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

Discussion in 'General Philosophy' started by Quantum Quack, Nov 2, 2013.

1. ### hansdaValued Senior Member

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What i meant to say is that distance between two consecutive points in geometry has to be the smallest, so that no third point can be placed in between these two consecutive points.

As there can not be any third point, between the two consecutive points, the distance between these two consecutive points has to be non-zero, non-divisible and infinitesimal.

3. ### someguy1Registered Senior Member

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727
No, exactly as many. If x is an infinite cardinal there is an obvious bijection between x and 2x. Just think about how you would correspond the natural numbers and the integers.

0, 1, -1, 2, -2, 3, -3, etc.

You can biject an uncountable set x with 2x also although you'd need a different proof.

5. ### Motor DaddyValued Senior Member

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5,425
No, twice as many!

If a length of 1 unit has a quantity of points, then 2 units has twice as many points!

http://en.wikipedia.org/wiki/Bijection

See, the difference is that the length of the radius is not mapped equally to the length of the diameter. It just happens to be a fact of nature that a diameter is twice as long as the radius, or a radius is 1 part of 2, 0.5, 50%, half as many.

If you have an option to pick a point on a radius does that mean you have access to every point on the diameter?

7. ### someguy1Registered Senior Member

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727
The idea that you would try to disagree with my point by quoting the Wiki article on bijections is hilarious. If you'd take the trouble to read the article your misunderstandings would be corrected.

8. ### Motor DaddyValued Senior Member

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5,425
What's even more hilarious is that you think every and any point on a diameter can be mapped to a point on the radius. Hint: There are points on the diameter that are not on the radius.

9. ### someguy1Registered Senior Member

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727
Well, let's examine this.

There are integers that are not natural numbers. For example, -3 is an integer that's not in the set of natural numbers.

But all the natural numbers are integers. The integers include all the natural numbers, plus all the negatives of all the natural numbers, plus zero.

Yet, we can biject the natural numbers to the integers, like this:

1 <-> 0

2 <-> 1

3 <-> -1

4 <-> 2

5 <-> -2

etc.

What do you think about that?

10. ### Motor DaddyValued Senior Member

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How about instead of coming up with more illusions and pulling more rabbits out of your hat, that instead you answer my question: If I ask you to pick a point on a radius that has an infinite quantity of points, are there any points along the diameter that you don't have access to?

11. ### someguy1Registered Senior Member

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727
I'm in complete agreement with you that the length of the diameter is twice the diameter of the radius. Two sets can have very different measure, yet the same cardinality.

The interval [0,2] has twice the length, or measure, as the interval [0,1]. But they have the same cardinality, as can be seen from the bijection x <-> 2x. They have the same number of points, yet different lengths.

I agree with you that this is counterintuitive. But it's been known for a long time.

It's even true that there are as many points in the unit interval [0,1] as there are in the unit square. You can go up in dimension and still have a bijection. Infinity's just funny like that.

12. ### Motor DaddyValued Senior Member

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5,425
You failed to answer the question. Why? Are you afraid of telling the truth? Afraid to admit that there are points on the diameter that are not on the radius?

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14. ### hansdaValued Senior Member

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2,424
A line consists of points. In a line, all the points correspond to real numbers. '1' and '0.999...' are also real numbers. So, these numbers will also correspond to some points in the line. I mean to say that, these two numbers and their corresponding points in the line will be two consecutive points; so that no third point can come in between these two points.

What point you are trying to prove here?

15. ### hansdaValued Senior Member

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Distance between two points can not be zero. Distance between two points has to be non-zero. If the distance between two points is zero, then both these points are at the same location.

As i explained above, this option is not true.

What do you mean by "zero points/point"? By definition all the points are dimension-less(ie radius zero).

If a point is having a non-zero diameter, it is no longer a point. It is either a circle or a sphere.

'Zero dimensionality' in a 3D space is a point(by definition). Where/What is the paradox/problem?

16. ### Motor DaddyValued Senior Member

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5,425
What number corresponds to the point that precedes the point that corresponds with the number .999...?

17. ### hansdaValued Senior Member

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It will be 0.999... - (1 - 0.999... ) .

18. ### Motor DaddyValued Senior Member

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What does (1-0.999...) equal?

19. ### someguy1Registered Senior Member

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727
I agreed that there are points on the diameter not on the radius; but that nevertheless, the points on the radius and the points on the diameter are in 1-1 correspondence if you assume there are infinitely many points. I said this several times already.

20. ### arfa branecall me arfValued Senior Member

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7,832
If no third point exists (can be found) between two points, then the two points cannot be consecutive, they must be the same point in that case.
You're confusing the integers with the reals: there is no integer between two consecutive integers and a finite number between any two distinct integers, the reals have an infinite number of points between any two points which are distinct.

What must hold, therefore, is that the integers are arbitrary 'distances' (or marks) on the real line; if there is no integer between any two of these then the "real" distance between them can be arbitrary, since none of these distances contains a number which is an integer.
Conventionally, unit distances are 'laid off' along a section of the real line, but this only corresponds to arranging the elements of a set (like, in a row) so they're easier to count.

21. ### Motor DaddyValued Senior Member

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What you are saying contradicts itself. You agree there are points on the diameter that are not on the radius, and yet in the same breath say there is a 1-1 correspondence. There is not a 1-1 if there are points on the diameter that are not on the radius.

Imagine a blue line about 1 meter long. Now imagine a red line .5 meters long that's matched to half of the blue line. All the red points are matched to a blue point, but only half of the blues are matched to a red. If blue points were males and red points were females, all the females would have a partner but only half the males would have a partner.

WHY? Because there are TWICE AS MANY points on a diameter as there are on the radius!

22. ### someguy1Registered Senior Member

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727
Since what you say contradicts well-known and widely accepted modern mathematics, the burden's on you to refute or modify the current theories. I showed you the example with the natural numbers and the integers. I have already said the same things to you several times, so I guess I have to let this one go. You are entitled to keep repeating things that are at odds with well established results. I can't add anything to what I've said.

23. ### Motor DaddyValued Senior Member

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5,425
Thanks for playing.