Considering mathematical points, lines and planes as opposed to something with substance (as a dot on this page) which has a certain volume. A mathematical point is a point on a line for which all length of the line on the left occurs up until that point and likewise all length of the line on the right occurs up until that point, that is the point separates the two parts of the line, but itself occupies no distance.
Likewise a mathematical line is the junction of 2 planes, but itself occupies no area. And a plane is the contact of two flat surfaces of two volumes, but itself has no volume.
The order of dimensions is:
lines (one dimensional) or length
planes (two dimensional) or length2
volume or area (three dimensional) or length3
Time is used sometimes as a 4th dimension, but I prefer, particularly in this hypothesis, to consider it a separate subject and to refer to dimensions as the geometry of (classical) space (see also More on Dimensions at end of this Appendix).
(From principle 10) any given quantity is infinitely sub-dividable. Any line therefore contains an infinite number of points. So it seems possible all infinities are not alike, as lines can be different.
Now I like to consider such infinities in terms of ratios, so that, for example, from the interval 1 to 10 (set 1) in units of 1 there are 10 units and from 1 – 20 (set 2) there are 20 units. If the units are subdivided (say from 1 to ½ to 1/10, etc ) equally the ratio of the number of units in each set remains equal, or 1 unit in set 1 for every 2 units in set 2, even up to infinity, such that an infinite number of points in set one corresponds (as a set) to 2 x an infinite number of points in set two.
Using the same reasoning and applying it at dimensions (using equal lengths, that is, 10, 102, 103) then,
1 point = 1 point
A line has an infinite number of points
A plane has a similar infinite number of lines each with an infinite number of points, therefore the plane has infinity2 number of points.
A volume has an infinite number of planes each with an infinite number of lines, each with infinite number of points therefore a volume has infinity3 number of points.
Now all numbers are infinitely long, that is even a whole number like 1 is actually 1.0000… to infinity.
All numbers that repeat in any fashion are called rational numbers and all numbers that don’t repeat (but are still infinitely long, as are all numbers) are called irrational numbers1.
The idea is this: that every point on a line is represented by a definite (exact) number, rather it is rational or irrational. However, there is no such thing as a succession of numbers, for the concept of successive amounts deals with area, but area itself is defined as the distance between points.
So between any two points there are an infinite number of other points, and each point represents a definite number, but no two definite numbers (that is points) are successive. This is a unique consequence of the property of points and infinite quantities.
Now why is this important? Because it is unclear rather contemporary math is contrary to the ideas expressed at the start of chapter 2, or in agreement. Therefore I have put forth the above reasoning to try to justify that position.
To discuss infinities further as regard Cantor’s infinite sets2; where it is postulated that the set of whole numbers and the set of even numbers is equal because there is a one to one correspondence as;
Now from 8 paragraphs back it was shown that for any interval of, for example, 1 (0-1) there are an infinite number of points and for 2 (0-2) there are 2 x the number of points in 0-1 or;
0-1 infinite number of points
0-2 2 x infinite number of points
0-3 3 x infinite number of points etc.
Now if Cantor’s sets concern the number of numbers (symbols themselves) in each set, then for a given interval, like 1-10, they are not equal as there are 10 symbols (numbers) in set 1 and only 5 in set 2 (see also Addendum 1). Therefore extending the interval to infinity there are 2 x the number of elements (symbols/numbers) in set 1 as in set 23.
As concerns not number of numbers but units for each correspondence (as each even number is twice as large as its corresponding whole number). It might be considered then that there are 2x as many even numbers as whole numbers per correspondence. However both Cantor's idea of a 1:1 correspondence and the above are wrong as it neglects the need to add the units over an equivalent interval. Doing so gives a fluctuating answer for each interval chosen, which (intuitively figured) tends toward a 2:1 ratio of whole numbers:even numbers, as each set continues infinitely (see also Addendum 1).
CHART I-1 Examples of infinite ratios
3/2, 10/6 etc are the ratio of values after each interval.
Likewise with other sets a ratio of their infinite quantities is arrived at by these methods. For example:
Or like whole numbers : perfect squares x a billion. Breaking this down into;
The question becomes one of how is a “set” defined. In the set of whole numbers the first element of the set is 1 and represents one complete unit. 2 represents two units, etc.
Now in the set 1-4, does this mean only the #4 (four units) or does it mean the value of each element added? Now in talking about apples at 1 there is 1 apple, at 2, 2 apples, etc ? The set 1-4 implies to me all the apples from each element, not just the #4, for if we wanted to talk about the number 4 we would just say, “the number 4”, but here we are saying “the set 1 to 4”.
Likewise the set of all even numbers would exclude all odd number elements or at 2, 2 units of 1, 4, 4 units of 1, etc. Therefore there would be 6 units in the set 1-4 of even numbers.
How are sets to be compared? To the point, how are infinite sets to be compared, regardless of whether we compare the quantities or just the number of symbols? Either way I propose the idea that;
Infinite sets must be compared in a like manner as finite sets.
For if we compare infinite and finite sets differently, then the consistency of mathematical operations is compromised.
If we apply Cantor’s hypothesis to this dictum as regards finite sets, we may compare single elements of the same or different quantities. Or we may compare finite groups or ranges of the same number of elements, of equal or varying quantities. But we may not compare groups, or ranges of numbers with a different numbers of elements.
For example, if we compare the set 1-4 of whole numbers and the set 1-4 of even numbers, by Cantor’s hypothesis there “is” a one-to-one correspondence, therefore for each element in the first set there must be an element in the second set, which there is not. To compare such finite sets is actually excluded by Cantor’s definition (which I claim is incorrect).
Cantor’s definition is a self-defining one that is by definition we need to compare element to element, thereby he excludes the possibility for anything other than a one-to-one ratio for the “total” set*.
Now approaching the problem as I would, first for finite sets one may compare element to element, same or different. Group to group, same or different, and a special case of this, range to range, same or different (for example for two sets of whole numbers compare a different range 10 to 20 with the range 5 to 35; or the same ranges 10 to 20 and 10 to 20).
Let’s now compare the set of whole numbers 1 to infinity and even numbers 1 to infinity. In comparing finite amounts variable sets can be compared, but no such liberty exists in infinite sets, as by definition they contain total amounts. These must then be both groups and ranges, and since the expressions 1 to infinity are equivalent, they must be equivalent ranges. In the finite ranges, 1 to 10 of whole numbers and 1 to 10 of even numbers, there are 2 x the number of elements in set 1 as in set 2. Therefore in the infinite ranges there must be a like ratio. That is the ratio of elements of two infinite sets is equal to the ratio of any finite equivalent range of each set, as the definition of infinite sets is that of an infinite range, and infinite sets must be compared in a like manner as finite sets.
Now the quantities are not equal over equivalent intervals because, I suppose, set 2 is a subset of set 1 and the progressive nature of the odd numbers to be counted that are left in set 1.
But this quantity can be figured easily, as the ratio of the infinite quantities is the same as the ratio of elements (see previous charts), therefore;
Ratio of elements set 1 : set 2 is 2 : 1
Ratio of quantities set 1 : set 2 is 2 : 1
Likewise a definition for all ratios of infinite sets is (Principle I-1 & 2)
The ratio of elements (number of symbols) of any two infinite sets is the same as the ratio among any finite subsets of equivalent intervals.
ratio of quantities of any two infinite sets is equal to the ratio of the
*An analogy is if we
match the sets of albino German Shepherds with Oak Trees. In a finite way there
are many more oak trees, but if we consider an infinite number of planets with
the same, then does Cantor's "mapping" mean there are equal number of
German Shepherds and Oak trees in an infinite universe as such? Or more to the
point if we map one infinite set to another, so supposedly the sets are equal
in elements then at this infinity. Now if we give one of these sets an infinite
number of subsets within it, but leave the other set as it was, does this mean
they are still equal because of one to one mapping? I have said, I think the
answer is no.
I would like to contrast my conception to my perception of Cantor’s conception of the infinite. Cantor’s idea seems to me to be as assuming as two sets converge to infinity, because of the infinite nature of infinity the number of elements in each set must become equal.
We have already seen in Addendum 1 that the “proof’ of this is really an open-ended conjecture, in that the definition excludes any possibility of disproof. That is by requiring one to one “mapping” anything other than a one to one correspondence is excluded.
My conception is more as infinite quantities are not mathematically expressible, in the sense we can only conceive of infinity as something consisting forever. That is, you start with something finite and then multiply or divide it forever. In this way two infinite sets are not equal but, as shown in Addendum 1, they can be ratioed by the finite expression that defines the set, but both extend infinitely.
Now other than the arguments given earlier, I would like to mention another. That is Cantor’s reversal of conceptions, that is, for some sets (non-denumerable) he uses conception #2. Now this is a contradiction to conception #1, and it would seem to me it can’t be both ways.
1 Dedekind, Richard, Irrational Numbers in the World of Mathematics, ed. By James R. Newman, (New York; Simon and Schuster, 1956) I, P. 528-536
2 Dauben , Joseph W., George Cantor and the origins of Transfinite Set Theory, Scientific American, CCXLVIII (June 1983), p. 128
So, if something expands I think it really MUST have a center. We are told if we say this about the expansion of the universe, we don't get it, its "all" expanding. But isn't the idea, it started in the big Bang and expanded outward, a centrist idea. You can’t take a point or area and say it expands to a larger size without the older size being more toward a center. BUT ITS ALL expanding, well from the orginial-ish smaller area particles or space from one side would be crossing over and through expansion from the other side, and particle going outward would be again separating from them, from another center. That would be violating laws of continuity. If expansion is only on the perimeter and the old area or particles or whatever only expands, you again still have a centrist notion. The very concept of expansion cannot escape the idea of a center to it. THEREFORE, if the universe is expanding it must have a center.
A balloon does have a center. The surface of a balloon is atoms, which break the balloon when their bonds lose strength as it expands too much. But space cannot expand on itself.
So, we say there are “three dimensions” But all mother nature really has is space. It does not have a coordinate system streaking though the universe. Points, lines and planes are our ideas, and they represent measuring tools only! Extension, as some have called it (Descartes?), is what is real. Nature has AREA, space is area of “vacuum” devoid of anything. Mass is area of substance superimposed on space. That is why everything is 3-D, because less is just the measuring tool, it does not represent real area itself. And that is why any “dimension” over 3 is unnecessary, because 3-D represents all area. And you should not superimpose space on space. You cannot superimpose mass on mass. You should not have another substance other than mass. One should to work from simple to complex, for as system of dynamic change to function. The simplest you can get is space, mass, and motion, and therefore time also as motion over time is what allows for more than one being to have intelligence. Also logic is a fundamental factor or we have chaos, in a natural universe, not order, and chaos has no measure, therefore no space, therefore no mass therefore no motion, therefore only time, well maybe not even time. Therefore, because of the reality of the universe logic has to be part of the mix. Logic, space, mass, motion and time, all in the classical sense. The fundamental laws of motion, of mass, space and of time, and even logic, must be the same universe wide, else we have lost logic. To superimpose extra dimensions, extra area that is on each other, and to interact mass from one parallel/superimposed area to another would require different laws of motion than within one space. As the different/extra dimensions are all part of the same overall universe, then that would violate consistent laws, else we have chaos.