Some Points About Nature of Infinity:
Posted by khuram on May 6, 2007
Some Points About Nature of Infinity:
Issue of the nature of infinity is quite confusing. There are theories in Mathematics, which suggest that a finite line and an infinite line, both will be having one to one, and onto correspondence between their respective points. It means that as per these theories (e.g. Georg Cantor; 1845-1918), number of points of a finite line has to be exactly equal to the number of points of an infinite line. Apparently, this theory made no proper sense to me. Actually I believe in the “discrete” nature of reality rather than considering it to be “continuous”. Soon I shall try to explain this point in my next posts. For the purpose of this post, we can just assume that reality is “discrete” and is not “continuous” in any way. Keeping in view this assumption, below I try to discuss some of the supposed characteristics of infinity:
1. The set of all positive integers (i.e. infinite) is smaller (in cardinality) than the set of real numbers between 0 and 1.
May be true. But I doubt in the “real” existence of real numbers. As I assume (which I shall prove later on) that reality is “discrete” and not “continuous”, so what I think is that continuous numbers can exist only in certain abstract mathematical relations but any such kind of continuity cannot exist in our Physical world. There cannot be infinite discrete numbers between 0 and 1.
2. The set of real numbers between 0 and 1 is equal (in the sense that there exists a one to one and onto mapping) to the set of real numbers between 0 and 2.
Actually this one to one and onto mapping of points is considered to be existing even between a finite line and an infinite line. It means that according to mathematics (theory of Georg Cantor; 1845-1918), the numbers of points on a finite line have its number of points exactly equal to the number of points of an infinite line.
It can be seen that this theory is older than the emergence of Quantum Physics. I no more consider this theory to be valid. A finite line must be having finite number of discrete points and an infinite line must be having infinite number of discrete points. There cannot be one to one correspondence between finite points and infinite points. Quantum Physics has even calculated the minimum possible (or absolute minimum) distance. There is no anything like perfect continuity in our physical world.
Secondly, Geometry still uses old Greek concept of point. It is defined something like an abstract point, which occupies no space. The same Geometry defines ‘line’ something like as a “combination of points”. What I think is that this pure abstract mathematics cannot be applied to the physical world. If a ‘point’ has no space at all then how any ‘line’ (i.e. combination of points) can have any space…??? I think that a combination of ‘space less’ points cannot have any length. Abstract Mathematics says that a ‘line’ has length but it does not have any width. Anyways, there is need to have a Quantum or Discrete Geometry as well.
Basic forms, and a definition of Infinity:
A line can be started from a definite point and can be considered to be extending to infinity on one direction. Such a line can be considered to be ‘infinite’ line. But remember that this ‘infinite’ line has a ‘definite’ origin. This line is infinite only on onward side but this line is not infinite on backward side.
On the other hand, there also can be a line, which can be considered to be extending infinitely towards both sides. This line is also infinite. But this line is infinite towards onward and backward sides both. So there can be one directional infinity as well as there can be two directional infinity. Similarly, there can be multiple directional infinity as well and in the same way, there can be an all-directional infinity also.
One type of infinity can be smaller or larger than other one. Meaning of infinity is Never Ending on one or more sides. A thing which can end on all sides (like a finite line), cannot have never ending points in it.
What is meaning of 1/0?
If you have to divide $100 among zero people, it only means that you are not going to disburse any sum to anyone at all. You can distribute $100 to as many (i.e. never ending) zero persons you like. 1/0 is only Abstract Mathematics. It is good only in abstractions. It cannot be as it is applied to real physical world. More precisely, 1/0 is not the case of “never ending”. Actually it is the case of “never happening”.
The following objection was raised on my above-mentioned points:
“There cannot be infinite discrete numbers between 0 and 1”. You may well be right. But if I take rationals to be discrete, as I can count them through a one to one and onto mapping with the set of integers, then there are indeed infinite discrete numbers between 0 and 1. So your frame of reference and mine are very different. We cannot discuss much and can only agree to disagree on our frames of references.
My response was:
I already have dealt with the issue that a smaller and a larger “line” have exact one to one correspondence between their respective number of “points”. I made diagrams to see if really there was such one to one and onto mapping of points or not.
The actual mistake in official theory lies in the definition of “point” in Geometry. Geometry considers “point” as a “space less” particular location. Since this “point” is space less, so it is having no “length” at all. The same Geometry considers “line” as a linier combination of “points”. The same Geometry also considers that a line possesses a non-zero length but it doesn’t possess any width. But if the constituents of line i.e. “points” had no length at all then how just the combination of those “points” could result in any non-zero length…???
Anyways, this definition of “point” may be right definition in pure Abstract Mathematics. But this is not right for our physical world because there can be no space less physical entity in our spatial world. A “physical point” would be having some “space”. In fact, all the “points” would be having same or uniform non-zero space. Let’s say the length of one point is 1 and the length of a finite line is 100. It means that this line has only and only 100 (discrete) points. Points of this line CANNOT have one to one and onto mapping with the points of that line whose length is 200 or infinite.
What is the mistake of official theory…???
The official theory actually draws one to one and onto mapping not between the individual points of a shorter and a longer (or infinite) line. It actually draws one to one and onto mapping between fractional parts of individual points of shorter line with the complete individual points of longer or infinite line.
Now I give you a task. Consider the “length” of a “point” to be 1. Now take two lines. First line being shorter and second line longer.
Length of first line = 100
Length of second line = 200
In this way, there are 100 “points” in first line and there are 200 “points” in second line. Now try to draw one to one and onto mapping between all the points of first line with all the points of second line. Be careful that do not take fractional or overlapping parts of individual points of shorter line!
Believe me, you will not be able to do it, because it is an impossible task. So come out of the fantasy of old Abstract Mathematics where length of “point” is zero and sum of many zeros (i.e. a “line”) is non-zero positive length. In a physical world, any real “point” will be having non-zero positive length.
This is not the case of just difference of frames of references. It is a matter of clear-cut mistake of 0 + 0 + 0 = 3