Infinitesimal Graph: Visualizing the Unit Interval

Briefly stated, infinitesimal graph is what we get when we consider an infinitesimal in the unit interval as a dedekind edge separating rational numbers above and below a number. The dedekind cut itself we imagine to be on the right side of the edge and call it a dedekind node.

Infinitesimal Graph. In representing the unit interval as a graph, we consider only the recursive numbers, that is, the numbers that are programmable on a computer. Since the programs in a computer can be listed, it follows that the cardinality of recursive numbers in a unit interval is \aleph_0. Thus the cardinality of the set of edges in the infinitesimal graph corresponding to a unit interval is \aleph_0.

Just as an infinite graph cannot be drawn in its entirety, we cannot draw a completed infinitesimal graph also. But we can imagine a limit process which will ultimately provide the infinitesimal graph in the limit, as described below.

Write the numbers 1,2,3,4,... in binary in the reverse order as 1,01,11,001,... and put a binary point in the beginning of each resulting in the infinite sequence .1,.01,.11,.001,... representing fractions in the unit interval. Mark the dedekind cut corresponding to these points and consider them as nodes. The graph we get as a limit is the infinitesimal graph for the unit interval (0,1].

The cardinality of the set of points in a dedekind edge can easily seen to be 2^\aleph_0. First of all, note that any fraction in the unit interval can be written as an infinite binary sequence. For example, we can write the sequence for 2/3 as 0.10101... , which we can use to have an infinite set representation for 2/3. If we take the positions at which the sequence takes the value 1, we get the set representation for 2/3 as the set of odd integers. Further, we can claim that the dedekind edge for 2/3 contains all those sets of positive integers which contain the odd integers. The cardinality of this set we can easily recognize as 2^\aleph_0.

The description above allows us to state the following facts. The unit interval is a set of dedekind edges, each edge representing a recursive number, and the cardinality of the set is \aleph_0. A dedekind node corresponding to a number cuts the unit interval into two pieces, each piece representing reals below and above the number. Each dedekind edge contains 2^\aleph_0 elements.

Real Cardinality. We may call an element of an edge a figment and the edge itself a relement (real element), and claim that not even the axiom of choice can pick a figment from a relement. We define real cardinality as the cardinality of a set when only relements in it are considered.

Skolem Paradox. These definitions allow us to resolve some of the vexing problems of set theory. Cantor's theorem asserts that every model of Zermelo-Fraenkel set theory (ZF) has to have cardinality greater than \aleph_0. On the other hand, Lowenheim-Skolem theorem (LS) says that there is a model of ZF theory, whose cardinality is \aleph_0. These two statements together is called Skolem Paradox. The notion of relements gives us a reasonable way to circumvent the paradox. We merely take the LS theorem as stating that the real cardinality of a model of set theory need not be greater than \aleph_0. Cantor's theorem is satisfied when we consider figments also as elements.

Nonmeasurable Sets. With the introduction of the concept of relements, we are in a position to dispense with the notion of sets which are not Lebesgue measurable. This is because a nonmeasurable set cannot be constructed without using the axiom of choice.

Further reading: See the paper http://www.e-atheneum.net/science/ist_visualization.pdf for some more details. Note that the axioms given in this paper are dated, even though nothing has been found to be wrong with them.

The Book: Visualizing the Intellectual Universe

The Book is an invention of Paul Erdos, one of the most prolific mathematicians of the twentieth century. This book lists all the proofs of mathematics, written in the formal language of mathematical logic, according to length and in lexical order. With our knowledge of formal logic today, we know that a computer can be set up to start writing this book, and the computer will write as many pages as we desire, but we also know that the computer will never be able to complete the book. Fortunately for us, the author of The Book is not any computer, but the handiwork of The Almighty, and what we have is a totally finished book.

While the initial part of the book can be visualized without any difficulty, getting a mental picture of the last pages of the book does present a problem. The following is a visualization of the book for those who have no difficulty with the cardinals and ordinals of Cantor.

Anatomy of The Book. The salient features and the physical appearance of the book can be visualized as below.

• The front cover and the back cover are each one millimeter thick, and the entire book, including the covers, is three millimeters thick.
• The first sheet of paper is half-millimeter thick, the second sheet is half thick as the first, the third sheet is half thick as the second, and so on.
• On every odd page is written a full proof, and in the next even page is written the corresponding theorem.
• All the pages corresponding to the ordinals after \aleph_0 are stuck together so that not even the axiom of choice can be invoked to open them. For this reason, unfortunately, we will never know what His Last Theorem is, in fact, we will not even know whether there is a last theorem.

From the description of the book, we can infer that any formula which is a theorem can be found in the book, by sequentially going through the pages of the book. The only difficulty is that, if a formula is not a theorem, we will be eternally searching for it.

Note that the lexical order is with respect to proofs and not with respect to theorems, a cruel joke by The Almighty. David Hilbert, the high priest of formalism, once had high hopes of rewriting the book with theorems in the lexical order, but it did not take very long for logicians to find out that He is not in favor of the project.

A scientist can use The Book for taking his oath in the Ultimate Court of Nature. Years of cogitation on the limits of his knowledge makes the oath somewhat subdued: I solemnly swear and affirm that, if I am sane, I will tell the truth, nothing but the truth, however, not the whole truth.

Obviously, the following facts must have weighed heavily in the mind of the scientist:

• Mathematical logic is unable to decide the consistency of even the Elementary Arithmetic of Goedel. We believe our mathematics has no contradictions, just because it has worked well for us for the last two thousand years. Thus, the scientist is unable to categorically vouchsafe for the sanity of his logic.
• The exhilarating experience of the scientist is that whenever he writes his musico-logical composition strictly conforming to mathematical logic, he finds that nature is faithfully dancing to his tune. Earth dances around the sun, because Newton's laws forces it to do so, and electromagnetic wave radiates rhythmically through the universe because Maxwell's equation says, let there be light. Considering all this, the scientist feels that he is telling the truth and nothing but the truth, about all his observations of nature.
• The incompleteness theorems of Goedel and other developments in mathematical logic make it clear to the scientist that he will never be able to tell the whole truth regarding any subject matter. The talk of subquarks after the discovery of the final sixth quark reinforces this conviction of his.

Epilogue. Essentially, what The Book teaches us is humility, and consistent with this, it is not surprising that Bertrand Russell has the following advice for us:

United with his fellow-men by the strongest of all ties, the tie of a common doom,.... Be it ours to shed sunshine on their path, to lighten their sorrows by the balm of sympathy, to give them the pure joy of a never-tiring affection, to strengthen failing courage, to instil faith in hours of despair. Let us not weigh in grudging scales their merits and demerits, but let us think only of their need, of the sorrows, the difficulties, perhaps the blindnesses, that make the misery of their lives; let us remember that they are fellow-sufferers in the same darkness, actors in the same tragedy with ourselves.

White Hole and Black Whole: Visualizing the Physical Universe

Here, white hole is the terminology we use when we cram a set of points of cardinality 2^\aleph_0 into an interval of infinitesimal length and black whole is the terminology for the entire space beyond the finite space.

We will consider only the real line in its entirety, infinite on both sides, with the understanding that we can have a visualization of the three-dimensional space, if we have a clear mental picture of the one-dimensional line.

White hole. First of all, let us note that corresponding to every real number it is possible to visualize a white hole attached to it. We will illustrate this with an example. Consider the number 2/3 written as an infinite sequence 0.101010... and its finite terminations 0.1, 0.101, 0.10101, ... which can be used to represent the intervals (1/2,2/3), (5/8,2/3), (21/32, 2/3), ... respectively. Note that the length of the interval decreases monotonically when the length of the termination increases and the cardinality of the set of points inside these intervals remain constant at 2^\aleph_0. Simply stated, we can say that a white hole is what we get when we visualize the interval corresponding to the entire nonterminating sequence, and this infinitely small interval contains 2^\aleph_0 points in it. From this description it should be clear that the term white hole does introduce us to the notion of an infinitesimal.

It is easy to see that any point in a unit interval can be represented by an infinite subset of natural numbers. For example, the representation 0.101010... shows that 2/3 can be represented by the set of odd integers, the places where the 1's occur in the sequence. Recognizing this, we can define the white hole or infinitesimal corresponding to 2/3 as the set of all those subsets of \aleph_0 which contain the odd integers. We will take it for granted that the cardinality of this set is 2^\aleph_0.

Black Whole. When we flip the real number xxx...xxx.xxxxx... around the binary point, the resulting ...xxxxx.xxx...xxx we define as a supernatural number and from symmetry considerations we assert that a transfinite stretch comes attached with every supernatural number. Using the notion of point at infinity of complex analysis, a transfinite stretch may also be called a black hole, without violating the conventional sense of the word. If we define the black whole as the set of all black holes and the white whole as the set of all white holes, we will have the universe neatly divided into the two halves, white whole and black whole.

Epilogue. According to our visualization, the infinitesimal corresponding to 2/3 is a dedekind edge between the rational numbers less than 2/3 and greater than 2/3. We do not call it a Dedekind Cut because of its nonzero length.

If we call the elements of a white hole, figments, and consider it as axiomatic that not even the axiom of choice can pick up a figment from a white hole, scientists will have the pleasant situation where they will not have to deal with sets which are not Lebesgue measurable. This follows from the fact that the axiom of choice is crucial for the creation of nonmeasurable sets. As a corollary, we can also claim that we will never be
able to walk along a transfinite stretch.

For some more details, see an earlier blog, Infinitude: the Infinite and the Infinitesimal.