Introductory Remarks
on Cantor

by Miles Mathis Topology makes the same mistake as the calculus in assuming that a mathematical line in R2 represents a one-dimensional subspace. But a mathematical line is not equivalent to a physical line. A physical line is a one-dimensional subspace. A mathematical line is a three-dimensional subspace. In my paper on the calculus I have shown that a mathematical line in R2 represents a velocity, which is not a one-dimensional subspace. In R3 a line represents an acceleration. In R4 a line represents a cition (Δa). Since velocity is a three-dimensional quantity—requiring the dimensions y and t, for instance, plus a change (a change or Δ always adds a dimension to any representation of a field)—it follows that a line in Rn represents an (n+1)-dimensional subspace.
The mistake underlying both fields goes all the way back to Euclid. The Greeks failed to differentiate between a mathematical line and a physical line. Therefore when Archimedes began solving the area under the curve, he mistakenly took these curves as straight representations of physical curves. They are not. Throughout the history of mathematics there has been confusion on this issue. Descartes added to it with his coordinate system, which also did not make it clear that points on the graph were not points in space.

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I have seen many proofs of Cantor's theorem that the irrationals (or reals) are uncountable, and none is at all convincing. They all set up some sort of one-to-one relationship between integers and irrationals, and then show that new irrationals can be manufactured in the given interval. But the logic in these proofs is very faulty. We are shown a chart like this one:
1 ...a11a12a13a14...
2 ...a21a22a23a24...
3 ...a31a32a33a34...
4 ...a41a42a43a44...

Line one is one irrational, each “a” standing for a digit. Then we assume we have a complete chart for integers to infinity, and we overlay that chart on this chart. This gives us an enumerated set to infinity. Then we simply find a new irrational. This is intended as proof that the set of irrationals is larger than the set of integers, as well as that the set of irrationals is not countable.
But we cannot possibly "count" irrationals in order, like we do integers. Notice that there is no way you could ever make a chart like this in the first place, since you could never choose a first irrational after zero. That first irrational in the chart has an infinite number of a's in it. And, whichever "first irrational" you choose, I can choose a smaller one.
This is proof not that the irrationals have a higher order of infinity, but that Cantor's methods are faulty. Counting requires an operation, and the operation of counting integers cannot be analogous to counting irrationals. If you were counting irrationals from 0 to 1, you would not start at the lowest and count toward 1. That is absurd. There is no lowest, for one thing. And since they are continuous, you could not find the "next" one, ever. To count them, you would have to start with a certain focus, and then proceed to a finer and finer focus. The first focus is arbitrary, say all the irrationals larger than 1/1000. Then you work your way infinitely down. As you count them, you take higher and higher integers, but your irrationals do not get higher and higher. Nor do they have any order at all, in a higher or lower sense. The only order they have is from your focus filter.
The primary operational fact here is that no matter how many irrationals you have to count, you will always have an integer available to count it with. Always. Therefore, the claim that there are more irrationals or reals than integers or rationals is nonsense.
Supporters of Cantor reply that this chart need not be in any order. It can be a random chart. That is, a11 can be higher or lower than a12 or a42. And line 2 can be a higher or lower irrational than line 1 or 3. But this is not to the point, since whether random or in series the chart can never be postulated to be complete. This fact is not proof that the interval is uncountable. It is proof that Cantor's assumptions are false.

Cantor begins by assuming that the irrationals are countable, and then proceeds to a proof by contradiction—a reductio. Meaning that he shows that his initial assumption cannot be true. But he proceeds by assigning false corollaries to the countability assumption. In this way he has created a strawman. By that I mean that he proceeds in this manner:
1) Imagine a person who believes that irrationals are countable. Let us call this belief [x].
2) This person would have to also believe [y], since [y] is a logical outcome of [x].
3) [y] is false, therefore [x] is false, therefore the person is wrong.
Cantor has created a strawman if it can be shown that he has manufactured argument 2. If I can show that [y] does not follow from [x], then I can show that Cantor has created an imaginary person with imaginary beliefs to argue against, and that his argument is therefore a fantasy. He has not created a proof, he has created a man of straw and an argument of bombast.
This is what I will do. The diagonal method (manufacturing the new irrational) is faulty because it relies on the assumption that, "If the irrationals were countable, we could enumerate all of the irrationals in a sequence. Then we get a sequence enumerating a given interval by removing all of the irrationals outside this interval."* This is the belief that Cantor assigns to his strawman. It is statement [y]. But is is not true. A person who believed that the irrationals were countable would not have to believe this at all. None of these definitions make any sense except at infinity. That is to say, we should expect to be able to enumerate all the irrationals in a sequence only if we are at infinity. But we are not at infinity, therefore we should not expect to be able to enumerate them—whether they are countable or not. The definition of "interval" here is also meaningful only at infinity. We will not have an interval until we reach infinity. The interval of any set of irrationals or reals is undefined at any number less than infinity.
Cantor is therefore guilty not only of manufacturing necessary connections, he is guilty of a contradiction. Countability and enumeration are real acts of a mathematician. They therefore cannot take place at infinity. Counting is like addition or subtraction. It is a definite action over a definite interval. But Cantor assumes that counting is both an infinite method and a finite method. He imagines an infinite set both in the process of being counted and as completely counted. It is like Zeno imagining that the arrow is both stopped and in motion. But you cannot imagine or postulate an infinite set completely counted. Nor can you say that your strawman imagines or postulates that an infinite set can be completely counted.
Cantor says, "My strawman, who believes that the irrationals can be counted, also believes that an infinite interval can be enumerated. Therefore he is mistaken and the irrationals cannot be counted." This is among the weakest and most dishonest proofs in history.

Beyond this, the diagonal method can be shown to be faulty in other ways. Cantor claims that the diagonal method yields a number not in any of the given series. Therefore the set is not only infinite, it is not enumerable. But it turns out that the diagonal method can be used to prove that finite sets are also not enumerable. It can be used to prove that namable sets are not enumerable. It can be used to show that sets that Cantor has already proved are countable and enumerable are not enumerable. The diagonal method (also called the bitnot method in binary math) is a flawed tool. The situation I have just described is not another paradox. It is a faulty explanation. The diagonal method does not manufacture a string of numbers not in the set. The diagonal method manufactures a string of numbers not in the partial set written down and diagonalized. Kevin Delaney** has shown that the manufactured string simply comes later in the finite or infinite list, somewhere beyond the partial list diagonalized. This falsifies Cantor’s proof once again.

*Wikipedia
**See http://descmath.com/frame4.html

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Cantor also claims that the irrationals are infinitely more dense than the rationals. But the rationals are already infinitely dense. You cannot get more dense than continuous. You cannot get more dense than infinitely dense. The rationals, by themselves, are continuous. They have no space between them on the number line. You cannot get more dense than that. You may ask, if the rationals are already continuous, how can you add the irrationals to them? Where do they fit on the number line? The answer is that they don't fit in that way. Numbers and number lines are abstractions. You do not "add" the irrationals to the rationals, like adding three oranges to two oranges. Rationals and irrationals are relationships between numbers—therefore they are not the same order of abstraction as the numbers themselves. Besides, at infinity, the irrationals and the rationals are the same thing. Just as .9 repeating is the same as 1 at infinity, the distance between some rational and some irrational goes to zero at infinity. There is no distance between .9 repeating and 1—that is what a continuous numberline means. It works the same way with the rationals and irrationals. You do not need to find room for the irrationals on an infinitely dense rational numberline. At infinity, one is an overlay of the other, just as .9 repeating is an overlay of 1.

This means that there is no such thing as an uncountable set.

This also affects the "completeness" of an ordered field. In topology, it is believed that the set of reals is complete and the rationals is not. But according to my logic, both the rationals and the irrationals are complete. You may say, what about √2? There is no rational number at that point on the line, therefore the rational number line must have a space at that point. I might say that the irrationals must have a space at 1, since 1 is not irrational. But the irrationals do not have a space at one, since .9 repeating fills that space. You may say, you are not only implying that rationals and irrationals have the same density on the number line, you are implying that there is a one-to-one correspondence between them. If they have the same infinite density, then they must overlay equally at infinity, one rational melding into one irrational. Yes, that is what I am saying.

The problem with irrationals is that they are unwieldy. They are not tidy. They are not convenient for our equations. But remember that most rationals are not wieldy either. Most rationals are huge, and would be exactly as inconvenient to us as irrationals. A supergiant rational is just as much beyond our precise use as an irrational. As both rationals and irrationals approach infinity, therefore, they both approach an infinite untidiness. At infinity, a rational is irrational. It is then an infinite ratio, which is just as untidy as an infinite decimal point. The only difference is that irrationals show an infinite untidiness in the vicinity of zero. Whereas rationals are infinitely untidy only at +/- infinity. Near zero, the rationals are tidy.

The strange thing is that the real world, being continuous, is like the number line at infinity. The real world is existence at infinity. The tidy numberline near zero—composed of the integers and smallest rationals—is the abstraction, because it is a simplification. It is a gross simplification of reality. Most often we calculate using these tidy numbers, far from infinity. But we live at a mathematical infinity.

I do not mean this to sound esoteric or avant garde. I am not suggesting any sort of mysticism. I am simply remarking that an assumption of physical continuity is equivalent in many ways to an assumption that reality is the number line at infinity. That is, all the infinite numbers, rational, irrational, real, etc, are in place. They exist simultaneously on the abstract numberline, either overlayed, or resolving one to one.
Some will say that quantum mechanics is the assumption of non-continuity, and therefore my assumption of continuity is iconoclastic. But quantum mechanics, properly understood, is the assumption of non-continuity in only one thing—the energy states of sub-atomic particles. Extrapolating from this to universal or mathematical non-continuity is a giant leap of logic. It is like saying that because I must climb stairs to my flat (which stairs each have a certain height) the space between the stairs no longer exists. It is like saying, "The stair height is four inches, therefore we may not imagine any height below four inches. A climber must therefore be on one stair or another; he may never be imagined to be between stairs, or at any intervening height."
In my opinion, the long-term findings of QED prove just the opposite. The continuing discovery of new subparticles tends to imply that whenever and wherever we search for subparticles or subdivisions, we will find them.
At any rate, the distinction between finite and infinite appears to me to be an ultimate distinction between measurement and reality. Reality is infinite. Measurement is finite. The value .9 repeating is equal to 1 only at infinity. Therefore, in taking a measurement or in solving any problem by any means whatsoever we must find some error. In measurement, .9 repeating is never equal to 1. Although we live at infinity we cannot calculate at infinity. What this means is that real bodies do in fact converge to the limit. They reach the limit. Achilles passes the tortoise, etc. But mathematical expressions are abstract signifiers of physical movements, they are not the movements themselves. I have shown elsewhere that it is logically impossible to graph a physical point [see my paper on the calculus]. This by itself proves that mathematics cannot fully express motion. It can partially express motion, with a remainder. With a margin of error. A mathematical term that expresses the motion of a body is logically a different sort of entity than the body itself. The body reaches the limit. The term, however, does not.
Mathematically this divergence can itself be expressed. You cannot graph a point because a real point and its mathematical expression are always one system away from eachother. A mathematical curve is a physical acceleration. A mathematical line is a physical velocity. A mathematical point is a physical distance. There is nothing left mathematically to express a physical point, you see. If the mathematical expression is in the field Rn, then reality is always in the field Rn-1.

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