Infinitude: the Infinite and the Infinitesimal

Prologue. Stated simply, we want to claim that corresponding to every infinitesimal on the real line, there is a transfinite stretch representing the space beyond the finite space.

Infinitude. Infinity has been always a difficult and intriguing subject for mathematicians, but they find themselves embedded in a universe which is infinite and hence forced to deal with it. Here, we want to talk about supernatural numbers which, hopefully, would make visualization of the space beyond the stars easier. We claim that corresponding to every real number, there is a supernatural number representing the infinite space.

Real Numbers. Note that one way to uniquely represent a number within the unit interval (0,1] is by an infinite binary string of the form .xxxxx... where the x's after the initial binary point represent either 0 or 1. As examples we have for the rational numbers 1, 2/3, 3/4, the unique representations .111111...., .101010...., .101111.... respectively.

Supernatural Numbers. The question we want to investigate here is that whether it is possible to give some meaning to these strings if we flip them around the binary point. Fortunately for us, computer science tell us that the appropriate meaning for ....111111. is -1 (minus one). The daring wrong argument used by the computer engineer to reach the right result is that the sequence here is the power series expansion of 1/(1-x), where x=2. Ignoring these arguments, we will accept the fact that the flipped sequences can have meaning and suggest that the flipped sequences ...xxxxxx.xxx...xxx corresponding to the real numbers xxx...xxx.xxxxx... should be called supernatural numbers.

Transfinite Stretches. Just as there are infinitesimals attached with the real numbers, we can claim that there are transfinite stretches attached with supernatural numbers. A little investigation shows that infinitesimals and transfinite stretches can be considered as duals of each other, reminiscent of the point at infinity of complex analysis. Also, it should be clear that corresponding to every transcendental number there is a supernatural number. Click here for some details. If your computer can download files fast, click here.

Epilogue. Tolerating some abuse of language, we can state that an infinitesimal is what we get when we compress and fuse a set of points of cardinality 2^\aleph_0. Similarly, we get a transfinite stretch when we keep the points of 2^\aleph_0 a finite distance apart from each other. Because of the duality between the infinitesimal and the transfinite stretch, it should be clear we need to study only one of them. In short, studying the reachable fused infinitesimal is as good as studying the unreachable transfinite stretch.

Proving Incompleteness Theorems Formally

Prologue. We are told that we should not trust our natural language arguments when it comes to mathematics, and we should follow the formal derivation rules of predicate calculus, if we want to prove any thing rigorously. Yet, when it comes to the most serious matter of incompleteness of Elementary Arithmetic (EA) of Goedel, what we do is to talk in plain English and conclude for certain that the theory cannot be complete.

Adding three derivation rules. The issue we want to consider is whether it is possible to rectify this flaw and make all our logical arguments totally formal. A little investigation suggests that this can be done by adding three more derivation rules to Predicate Calculus. For details of these derivation rules, click here. Of course, adding derivation rules in a theory is a risky enterprise, and to make it worse, Goedel tells us that there is no logical way to prove the consistency of a significant axiomatic theory. The time tested method is to wait and see whether a given theory eventually turns out to be inconsistent. Recall, Bertrand Russell's devastating letter to Gottlob Frege about set theory.

Introversion and profundity. If what we are suggesting is indeed a solution, every formula in a theory would fall in one of four categories: theorem, falsehood, introversion, profundity. To give a rough idea of the classification, an example of an introversion is the Consistency of EA, and an example of a profundity is the Continuum Hypothesis of Zermelo-Fraenkel theory. Introversions arise when we attempt to use the theory to investigate itself, and profundities are the profound concepts which we are not sure to choose or not to choose as axioms.

Elevating Goedel's Incompleteness Theorem. It is worthwhile to note that what Goedel has shown is that there are introversions in EA. A more significant result would be to show that there will always be profundities in any significant theory. Whether this is what Goedel really wanted to show is a moot question. Shankara, a philosopher of eighth century, in what he calls "maya" and what I call Shankara's Incomprehensibility Thesis, claims that humanity will have profound questions facing them perennially and our understanding of nature will always be imperfect.

Inescapable Question. By allowing four categories of statements, we are in essence giving freedom of expression to a theory. Throughout civilization humans have indulged in introversion, mostly with disastrous results. Wise men have always been preoccupied with profundities and have enjoyed the consequent agony and ecstasy. A conspicuous example is provided by Mahatma Gandhi's autobiography, The Story of my Experiments with Truth. What if, we decide to whole-heartedly accept and live contented with Goedel's incompleteness theorem and Shankara's incomprehensibility thesis. The fact is that we will still have no peace of mind when the following question about religion, in the best sense of the word, crops up. It is as though God wants us to be permanently on tenterhooks when it comes to important issues.
Humanity's Inescapable Question: Is religion, an introversion or a profundity?

Epilogue. Those who may consider this as a frivolous question should look into some facts of history. It is easy to see that, if an introversion is accepted as an axiom in a theory, the theory will be destroyed by contradictions. From the gruesome violence prevalent in the world today in the name of religion, it is tempting to classify religion as an introversion. But on the other hand Mahatma Gandhi, deeply religious, spent all his life experimenting with Truth, which he considered the same as Reality or God. His autobiography, makes it clear that he considers religion as a profundity.

An Axiom to Derive Continuum Hypothesis

Prerequisite. No more than a passing acquaintance with the cardinals of Cantor is required to read what follows. However, the notations here are a little clumsy since Internet still lacks math symbols.

Axiom of Combinatorial Sets. This blog entry is about the most fascinating problem I know in metamathematics, the Continuum Hypothesis (CH) of Cantor. CH states as a guess that \aleph_1, the cardinal next to \aleph_0, is equal to 2^\aleph_0. It is known that neither CH, nor its negation, can be derived from Zermelo-Fraenkel theory (ZF) and hence it is necessary to introduce an axiom into ZF, if we want to derive CH. The suggestion we want to make here is that the The Axiom of Combinatorial Sets (ACS) can serve as an appropriate axiom for the purpose. Here, by combinatorial set, \aleph_0 \choose \aleph_0, is meant the set of all subsets of \aleph_0 of cardinality \aleph_0. Click here for details.

Can we hope for a better axiom? It is easy to see that ACS can accomplish the job, but the real issue is whether we can have any hope of getting a simpler or more elegant axiom for the same purpose. In other words, is it possible to generate an infinite subset of 2^\aleph_0 substantially different from the combinatorial set which has cardinality less than or equal to that of 2^\aleph_0? Or, is there a simpler way to generate a superset of \aleph_0 from \aleph_0, other than the combinatorial set, whose cardinality is less than or equal to that of 2^\aleph_0?

A fact that may be of some interest here is that the value of, \aleph_0 \choose k, is respectively 1, \aleph_0, 2^\aleph_0, for k=0, finite, \aleph_0.

Internet Principia Mathematica. For a Principia Mathematica type proof of the equivalence of combinatorial set and power set of \aleph_0, see Metamath Proof Explorer (MPE). In my opinion, this MPE is indeed a spectacular achievement of mathematical logic since Principia Mathematica. For more of my comments, see my review.

For some: proof is music to their ears. Did you ever feel that there is something musical about mathematical proofs? If you did, you are absolutely right. To hear a musical version of the proof mentioned above click here (courtesy: Norm Megill).