sqishy
September 24th, 2015, 07:49 PM
After having a set of ideas a few days ago on finite V infinite and bounded V unbounded, I'm just goint to put this out in a few words (a few mostly because it's 1 am here). This is somewhat mathematical by the way. I'm relatively not good at explaining what goes on with ideas in my head, so I'm also doing this to see how well I can try to get it out. It's not a finished idea by the way.
Let there be a 1-D (D for dimensional) object in a 2-D space. We will define this object by two aspects of it: it's length, and if it has an edge/bound to it. (Before going on, I'm not talking about mathematical sets and their cardinalities etc for those in the know. This is why I'm introducing independent definitions below.)
Imagine there is a point (a 0-D object) that exists on the object, and only on it. This point is free to move along the object in either one direction or the other (let's assume that there are only two possible directions to travel in 1 dimension), and it has to stay on the object. It is not allowed to move 'outside' it. The point is always a part of the object. The point also can only move at a certain constant (but arbitrary) speed.
The length of the object can either be finite, or infinite. If it is finite, it will take a set finite amount of time for the point to travel across all of the object.
If the length is infinite, it will take forever for the point to travel across all of the object.
Short story: either it takes you some amount of time to cover the object, or it takes forever.
The 1-D object either has edges, or it does not. If it has an edge, the point will be 'stopped' from moving further in either one or both of the two possible directions it can travel across the object. In other words, the point's motion is at some point forbidden because it would thereafter travel 'outside' the object.
If the object has no edge, both of the possible directions the point can take will never reach a 'stop'; the point will always stay part the object.
Short story: either your motion can possibly make you hit the edge of the object, or it can't.
(Hope this makes sense so far)
So, if there are two ways to define this object, and two options for each, that means that there are four permutations of what way this object can turn out to be:
1. Finite and bounded (with edges)
2. Finite and unbounded (no edges)
3. Infinite and unbounded
4. Infinite and bounded
Number 4 sounds paradoxical. The usual definition of infinity implies that there are no edges or bounds. Anything infinite goes on forever, so how can it have any edges to it?
Let me give suggestions of examples for each, including 4.
1. Finite and bounded: LINE SEGMENT
https://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/2a6ad8b5-093a-4600-aab7-86b9a9cbbf81.gif
2. Finite and unbounded: CIRCLE
https://www.sharelatex.com/blog/images/tikz/1/circle.png
3. Infinite and unbounded: LINE
http://etc.usf.edu/clipart/41700/41722/fc_line_41722_lg.gif
4. Infinite and bounded: RAY
http://www.homeschoolmath.net/teaching/g/angles/ray1.gif
The points define edges/bounds, and arrows defines the indicated lines going on forever.
Hopefully what I said already makes 1,2 and 3 understandable with their examples. I only came up with relevant ways to define finiteness and boundedness as I was typing this out.
What I did this all for, was to show the ray as something that is infinite, but also bounded. The ray has an infinite length, yet there is an edge to it. Denying either means denying the idea of what a ray is, and there is an image of that very thing.
This is only an idea, like everything I think of, and what my opinions on most things, if not everything, are too. I came here only to see if it gives a new perspective for some people. Hope it's something.
Let there be a 1-D (D for dimensional) object in a 2-D space. We will define this object by two aspects of it: it's length, and if it has an edge/bound to it. (Before going on, I'm not talking about mathematical sets and their cardinalities etc for those in the know. This is why I'm introducing independent definitions below.)
Imagine there is a point (a 0-D object) that exists on the object, and only on it. This point is free to move along the object in either one direction or the other (let's assume that there are only two possible directions to travel in 1 dimension), and it has to stay on the object. It is not allowed to move 'outside' it. The point is always a part of the object. The point also can only move at a certain constant (but arbitrary) speed.
The length of the object can either be finite, or infinite. If it is finite, it will take a set finite amount of time for the point to travel across all of the object.
If the length is infinite, it will take forever for the point to travel across all of the object.
Short story: either it takes you some amount of time to cover the object, or it takes forever.
The 1-D object either has edges, or it does not. If it has an edge, the point will be 'stopped' from moving further in either one or both of the two possible directions it can travel across the object. In other words, the point's motion is at some point forbidden because it would thereafter travel 'outside' the object.
If the object has no edge, both of the possible directions the point can take will never reach a 'stop'; the point will always stay part the object.
Short story: either your motion can possibly make you hit the edge of the object, or it can't.
(Hope this makes sense so far)
So, if there are two ways to define this object, and two options for each, that means that there are four permutations of what way this object can turn out to be:
1. Finite and bounded (with edges)
2. Finite and unbounded (no edges)
3. Infinite and unbounded
4. Infinite and bounded
Number 4 sounds paradoxical. The usual definition of infinity implies that there are no edges or bounds. Anything infinite goes on forever, so how can it have any edges to it?
Let me give suggestions of examples for each, including 4.
1. Finite and bounded: LINE SEGMENT
https://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/2a6ad8b5-093a-4600-aab7-86b9a9cbbf81.gif
2. Finite and unbounded: CIRCLE
https://www.sharelatex.com/blog/images/tikz/1/circle.png
3. Infinite and unbounded: LINE
http://etc.usf.edu/clipart/41700/41722/fc_line_41722_lg.gif
4. Infinite and bounded: RAY
http://www.homeschoolmath.net/teaching/g/angles/ray1.gif
The points define edges/bounds, and arrows defines the indicated lines going on forever.
Hopefully what I said already makes 1,2 and 3 understandable with their examples. I only came up with relevant ways to define finiteness and boundedness as I was typing this out.
What I did this all for, was to show the ray as something that is infinite, but also bounded. The ray has an infinite length, yet there is an edge to it. Denying either means denying the idea of what a ray is, and there is an image of that very thing.
This is only an idea, like everything I think of, and what my opinions on most things, if not everything, are too. I came here only to see if it gives a new perspective for some people. Hope it's something.