Deceptive Reals

Slippery path - Deceptive Reals

The axiom of crowding, that is: between any two real numbers, there is yet another real number, holds true for rationals as well as irrationals placing them at par.  So does the concept of denseness; in other words. But, Cantor's concept of countability reduces rationals to the level of baby infinity and the concept of Lebesgue Measure reduces it to triviality. This descent to weightlessness from denseness for rationals is the beginning of shocks. It is hard to chew stuff, and painful to swallow. Let us consider the following: 

arrange the rationals in the set (0, 1) as:

\[ 1/2,\, 1/3,\, 2/3,\, 1/4,\, 3/4,\, 1/5,\, 2/5,\, 3/5,\, 4/5,\, 1/6,\, 5/6,\, 1/7,\, 2/7,\, \ldots \]

Let us enclose $1/2$ in the open interval $(1/2 - 1/2^3,\, 1/ 2 + 1/2^3)$, $1/3$ in the open interval $ (1/3-1/2^4,\, 1/3+1/2^4)$, $2/3$ in the open interval $( 2/3-1/2^5,\, 2/3+1/2^5)$, and so on. This countable family of open intervals containing all the rationals in $(0,\, 1)$ covers at most measure equal to $( 1/ 2^2 + 1/2^3 + 1/2^4+ \cdots = 1/ 2$. The irrationals outside these open intervals have measure at least $\frac{1}{2}$. Shall we know at least a few of these?!!  Too manyedness of the answer and representability of irrationals are probably the problem.  Let us look at the Cantor's set. We trisect the closed interval $[0,\, 1]$ and remove the middle open interval $(1/3, 2/3)$. We trisect the remaining two parts $[0, 1/3]$, $[2/3, 1]$; and remove their open middle parts $(1/9, 2/9)$ and $(7/9, 8/9)$. The process is continued infinitely. The left over is Cantor's set. The interesting thing is that it has many more points than just the terminal points. What it amounts to is that after the trisection, if the closed mid parts like $[1/3, 2/3]$, $[1/9, 2/9]$, $[7/9, 8/9]$ ... are removed even then an infinite number of points, uncountably infinite indeed, are left over after the removal process. Proofs apart, I don't fail to admit to my students that I am not able to view the way Cantor did. My eye lenses don't have the number required to SEE THROUGH the real line as Cantor's and the great master's did. The proof without vision becomes mechanical verification.

In the ternary representation points of the Cantor set are given by $c_1,\, c_2$,   $c_3,\, \ldots$, where $c_i$'s being $0$ or $2$. The geometric meaning of  $c_i$ being $0$ or $2$ is that in the $i$-th trisection the selected point is on the left-side one-third or the right-side one-third. It may be convenient therefore to replace $0$ by $L$ and $2$ by $R$. Then a point $.2202200 \ldots$ of the Cantor's set could be represented by the sequence $\{RRLRRLL.....\}$. The point seems to be calling out to the rescuer, as if from the debris of demolition of removals, yelling out its coordinates: first on the right one-third (after first trisection) and again on the right one-third (after second trisection), and then on the left one-third (after third trisection), then right, again right, left now, and again left and so on. All $L$'s or all $R$'s or with a recurring $L$'s or with a recurring $R$'s after a fixed stage will represent rational points of the Cantor set. The real crowd of the set, that of the irrationals in the set can be guessed (nicely or badly) as limits of non-monotonic convergent sequences. Don't ask, what limit. The infinite expansion is the limit. That is not the end of surprises. Each of these (intuitively non-existent points) is crowded around by the points of the set, none an isolated point.

I am reminded of the answer pages of Chapter on Convergence of series with  different set of examples based on convergence tests. The `Answers' read Convergence, Divergence, Divergence, Convergence, for different questions. In case of convergent series, it is not asked, and luckily so, what it converges to. Quite often, it would be too hard to answer.  Slowly and surely, you would be led to: a real number is a Cauchy sequence (convergent sequence) of rational numbers. The existence of the limit follows from the axiom of `no gap in reals', called Completeness of reals.

 My frustration of not being able to place some real numbers on the number line was attempted to be diluted by my friend, Prof. G. Das of Utkal University. He explained: he had a great grandfather, whose name is not known to him, yet he existed, a very definite person. Our very being proves a definite existence of our (each of ours) great grandfather of exact name, address, etc. Our inability to tell his name or specification cannot dilute even minimally the surety of his definite exact existence. So a convergence series represents a definite real number: we may or may not be able to describe it otherwise.

Lebesgue has proved that all real numbers cannot be represented. The  difficulty in the representation of the real numbers is one part of  the story. Ascertaining the nature of the real numbers, whether it is transcendental or not is no easy. The Hilbert's conjecture of transcendence of $2^{\sqrt{2}}$ was proved only in $1934$. The proofs of transcendence of $\pi$, $e$; and others; and characterization of transcendental numbers, the work of Hermite,  Hilbert, Lindemann, Gelfand, Schneider, Liouville, Siegel and Roth are only $50$ to $150$years old. And the era also witnessed the rejection of their work and that of Cantor's particularly by Kronecker.

Is it not amusing that algebraic numbers are countably few; while transcendentals we are not comfortable with. It is the vision of great men like Lebesgue and Cantor that makes all the difference. Lebesgue observed that irrationals $\sqrt{2},\, \sqrt{3},\, \sqrt{5}, \, \ldots$ belong to different castes in the sense that their difference is not rational; and that such castes are uncountable in number. By picking one member from each caste, the set obtained he could observe is unmanageable to his definition of `measure'.

Denseness means crowding, that however does not ensure some weight (in the sense of measure). Nowhere-dense means hollow in the sense of having holes all through. A countable union of nowhere dense sets (called a set of the first category) may or may not have holes but it has a dense set of gaps. A decomposition of reals is obtained; one part (a thin sandwiching of rationals) having measure zero ( hollow weight wise) and the other part being of the first category (hollow having gaps everywhere) So the reals is hollow plus hollow, while we started having a decomposition of reals in parts crowded and its complement also crowded ( in the sense of denseness). Amazing discovery resulting from a fine surgery!! The progress in surgery goes on.