Good. Then shut up.
0d Lines and Circles
Re: 0d Lines and Circles
Re: 0d Lines and Circles
wtf wrote: ↑Fri Oct 18, 2019 4:45 amNo. Two points may have a distance between them. They don't "constitute" a length, that doesn't mean anything. But two points (in a metric space such as the real numbers) always have a distance (or length, if you like) defined between them.
So points have a distance between them but this distance has no length? That is a contradiction.
In a metric space, infinite or finite, there is a distance defined between any two points.
The unit interval typically refers to the interval between 0 and 1. But you could view the real number line as being composed of infinitely many length-one intervals end to end. Is that what you mean?
That unit interval is composed of fractions, as composed of fractions, where each fraction is a whole number in itself (ex: 1/3,1/3,1/3 observes three lines, thus 3.)
This line as composed of many lines, which each line composed of lines, necessitate infinite 0d point which have a distance between them thus length equivalent to a number.
If I have a number line divided in half, it is 2 lines as one number line. Each set of points has a distance between them with this distance being the number as a ratio itself. So each line, on the one line is 1/2 and 2. For three lines it is 1/3 and 3. So on and so forth. The distance between the points, as the line segment, is a quantity.
A line composed of infinite points is infinite lines. Each number is an quantified infinity. 2 lines is 2 infinities, 3 is 3 infinities, so on and so forth.
Re: 0d Lines and Circles
I said you can call the distance a length if you like. That's exactly what I said. Formally, in the theory of metric spaces, we assign a distance between any two points. If you like, you may informally call that a length; or you can think of it as the length of a line segment between the points.
Yes the real numbers contain fractions, or rationals; and also a lot of irrationals, numbers that can not possibly be represented as the ratio of integers. Old Pythagoras discovered that.
I'm afraid I could not understand this sentence. Are you saying that each pair of points has a length between them? Yes I agree.
Yes ok. The unit interval [0,1] = [0, 1/2) ∪ [1/2, 1}. The square brackets mean the point is included, and the paren means that point is excluded. I'm notating it this way so that no point is in both segments; that is, this is not only a union, it's a disjoint union. So ok, yes we can do this. We can subdivide a line segment into a disjoint union of smaller segments.
You lost me here. But if you mean the lengths of the segments get smaller, ok. Or if you mean that you could divide the original segment into thirds, or fourths, or fifths, yes you could do that.Eodnhoj7 wrote: ↑Fri Oct 18, 2019 5:16 am Each set of points has a distance between them with this distance being the number as a ratio itself. So each line, on the one line is 1/2 and 2. For three lines it is 1/3 and 3. So on and so forth. The distance between the points, as the line segment, is a quantity.
Otherwise I have no idea what you are saying here.
Meaning what? For example I could subdivide the unit interval as [0,1] = [0, 1/2) ∪ [1/2, 1/4) ∪ [1/4, 1/8) ∪ ...
Is that what you mean? That's a subdivision of the unit interval into infinitely many subintervals. Interestingly enough, the lengths add! 1/2 + 1/4 + 1/8 + ... = 1. That's countable additivity!!
But if we wrote the unit interval as the union of "degenerate" line segments of zero length, in other words the union of all the singleton elements, the lengths would no longer add. Each length is zero but the total segment length is 1. That's because there are uncountably many points. We can only add lengths of a countable infinity of segments. Do you know countable and uncountable infinity?
Yeah ok. I fail to see the point you're making.
Oh do you mean each subsegment also has infinitely many points? Yes most definitely. But the cardinality never changes. Each segment is uncountable infinity and so is the union. Is that what you're talking about?
Re: 0d Lines and Circles
wtf wrote: ↑Fri Oct 18, 2019 5:35 amI said you can call the distance a length if you like. That's exactly what I said. Formally, in the theory of metric spaces, we assign a distance between any two points. If you like, you may informally call that a length; or that you can think of it as the length of a line segment between the points.
Of course Pythagoras showed your claim to be false. The unit interval and the real numbers in general contain fractions, but they also contain many number that can not be fractions, such as the square root of 2.
Ha, Ha, that is where it gets interesting the square root of 2 (1.41...) is an irrational number as an irrational number it is perpetipually changing...it is dynamic as the fraction itself observes the number line still approaching infinity. The number line is always approaching infinity, as each line composed of infinite lines. The number line itself, continually manifesting whole numbers as 1 infinite set, necessitates itself as dynamic.
The line is continually expanding, but we measure this expansion by multiple other expansions.
To picture this from a different point of view if a train is moving at 60 miles an hour and I am moving at the same speed it appears still.
So the two lines of length one forming the right angle are the observer and the training, the 1.41.. is the surrounding environment. The "hypotenuse" (some would not call it that) is continually changing but this change is contained by the horizontal and vertical lines of the right angle.
That's a number, or length as the Greeks thought of it, that arises as the diagonal of a unit square. Pythagoras discovered that sqrt(2) can NOT be written as a fraction. So the real line contains the fractions along with the irrational numbers.
The number line itself is irrational, and yet 1 line, thus the number line is dynamic.
It appears static using the above example. The line is a dynamic entity, one is dynamic...it is always changing yet this change maintains itself as one.
I'm afraid I could not understand this sentence. Are you saying that each pair of points has a length between them? Yes I agree.
Yes I am saying that, and since that is the case a line as defined by infinite points has infinite lines.
Yes ok. The unit interval [0,1] = [0, 1/2) ∪ [1/2, 1}. The square brackets mean the point is included, and the paren means that point is excluded. I'm notating it this way so that no point is in both segments; that is, this is not only a union, it's a disjoint union. So ok, yes we can do this. We can subdivide a line segment into a disjoint union of smaller segments.
Eodnhoj7 wrote: ↑Fri Oct 18, 2019 5:16 am
Each set of points has a distance between them with this distance being the number as a ratio itself. So each line, on the one line is 1/2 and 2. For three lines it is 1/3 and 3. So on and so forth. The distance between the points, as the line segment, is a quantity.
You lost me here. But if you mean the lengths of the segments get smaller, ok. Or if you mean that you could divide the original segment into thirds, or fourths, or fifths, yes you could do that.
The number line as infinite points means the line is composed of (2,1/2,1)(3,1/3,1) etc. The number line necessitates that is progressive fractals as whole numbers as it contains all fractions and fractals.
Otherwise I have no idea what you are saying here.
Meaning what? For example I could subdivide the unit interval as [0,1] = [0, 1/2) ∪ [1/2, 1/4) ∪ [1/4, 1/8) ∪ ...
Is that what you mean? That's a subdivision of the unit interval into infinitely many subintervals. Interestingly enough, the lengths add! 1/2 + 1/4 + 1/8 + ... = 1. That's countable additivity!!
Yes, but each line as a fractal is [0,1]. So each line, as composed of infinite points is infinite fractals with each fractal being a whole number and number line simultaneously.
But if we wrote the unit interval as the union of "degenerate" line segments of zero length, in other words the union of all the singleton elements, the lengths would no longer add. Each length is zero but the total segment length is 1. That's because there are uncountably many points. We can only add lengths of a countable infinity of segments. Do you know countable and uncountable infinity?
All numbers are composed of infinite numbers on both the number line and the basic tautological nature of arithmetic (ex: 3= 1+1+1,1+2,4-1,5-2,6-3...)
Infinity is always countable and uncountable as all numbers are infinities.
Yeah ok. I fail to see the point you're making.
You have 1 line, it is divided into 3rds. 3 lines exist as one Line. Each line is an infinity as it is composed of infinite lines. The 1 line is infinite through each respective of the 3 lines being infinite.
Finiteness is multiple infinities.
Oh do you mean each subsegment also has infinitely many points? Yes most definitely. But the cardinality never changes. Each segment is uncountable infinity and so is the union. Is that what you're talking about?
Read above.
Ignore the red, I messed up and forgot you posted in read. Sorry, you are going to have to sort it out...I am using an ipad.
Re: 0d Lines and Circles
It's all red. I never posted in red. YOU posted in red and I had to sort it out. I'll take a look at what you wrote but meanwhile the quote tags are very straightforward on this board, it would be more clear if you used them as intended. IMO.
Ok I read a little. Irrational numbers don't "keep changing." They're a point on a line. sqrt(2) is a particular point on the number line, it doesn't change. It's simply not the ratio of integers.
Your use of "fractals" is out of place, there are no fractals being discussed.
I don't understand the notation in "The number line as infinite points means the line is composed of (2,1/2,1)(3,1/3,1) etc." What do those number mean? They're ordered triples. I don't follow your notation.
Ok I finished reading your post. A lot of your jargon simply makes no sense to me. What exactly is the point you're making?
Re: 0d Lines and Circles
wtf wrote: ↑Fri Oct 18, 2019 7:21 amIt's all red. I never posted in red. YOU posted in red and I had to sort it out. I'll take a look at what you wrote but meanwhile the quote tags are very straightforward on this board, it would be more clear if you used them as intended. IMO.
Ok I read a little. Irrational numbers don't "keep changing." They're a point on a line. sqrt(2) is a particular point on the number line, it doesn't change. It's simply not the ratio of integers.
Irrational numbers cannot be marked on a number line as that would make them rational and finite....they would have to be rounded and in a fixed postion.
Your use of "fractals" is out of place, there are no fractals being discussed.
No, a number line is a line composed of many lines, these lines are fractals.
I don't understand the notation in "The number line as infinite points means the line is composed of (2,1/2,1)(3,1/3,1) etc." What do those number mean? They're ordered triples. I don't follow your notation.
2 on a number line.
2 lines are present
1/2 line is present
1 line is present.
The same occurs with 3 on a number line.
And 4,5,6,etc.
The line as infinite points, hence infinite lines, is the number line itself.
Ok I finished reading your post. A lot of your jargon simply makes no sense to me. What exactly is the point you're making?
You cannot have a line composed of infinite 0d points without having infinite lines.....your "mystery" is just bullshit....it is made up.
Re: 0d Lines and Circles
Everyone knows the first person to throw the insults ran out of the ability to present a point....thanks.
Anyhow, the 0d lines allow for 1d lines to exist as condensed matrixes.
Re: 0d Lines and Circles
Re: 0d Lines and Circles
That's silly. Take a square of side 1. Its diagonal has length sqrt(2). That's a fixed, unchanging length of sqrt(2). You could translate the diag to the number line and you'd see that sqrt(2) is a specific point. You're just making things up. You'd benefit from studying some math.
Re: 0d Lines and Circles
Re: 0d Lines and Circles
Really, a line is composed of infinite x 0d points is a mystery we must accept because someone told us too...wtf wrote: ↑Fri Oct 18, 2019 7:02 pmThat's silly. Take a square of side 1. Its diagonal has length sqrt(2). That's a fixed, unchanging length of sqrt(2). You could translate the diag to the number line and you'd see that sqrt(2) is a specific point. You're just making things up. You'd benefit from studying some math.
But saying this is wrong:
You cannot plot an irrational continuous number on a chart without rounding it.
It is a continuous fraction.
It is like saying pi is 22/7 but it progressively get more an more accurate at 355/113 and more accurate from there.
Whatever is charted is rounded. If the number is truly charted, the line would slowly change because it would be dynamic.
Re: 0d Lines and Circles
Re: 0d Lines and Circles
Good, if I am pretender, address the OP question then....or are you going to hide behind ad hominums to fake some intellectual superiority over everyone. I mean let's face it, you think you are better than everyone here.
So answer the question:
You have 2 1d lines
_______________
_______________
A line is formed between them as 0d. It has no form or direction. It is isomorphism at is simplest.
Does a 0d line exist or not exist?
Re: 0d Lines and Circles
My intellectual superiority is a fact. I don't even remember the last time I lost a major philosophical debate, was probably 10+ years ago.Eodnhoj7 wrote: ↑Fri Oct 18, 2019 8:53 pmGood, if I am pretender, address the OP question then....or are you going to hide behind ad hominums to fake some intellectual superiority over everyone. I mean let's face it, you think you are better than everyone here.
So answer the question:
You have 2 1d lines
_______________
_______________
A line is formed between them as 0d. It has no form or direction. It is isomorphism at is simplest.
Does a 0d line exist or not exist?
Very well, my answer to your question is (I didn't read the OP, just this short version): you don't understand what a line is, what dimensions are, and how to use 'exist' in such a context. The way you are confusing things hints at a possible psychosis. Especially this idea of the 'act of forming'.