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

MD, it is indeed the case that hansda has no idea what he's talking about. There is no such mathematical entity as .000...1. I would simply ask hansda to name the index at which the '1' appears.

To elaborate on this point: The decimal notation .abcdef... means by definition the sum of the infinite series a/10 + b/100 + c/1000 + ...

In other words given a positive integer position n, we know what digit a_n is in decimal position n. A number in decimal notation is really a function whose domain is the natural numbers 1, 2, 3, ... and whose range is the set of decimal digits zero through nine.

The problem with the .000...1 idea is that you can't tell me which index position the '1' is in. This notation is simply not defined. It's meaningless.

 

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We both know he's lost, but will he admit it when the light bulb finally lights up, or will he lie to himself and kill the light and pretend it never went on? I'd like to see him get it, but I also enjoy watching people bask in their own ignorance, so there's entertainment value in it for me if he either doesn't get it, or he gets it but lies to himself. Either way, the joke is on him!

 

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In the decimal number "0.000...1", the '1' appears at the "infinite-th" position.

In your example above of the number "0.abcdef...", how you will write it if 'a' appears infinite times or say if 'e' appears infinite times?

I already told it is infinite-th location.

0.000...1 = 1/(1000...infinite times 0).

 

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All you are talking about with this 0.000...1 is 1/infinity or typically referred to as an infinitesimal.

This issue of an infinitesimal being left over when you 1-0.999... is the bane of the issue about 0.999... = 1
and I believe it can only be resolved when one considers that the zero has a volume of 1/infinity,
Therefore there is no difference between 0.999... and 1, but in saying so they are not exactly equal either due to the paradox set up by the number system employed.

I refer again to the diagram I made and posted earlier which clearly shows what i mean when I say that there is no difference between them yet they are not equal ...

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The other thing to realize, I believe, is that infinity is axiomatically "undefinable" so "1/infinity" is actually "1/undefined" and attempting to define by granting the infinitesimal a value as a residual "after 0.999... and before 1 presents the same problem as attempting to terminate any infinite sequence.
To suggest that the 0.999... terminates with a left over 0.000...1 [ infinitesimal] is declaring that infinity has been terminated and therefore "finite", which is an axiomatically flawed position.
Therefore a process of limits is employed to simply state that the magnitude of difference is less than any real number. Thus containing the argument in a way that is indisputable or irrefutable according to the axiomatic treatments given by that number system.

 

Infinity as a number makes little sense. However there is a clear definition of an infinite set.

Any set that can be mapped 1-to-1 with a proper subset of itself is infinite.

For example the map \( x \mapsto 2 x\) is a bijection between the integers and a proper subset of the integers that contains no odd integers.
The map \( x \mapsto x + 1\) is a bijection between the non-negative integers and a proper subset that does not contain zero.
The map \( x \mapsto e^x\) is a bijection between the real numbers and a proper subset that does not contain zero or any negative numbers.

Thus the integers, the non-negative integers and the reals have a property that finite sets never have.

 

Here you are considering as if the point has an infinitesimal radius. This is against the definition of a point which is dimensionless ie zero radius.

If you consider a point as a circle having infinitesimal radius; this implies two points can touch each other the way two circles can touch each other. When two circles touch each other, the line joining the two centers of the circles, contain at least three points. If two consecutive points touch each other, the third point will come in between the two consecutive points. This is not possible. So, a point can not have infinitesimal radius. A point being dimensionless or having zero radius is the correct assumption.

 

So you have two distinct points in space, the point at location (0,0,0), and the point at location (1,0,0), the numbers representing distance in units of light seconds along each axis (x,y,z) from a chosen point in space (0,0,0). So the points (0,0,0) and (1,0,0) are separated along the x axis by a distance of 1 light second, or 299,792,458 meters.

As a stop watch elapses time, starting at t=0 at the point in time the watch is started, and elapsing one second per second of duration after it is started, when the stop watch starts light is emitted simultaneously from each of the two points in space (points at locations (0,0,0) and (1,0,0)). In exactly .5 seconds the lights meet each other at location (.5,0,0) along the x axis. Each light traveled a distance of .5 light seconds in the duration of .5 seconds. Each of the two points are center points to light spheres that have a radius equal to a length of ct. Three points in space verified by the definition of the meter, and the fact that each light travels at the same speed from their original location. 3 points (0,0,0), (.5,0,0), (1,0,0) and 1 of them doesn't have a radius!

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Would you agree that a point with zero radius can not possibly exist in three dimensional space except as an imaginary artifact?
So when you are placing a point on a line what are you doing?
What are you describing?
If it is non-existent...
===
The problem is the merging of pure abstraction [ imagination ] with reality.
Zero is non-existent where as an infinitesimal diameter can exist in 3 dimensional space.
By granting zero the reality of having an infinitesimal diameter, which one has to grant it, if it is to exist other than mere imagination [ pigs that fly] zero is grounded in 3 dimensional space as being "real" and not pure abstraction.
By reducing a ball to the smallest it can be [ >0 ] proves that zero has an infinitesimal diameter.
Nothing can be smaller than an infinitesimal diameter means just that.. Nothing (Zero) can and is the only "thing" smaller than an infinitesimal.
So the paradox of abstraction and reality is highlighted, in a very real way.
Infinite reduction does not terminate at zero.
Zero does indeed exist but only if it has dimension in a 3 dimensional space yet paradoxically this appears, on the surface, to defeat the notion of zero. However it depends on perspective. Using an inverse sphere or ball clarifies how zero space can be discovered. The ball however has to be inverse and not converse to properly do so. Thus zero MUST BE derived by default and not directly IMO.

The question I asked years ago was:
"How can we prove zero to actually exist in 3 dimensional space?"
...the above is the only solution I have found to make sense.

Especially as it leads to a possible solution to QM's entanglement "mechanism" problem and a solution to the universal constancy of gravity.

"Zero is the instantaneous "worm hole" the "tunnel" that connects all things in a form of universal entanglement. It is via zero space that quantum entanglement can find a mechanism for instantaneous communications regardless of 3 dimensional distances. If all things of substance share the same zero "wormholes" or "tunnels" then by default all things are entangled"
Zero is the only true absolute universal constant, so therefore....blah blah blah!

 

Maybe.

Just locating a position on the line.

I think your concept of a point with a 'non-zero infinitesimal radius' has a very fundamental flaw.

Lets say a point has a non-zero infinitesimal radius. So, this point will behave like a very small circle. Every circle has a center-point. So, what about the dimension of this center-point? Will the center-point also have a non-zero infinitesimal radius or it will be dimension less? If the center-point has a non-zero infinitesimal radius again it will have a center-point and the problem will continue. So, this is a big fundamental flaw in this concept of a point having infinitesimal radius.


I think the only solution would be the concept of consecutive points, where the points are dimensionless and they have a non-zero, non-divisible, infinitesimal distance.

 

Except consecutive points are forbidden by Euclidean geometry which says any two distinct points define a line segment and every line segment has a distinct midpoint.

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI10.html

This property of "ubiquitous midpoints" is well-captured by the rational numbers, algebraic numbers and real numbers, but not by discrete systems that have the concept of "the next larger number" like the integers.

 

That is meaningless until you tell me what that means.

I can tell you what I mean by decimal notation. Decimal notation is a function that inputs a counting number 1, 2, 3, ... and outputs the decimal digit d(n) that appears in that decimal place [to the right of the decimal point, it's always understood.]

In other words, decimal notation is a function

d : N -> {0,1,2,3,4,5,6,7,8,9}

where N is the natural numbers starting with 1.

We can define the value of a given decimal expression {d(n)} as d(1)/10 + d(2()/100 + d(3)/1000 + ...

where we have already developed the theory of the real numbers, infinite sequences and series, and limits. You can refer to any standard textbook on real analysis for the details.

The key point here is that "infinity" is not a natural number. All the natural numbers are finite, even though there are infinitely many of them. There is no such thing in standard math as the infinitieth decimal position. It's simply not defined and there's no place for it in the accepted system.

Now it's fair for you to say that you have a different theory of the real numbers and that you don't believe in real analysis as it is presently understood. I have no problem with that. Real analysis is not the last word on the mysterious nature of the infinite. It's just a clever body of ideas and techniques that have served us well. Intellectual ideas come and go. I'm not emotionally wedded to standard math if something better comes along in a a few hundred years. Or tomorrow morning. You never know what genius is out there putting the finishing touches on the next revolution.

In other words I'm open-minded about alternative frameworks for the real numbers.

But when you say that a given digit is in the "infinitieth" place, as you can see that is meaningless in accepted mathematics, namely the fact that the n-th digit is defined for all n in the natural numbers, and infinity is not a natural number. So I have no idea what you mean by the infinitieth digit; and if you seek to convince me that you have thought through the implications of your own ideas, then I'd ask you to simply define for me:

What do you mean by the "infinitieth" digit?

 

Well.... does it or doesn't it?
If it does then please explain how you feel it could.

but how so if it is non-existent?
How can you use a non-existent thing to locate anything?

The infinitesimal can never be divided as to do so means only deriving another infinitesimal.

If we reduce a physical ball so that it's diameter is 1/infinity of course the volume it holds with in has space yet no dimension [zero space] because it is less than 1/infinity.

This is a highlighted example of the nature of the paradox between zero dimensional and 3 dimensional volumes.

Zero dimensions can then be "proved" to have a "positive" volume that has no actual dimension. [paradox]

Imagine you have zero and you expand it so that it just acquires dimension.

How big is it's diameter?

The boundary between 3 dimensions and zero dimensions MUST be 1/infinity yet in saying so we have also the need to declare a paradox exists.

 

1. Does your concept of an expanding zero diameter sphere, from a point with zero dimension to a 3 dimensional sphere mean that the location of the original zero point is always at the center of the expanding sphere, or always on the outer surface of the sphere?

2a. If the original location of the zero dimensional point is always at the center of the expanding sphere, then the radius is increasing in length at the speed of light, 299,792,458 meters per second. Since the diameter is twice the length of the radius that means the diameter of the expanding light sphere, from a zero dimensional point to a 3 dimensional sphere, is equal to 2c. Take note, the diameter increasing in length at the rate of 2c does not mean anything traveled faster than c, it means that light traveled away from the center in one direction at the rate of c. A separately traveling light from the same point in the opposite direction adding to the length of the diameter at the same rate in the opposite direction of travel as the other light is adding to the length of the diameter while creating the radius at a rate of c. If the radius is being created at the rate of c then the diameter is being created at the rate of 2c.

2b. If the original location of the zero dimensional point is always at the surface, then the diameter of the sphere is increasing at the rate of c and the radius is increasing at the rate of .5c.

 

Happy New Year btw!

You pose some interesting questions and thoughts to go with them...

My thoughts at the moment are that because the zero point with zero dimension is non-existent the location of that point is impossible to find unless it has a dimension of 1/infinity. This means that the zero point volume described by the dimension 1/infinity is entirely the central point. There is no central point to this void -Neither at the center nor at the perimeter.
It is a bit like taking the perspective that zero IS 1/infinity from a 3 dimensional space perspective, but is not truly zero because if it were it would not be "existent" even as a location.

If you take a 3 dimensional length of "string" [a 3d line ] and note that it has a left side and a right side.
To "travel" from the left side to the right side one MUST pass through the mid point of the string. Yet when applying infinite reduction that mid point can never be defined exactly. It can be known to be there but it can never be defined exactly in absolute terms other than by default. Hence the use of an inverse ball or sphere instead of a converse one to find that point.

The thing is the mid point [ Lagrangian point **] is present but impossible to locate with absolute precision. [ due to infinite reduction to 1/infinity ] except by default.
This appears to be in part associated with the Uncertainty principle when taken to the extremes we are taking it to.

The radius is the only actual dimension and the diameter is cumulative [calculated] dimension, are my only thoughts at the moment. [NYE was a long night

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The "light cone" in this example is a spherical shell expanding outward from a central point, the rate of expansion could only be deemed to be 'c', I believe. To bring diameter in to it only confuses the issue unnecessarily IMO.

However as suggested in the preceding item, the zero point must have a diameter of at least 1/infinity from a "material 3 dimensional perspective" for it to exist in 3 dimensional space to begin with. So the light shell can not expand from a lesser "sized" point.

If I understand what you have written correctly, I think you are quite correct, in your assessment however I wonder how it adds value to the discussion. Can you explain how?
The key difference is that we are talking about a light shell [ sphere ] and not cone... I feel.

**Edit: I am referring to Lagrangian points in a way that is simplified to a two body system. [possibly incorrectly]
Where the gravitational attraction between two masses is found to be in a state of equilibrium. eg. (small G mass) m---0---------M (larger G mass) - 0 indicates the L point or the Center of Mass for the two body system.

 

Happy New Year!

Is it impossible to start a stop watch at t=0? Is it impossible to place one end of a meter stick at the starting point where you want to start measuring length along that meter stick? Is it impossible for the x and y axis to have a point of intersection? How about the z axis? Does that axis intersect the x axis? Is it possible that the x, y, and z axis intersect at one point? If so, is that point qualified to be the (0,0,0) point? Does the point have dimension, or is it an intersection of three axis?

We are speaking about the 3D grid system and math that accompanies that system, which is our system of measures in order for us to have a grid location system that has distance and locations that mean something. You can go to any point in space and measure a distance from that point. You can declare that point the location (0,0,0) in the grid system in which we humans have designed as a measuring system. Imaginary lines intersect and form an imaginary point in that imaginary grid system. There is real distance between each two distinct points (locations).

The mid point between any two points is not any more impossible to find than the one mile marker on a two mile run!

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A point doesn't have a diameter. You are confusing an infinitesimally small object (which its size is greater than zero) with zero. Zero is the starting line of the race (the point where the y and z axis intersect the x axis). The shell expands from the zero point like the race car expands from the starting line of a 1/4 mile run. The car travels a distance away from the zero point starting line on the x axis in a duration of time. It really is that simple.

I am explaining to you how points work. You seem to think they can be divided, and I am telling you they are locations in the grid, defined by the grid as to distance between another point.

 

well ...then it should be easy to find zero on a 3 dimensional line. Go for it and report you findings ... "ver oh ver tis dat zero gone?"

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Given that zero is non-existent how can anything "start", "pass though" or "end" with it?

The issue is the same as ex-nihilo in philosophy. How can anything come from nothing?

Challenge:
Find the absolute exact location of the Center of Mass [ center of gravity ] of any given object.

When you realize why you can't you start to appreciate "metastability", "A universe that is in constant motion" etc.
Follow the logic to it's natural conclusion and you may realize, as I have, that it is this paradox of zero dimension in a 3 dimensional space that forces universal existence [time/energy/ movement] as it's only solution.

The logical reality is that at t= 0 duration [ your zero point on a line ] distance must also equal zero... So therefore the line itself is non existent at that point. So at any given t using your point system the universe is non-existent, and as the line is made up of all these zero points then the line can NEVER be existent...no matter what t you use. [unless of course you grant the t a duration value greater than zero.]
So for the line [universe] to exist the duration of any t point MUST be greater than zero, and if duration is greater than zero accordingly distance MUST also be greater than zero.
0,0,0,0 = 0 [nihilo]
1/infinity,1/infinity,1/infinity,1/eternity = 1/infinity x 1/eternity [universe in motion]

Therefore by stating that a a zero point has a diameter of 1/infinity allows the universe to exist where as to state that it has zero diameter means that the universe is nihilo.

This is why I am laboring overt this issue of 1-0.999... = 0, but only if zero has a diameter of 1/infinity. [in Physics]

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You came into this world with zero money. Do you have money now? If you do have money now, how is that possible?

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ha ..not true.
I came into this world with the money my parents had, but will certainly "Go to heaven with an empty wallet" [chuckle]
As zero is the sum of all things [ 0= (-)x + (+)x ] I came into this world from zero owning everything and lost it all on the "gamble" called life... [the poetic muse has struck again! oh no!

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That is your parents money until if/when they give you some, and until that time you have a balance of $0.

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Unless of course someone left you a bank account before you were born, which had a balance greater than zero. In that case, the money is yours when you go get it, but not until then!

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Oxygen is money, water is money, begging in the streets cost money...
or
I may have no money by I damn well cost heaps... [ Said new born baby to his Dad... just ask Mom...hee hee]