Point Circle Paradox

[Eodnhoj7]

Point-Circle Paradox:

1. A circle with a circumference of one exists.

2. This circle contains a series of fractal circles progressing from 1/2 to 1/3 ad infinitum.

3. An infinite number of potential circles is contained by the dot, as zero, at the end.

4. 0 = (1/n-->inf)

[Scott Mayers]

Point-Circle Paradox:

1. A circle with a circumference of one exists.

2. This circle contains a series of fractal circles progressing from 1/2 to 1/3 ad infinitum.

3. An infinite number of potential circles is contained by the dot, as zero, at the end.

4. 0 = (1/n-->inf)

What are you stating here, Eodnhoj7?

This appears to be about the 'continuum'. This is about infinites upon infinites rather than a simple infinity problem. When you 'account' for each point of a given circle, any other larger circle intuitively has 'more' points than the smaller. When taking any two points of the given circle, drawing a line through each from its common center demonstrates that should you have a corresponding set of points to each larger circle.

(Edit addition: And of course, if we treat a point as being 'one' thing, then anything surrounding it that is 'greater' than it has to have at least 'two'. See below for the concept, "idempotence". The 'continuum' is a power of powers of some operation that leads to indeterminately more infinities and so relates better to what you mean here.)

The question then is if you recognize that points are 'next' to one another, take two such imagined points on that same given circle. If they are 'next' to each other, then we expect at least some 'angle' created from two rays created from the common center. Yet, for a larger circle, this angle should diverge more leaving 'new' points that exist on the outer circles. This means that if given one circle is defined as having an 'infinity of points', every other circle from the same center cannot be measurably accounted for because they each have different continuous levels of infinites (or infinitesimals).



[And I'm thrown off by your use of 'fractal' here, even though I understand what you mean. A 'fractal' is more specifically related to repeated patterns on the whole but DIFFERS in some part. So while the set of circles surrounding a point differ upon radii, it would only be coincidental to define it as a fractal and confuses the traditional meaning of it as it was invented to describe: those whole patterns that are CONTAINED within the same identical shape or structure that has a repeated pattern. For concentric circles, each circle is DISTINCT. If you then treat a point as a kind of 'area' with the other circles, then this would be a perfectly infinite and infinitesimal 'fractal' but hides the relevance of the meaning of 'fractal' as a pattern because within one given point (as a conceptual area, volume, etc) it consists of only one 'substance' and so is indeterminate as to being a 'pattern' other than something like comparing equality of the meaning of 'zero' itself: 0 = 0 + 0 = 0 + 0 + 0 = ... This though is then just a logical property called, "idempotence" (the powers of any part equals the identity of the whole). ]

[Eodnhoj7]

Point-Circle Paradox:

1. A circle with a circumference of one exists.

2. This circle contains a series of fractal circles progressing from 1/2 to 1/3 ad infinitum.

3. An infinite number of potential circles is contained by the dot, as zero, at the end.

4. 0 = (1/n-->inf)

What are you stating here, Eodnhoj7?

This appears to be about the 'continuum'. This is about infinites upon infinites rather than a simple infinity problem. When you 'account' for each point of a given circle, any other larger circle intuitively has 'more' points than the smaller. When taking any two points of the given circle, drawing a line through each from its common center demonstrates that should you have a corresponding set of points to each larger circle.

(Edit addition: And of course, if we treat a point as being 'one' thing, then anything surrounding it that is 'greater' than it has to have at least 'two'. See below for the concept, "idempotence". The 'continuum' is a power of powers of some operation that leads to indeterminately more infinities and so relates better to what you mean here.)

The circles become progressively smaller until you reach a point, this point contains an infinite number of circles

The question then is if you recognize that points are 'next' to one another,
Any three points, stemming from the center appear as angle, yet the infinite number of circles within circles means the center circle contains an infinite number of circles within it.

take two such imagined points on that same given circle. If they are 'next' to each other, then we expect at least some 'angle' created from two rays created from the common center. Yet, for a larger circle, this angle should diverge more leaving 'new' points that exist on the outer circles. This means that if given one circle is defined as having an 'infinity of points', every other circle from the same center cannot be measurably accounted for because they each have different continuous levels of infinites (or infinitesimals).

False, a paradox occurs as the center post always exists no matter how far you go within the circle.



[And I'm thrown off by your use of 'fractal' here, even though I understand what you mean. A 'fractal' is more specifically related to repeated patterns on the whole but DIFFERS in some part.


So while the set of circles surrounding a point differ upon radii, it would only be coincidental to define it as a fractal and confuses the traditional meaning of it as it was invented to describe: those whole patterns that are CONTAINED within the same identical shape or structure that has a repeated pattern.
The repeated pattern is the circle, each circle as a fraction of the diameter, 1/2, 1/3, etc. is a fractal..

For concentric circles, each circle is DISTINCT. If you then treat a point as a kind of 'area' with the other circles,
All points are areas, we only see them as points relative to other points.

then this would be a perfectly infinite and infinitesimal 'fractal' but hides the relevance of the meaning of 'fractal' as a pattern because within one given point (as a conceptual area, volume, etc) it consists of only one 'substance' and so is indeterminate as to being a 'pattern' other than something like comparing equality of the meaning of 'zero' itself: 0 = 0 + 0 = 0 + 0 + 0 = ... This though is then just a logical property called, "idempotence" (the powers of any part equals the identity of the whole). ]

Each point, as an area, contain and infinite number of circles. Take for example you only see x numbers of circles, move further and still only see x number of circles...no matter how many circles you see they all converge at a single point.

[Scott Mayers]

I'm still uncertain what you are asserting uniquely. Are you proposing something new or just stating the fact of the difficulty of making sense of certain infinites/infinitesimals?

I see that you were assuming the 'point' as a two-dimensional concept rather than a single dimensional idea. A point within two or more dimensions is still one-dimensional unless you are speaking of a circle itself. The problem of the continuum begins with the circle....NOT a point. The paradox of the enclosed circles IS defined as the Continuum.

Note that you can define infinity as any set that contains itself. The 'continuum' is an extension of this by what I believe you are meaning. Cantor thought of this by beginning with a circle as I said above, and asked whether the same number of infinite points of one given unit circle is the same infinity. It is not.

Another way to think of the problem that you MAY be thinking of instead is to begin with an area square and ask how many 'points' it contains in comparison to other sized squares. For example, if you begin with a square that is 1 cm square, imagine dividing it in half and then aligning each half length-wise (to become a (0.5 cm X 2 cm). The area remains the same but the shape is an elongaged rectangle.

Now repeat this process by dividing the new rectangle in half by the same way and reshaped to be (0.25 cm X 4 cm).

If you continue to do this infinitely, you get a single line that is
[1] (0 cm X ∞ cm) = 1 cm^2

This turns the area into a single dimension line. Next take a new square of 2cm to each side. Repeat this and you also have another line that is
[2] (0 cm X ∞ cm) = 2 cm^2


Now combining these, we have,

[1] == [2] because

(0 cm X ∞ cm) == (0 cm X ∞ cm)

But by substitution, we also have,

1 cm^2 == 2 cm^2 [a contradiction]

This demonstrates the problem that I believe you are thinking. You can only resolve this if one infinity differs from another. This is mathematically resolved by the concept of 'derivatives' in calculus. This particular problem is the derivative of x^2 which equals 2x. This is a relative concept that requires some standards but should give you an idea of how Calculus was developed.

[Eodnhoj7]

I'm still uncertain what you are asserting uniquely. Are you proposing something new or just stating the fact of the difficulty of making sense of certain infinites/infinitesimals?

I am proposing that the point zero, at the end of the series of circles within circles, contains an infinite number of circles within circles and that if you imagine the circles progressively moving they would be unfolding from point zero. The point is not a 2 dimensional construct as that requires limits to the field.

If you take a single point, with no contrast, it becomes a boundless field. However if that single point is compare to another backdrop or if it individuated into multiple points (with a line connecting them) then you have specific boundaries. A single point against no backdrop is a field.

Now if you have a series of fractal circles within circles the point contains these fractal circles as you continue progressing to the center. The point contains everything's and all the circles unfold from it. The point is 0d, not even single dimensional (unless you are arguing a theoretical single dimensional point, which I do argue elsewhere on the forum).

I see that you were assuming the 'point' as a two-dimensional concept rather than a single dimensional idea. A point within two or more dimensions is still one-dimensional unless you are speaking of a circle itself. The problem of the continuum begins with the circle....NOT a point. The paradox of the enclosed circles IS defined as the Continuum.

A series of infinite circles, one inside the other as fractions, progresses to a point eventually.

Note that you can define infinity as any set that contains itself. The 'continuum' is an extension of this by what I believe you are meaning. Cantor thought of this by beginning with a circle as I said above, and asked whether the same number of infinite points of one given unit circle is the same infinity. It is not.

Each circle's circumference has an infinite number of equilateral geometric shapes which occur within it, as such each circumference has an infinite number of points.

My evidence is strictly counting the number, you cannot stop.


Another way to think of the problem that you MAY be thinking of instead is to begin with an area square and ask how many 'points' it contains in comparison to other sized squares. For example, if you begin with a square that is 1 cm square, imagine dividing it in half and then aligning each half length-wise (to become a (0.5 cm X 2 cm). The area remains the same but the shape is an elongaged rectangle.

Now repeat this process by dividing the new rectangle in half by the same way and reshaped to be (0.25 cm X 4 cm).

If you continue to do this infinitely, you get a single line that is
[1] (0 cm X ∞ cm) = 1 cm^2

This turns the area into a single dimension line. Next take a new square of 2cm to each side. Repeat this and you also have another line that is
[2] (0 cm X ∞ cm) = 2 cm^2


Now combining these, we have,

[1] == [2] because

(0 cm X ∞ cm) == (0 cm X ∞ cm)

But by substitution, we also have,

1 cm^2 == 2 cm^2 [a contradiction]

This demonstrates the problem that I believe you are thinking. You can only resolve this if one infinity differs from another. This is mathematically resolved by the concept of 'derivatives' in calculus. This particular problem is the derivative of x^2 which equals 2x. This is a relative concept that requires some standards but should give you an idea of how Calculus was developed.

The problem of one infinity being greater than another is solved by geometry. You can have 2 infinite lines, divide one in half and the other line is a greater infinity than the prior.

[Scott Mayers]

You missed the argument. You don't have a proposal that is defined clearly nor is 'unique' to the issues. The last part I presented on the topic is not a proof related to a line but an area (2-dimensional concept) reduced to a line (1-dimension). This is needed to relate the different areas to a single line.

Your assumptions lack foundation with no dimensions nor one dimension. You are "intuiting" an area. No paradox exists otherwise. And a 'field' is a set of distinct points, sets, or objects that are involved collectively in some expression. The simplest 'field' would be a plane (as a 'Cartesian product of two sets) AND therefore at least 2 dimensions.

[Eodnhoj7]

You missed the argument. You don't have a proposal that is defined clearly nor is 'unique' to the issues. The last part I presented on the topic is not a proof related to a line but an area (2-dimensional concept) reduced to a line (1-dimension). This is needed to relate the different areas to a single line.

Your assumptions lack foundation with no dimensions nor one dimension. You are "intuiting" an area.

False, an area is a boundary between points. The center point in contrast to the circle is an area. Now matter how many "circles" you progress inward to the point there are infinite more, while the point is always present. The point is 0d, and as 0d has an infinite number of fractal circles progressing from it.



As to the second point, false, you can draw the proof yourself. If you draw a circle, then a circle of 1/2 circumference inside of that one, followed by a circle of 1/3 inside that one...and keep repeating, you end up with a simple dot, the circles progress to point zero. Now take a circle of 1/× dimension, use it ad a starting point you have a series of circles (1/(x+1), 1/(x+2), etc) and they continue moving towards a simple dot.

The circles are fractals of the original circle.

If you take the circumference, stretch it out into a line, you have a series of linear ratios that eventually move to point zero as well.

All of this can be proven by drawing, it always ends in a point no matter how far you go.


No paradox exists otherwise. And a 'field' is a set of distinct points, sets, or objects that are involved collectively in some expression. The simplest 'field' would be a plane (as a 'Cartesian product of two sets) AND therefore at least 2 dimensions.

I said "boundless field" not "field". A single point with no back drop equates to a blank slate. The evidence is real simple, draw a point with no other point to reference, not even the outside edge of the paper, and you cannot do it.

[Scott Mayers]

A circle is TWO-dimensional by necessity! A point is NOT.

I think that if you want to propose some theorem, you need a particular system of logic to argue from or require developing one yourself from scratch. I can't relate to your language and will stop here. I probably shouldn't have said anything.

[Eodnhoj7]

A circle is TWO-dimensional by necessity! A point is NOT.

I think that if you want to propose some theorem, you need a particular system of logic to argue from or require developing one yourself from scratch. I can't relate to your language and will stop here. I probably shouldn't have said anything.

Yes, I know. That is why it is called a paradox. The progressive shrinking of the circles results in an infinite number fitting inside of a dot. The proof is strictly in drawing it naturally or through a computer. I don't need an advanced logical system to convey that.