − (1/2) × ϵ ≠ (1/2) × ϵ follows from the same assumption that ϵ is the smallest positive number.

We already have a number system where there is a smallest positive number -- the integers. And it gets around this problem because 1/2 is not a integer and therefore multiplication by 1/2 is not defined when integers are the only applicable concept of number. With the integers, any sum, difference or product of two integers is guaranteed to be an integer -- a property of closure that the rational and real and complex and hyperreal and surreal numbers all share.

 

[Quantum Quack]

Consider a straight line. Consider any single point on this line. Denote this point as mathematical zero(0).

Consider another point at the right hand side of this zero point. Denote this point as mathematical one(1).

Distance between '0' and '1' will be finite distance but there can be infinite number of points between '0' and '1'.There also will be infinite numbers between '0' and '1'. So, every point on this segment will correspond to a number and the same point can not correspond to two numbers.

Now consider two consecutive points in this line segment between '0' and '1'. Consecutive means there can not be any third point between these two consecutive points. What will be the distance between these two consecutive points?

(Option-a) If the distance is finite, there will be infinite points between the two consecutive points. This is not possible by definition of consecutive points. So, this option is ruled out.

(Option-b) So, the distance between two consecutive points has to be infinitesimal so that no third point can stay in between these two points. This is the right option.

So, if one of the two consecutive points is '1'; What will be the other point between '0' and '1'?

Isn't it 0.999... .

Hansda, I feel you are drawing your points as mere points when the question is about the value of 1 and 0.999... and not just mere points.

if you draw a point that indicates the sum value of something... say one point is made at 1/2 we are already implying a point at 1.


But we are not talking about merely two consecutive points. We are talking about a sum value of 1/2 of 1

The point at the 1/2 is the sum value of half of one.

When considering 0.999...= 1 the same issue presents. IMO

Which is why I drew the diagram below as posted earlier:

where by the sum total of 0.999... of 1 = 1

For 0.999... to equal 1 it can not follow 1 ... it it has to equal 1

 

[Quantum Quack]

Say for example you have a 3 dimensional volume that has an infinite number of points located with in it and the volume is filled with these points.

Qu: What is the distance between each point?

Is it zero or is it 1/infinity?

If it is zero then the volume of space can NOT be filled with points. [ no points can exist other than one zero point ]
If it is 1/infinity (infinitesimal) the volume of space CAN be filled with zero points to make up the volume...

Therefore my case is that a zero point must have the diameter of 1/infinity for it to exist in any form, as shown in this diagram:

The problem [paradox] of zero dimensionality in a three dimensional space will not go away...

 

[Layman]

A Planck length, $$\sqrt{\frac{\hbar G}{c^3}}$$, is not the smallest length of the universe. It is merely the scale where our imperfect understanding of quantum mechanics collides with our imperfect understanding of gravitation, and represents a lower limit on the applicability of our imperfect understanding of the universe beyond which we would certainly need a better theory of quantum gravity to talk about the universe with precision.

Blasphemy, it is the smallest length that can be theoretically measured via any type of experiment, because it would require an infinite amount of energy to measure a distance smaller than that all concentrated at one point. It is the smallest length in the universe that will have any effect on anything.

 

[Motor Daddy]

Say for example you have a 3 dimensional volume that has an infinite number of points located with in it and the volume is filled with these points.

Qu: What is the distance between each point?

Distance being in one dimension (axis), if you pick a point in space and emit a light sphere from that point, light moves away from that point in every direction at the same rate. At every time t there is an exact distance from the center point, which is the radius. The radius has an infinite amount of points in and of itself, but the diameter of the light sphere has twice as many points as the radius does!

So, will you please restate your question in terms that reflect the nature of the diameter having twice as many points as the radius does? Also, you'll need to be specific as to which point(s) you are referring to when you ask for the distance between them, so I'll also need the (x,y,z) coordinates in units of light seconds so that I can be sure to calculate the exact distance to the precision allowed by the data which you provide.

Distance between points in space is in mathematical terms: ct!

 

[someguy1]

The radius has an infinite amount of points in and of itself, but the diameter of the light sphere has twice as many points as the radius does!

If there are infinitely many points in a line segment, there are the exact same number of points in a line segment of twice the length. That's because there's a bijection between the interval [0,1] and the interval [0,2]. And in fact a bijection between any infinite set and the union of two copies of itself.

 

[Quantum Quack]

So, will you please restate your question in terms that reflect the nature of the diameter having twice as many points as the radius does? Also, you'll need to be specific as to which point(s) you are referring to when you ask for the distance between them, so I'll also need the (x,y,z) coordinates in units of light seconds so that I can be sure to calculate the exact distance to the precision allowed by the data which you provide.

Distance between points in space is in mathematical terms: ct!

so the distance is 1(c)/infinity... or infinitesimal...
You see if it were a Planck length then the number of points in a volume of space would be finite not infinite.

A theoretical Planck length is finite.
A theoretical infinitesimal length is 1/infinity [non-finite]

 

[Motor Daddy]

If there are infinitely many points in a line segment, there are the exact same number of points in a line segment of twice the length. That's because there's a bijection between the interval [0,1] and the interval [0,2]. And in fact a bijection between any infinite set and the union of two copies of itself.

A line segment of twice the length would take light twice the time to traverse that line as it would if it were half as long. Twice the length means twice as many points. If the radius is represented by the symbol r and has x quantity of points, then the diameter, represented by the symbol d has 2x quantity of points.

 

[James R]

Twice the length means twice as many points. If the radius is represented by the symbol r and has x quantity of points, then the diameter, represented by the symbol d has 2x quantity of points.

The mathematical definition of a point is that it has zero size (length). Therefore, in any line segment there is an infinite number of mathematical points. It can be shown that the cardinality of the number of points on a segment of length x is the same as on a segment of length 2x. Both infinities are denoted "Aleph-1".

 

[Motor Daddy]

The mathematical definition of a point is that it has zero size (length). Therefore, in any line segment there is an infinite number of mathematical points.

That is true that a line segment has an infinite quantity of points. What is also true is if a line segment of length L has x quantity of points, then the line segment of length 2L has 2x quantity of points.

 

[rpenner]

The mathematical definition of a point is that it has zero size (length). Therefore, in any line segment there is an infinite number of mathematical points. It can be shown that the cardinality of the number of points on a segment of length x is the same as on a segment of length 2x. Both infinities are denoted "Aleph-1".

The cardinality of the continuum is denoted $$\aleph_1$$ IF the continuum hypothesis, $$\aleph_1 = 2^{\aleph_0}$$, or generalized continuum hypothesis , $$\aleph_{n+1} = 2^{\aleph_n}$$, is accepted as true. The proofs about infinity and the real numbers from ZFC set theory are independent of the truth of either hypothesis, so they may or may not be taken as axioms without affecting most of mathematics.

 

[someguy1]

That is true that a line segment has an infinite quantity of points. What is also true is if a line segment of length L has x quantity of points, then the line segment of length 2L has 2x quantity of points.

Yes this is true, but if x is an infinite cardinal, then x = 2x.

 

[Motor Daddy]

Yes this is true, but if x is an infinite cardinal, then x = 2x.

Like I said, twice as many!