Is a line segment divisible without limit? Take a segment of finite length, call the first “point” on it zero and the last “point” one. The distance from the origin (zero) to any point on the segment is given by a (standard) real number of the form: zero-point (0.), followed by an infinite expansion (sequence) of digits (for instance, 0.2854618326580009276492651648206517848...). This is the mathematical version of Zeno’s axiom, inaccessible to the Greek who did not know the zero.
Accepting the “existence” of the real numbers is a strong hypothesis and, consequently, their study has become a key element in the search for a set of axioms or fundamental assumptions on which elementary number theory, and by extension, the whole of mathematics, might be firmly based. It is generally accepted that the set of 10 or so statements supporting most mathematical systems is the Zermelo-Fraenkel set theory. To these statements the nonstandard approach adds three additional axioms. They are based on the definition of a new nonstandard number: the infinitesimal. An infinitesimal nonstandard number is a new type of number: by definition, it is greater than zero but always less than any standard real number, however small.
These infinitesimals possess a thoroughly elusive character because they can never be captured through any possible measurement. The reason: measurements have always as result a standard real number. Furthermore, the difference between two standard real numbers can never be a nonstandard number, which is by definition always less than any standard number. Thus the interval between two nonstandard points on the line, or two nonstandard intervals of time, can never be measured, and so these intervals are forever beyond the range of observation. They exist only by axiomatic (Platonic) definition but can never become actual in Aristotle’s sense.
The nonstandard theory adds two more nonstandard numbers as axioms. The nonstandard unlimited number, which is the inverse of an infinitesimal number, is greater than any standard number but nevertheless smaller than infinity. The nonstandard unlimited numbers are thus very large, larger than any standard number, but always finite, that is, always less than the truly infinite numbers. The nonstandard mixed numbers are so to speak in between: around each standard number, on both sides of it on the line segment, a particular set of nonstandard numbers is, on the left, greater than any other standard number but less than this particular standard number; on the right side, it is smaller than any standard number greater than this particular number, but it is still greater than this number. In summary, between zero and infinity, a new infinity of nonstandard numbers has been added by axiom.
How does this nonstandard mathematics “solve” Zeno’s paradox? Achilles, as he gets closer and closer to the tortoise, will be traversing an infinite series of ever smaller space segments until eventually he will be at nonstandard infinitesimal distance from the tortoise. From this point on, his progress until he catches his prey escapes all possibility of measurement: all the final segments being nonstandard infinitesimal distances. In other words, what “really” happens when Achilles catches the tortoise can never be known, by definition, and so the case rests.
In the early twentieth century some philosophers have been particularly intrigued by the time continuum: if the instant is zero, when do we exist? The past is already gone, the future not yet here, and the present instant zero: when can we claim that we are? Martin Heidegger’s attempt to crack this problem in Being and Time, first published in 1927, may in a certain way be considered to be one more “solution” to Zeno ́s paradox. This philosopher suggests that humans are never authentically “being"; instead, from the very moment one is born, one is already dying, i.e., not-being. “The moment you are born you are old enough to die”. He furthermore claims that the only “time” that has a sense is the unknown period a human still has before he dies, that is, only the yet non-existent future is real. If one is asked: Will you die? The answer is: Yes, of course, but not yet.
Like Achilles: will he ever catch the tortoise? Yes, of course, but not yet. So, Heidegger states, there are two manner of being: the inauthentic and the authentic. In the former, which is where most of us choose to be, our allotted time-span is this not yet, this unknown future which allows us to escape from, to conceal, the unbearably displeasing fact: we are mortal. In this manner of being, we never actually die, we are always alive; death is “only” a potentiality. And we can say such strange things as: I have no time, don’t waste your time, etc. Whereas being authentically is equivalent, in a sense, to dying! That is, to fully accept human mortality. Put differently, inauthentic being is equivalent to live, to be, in the standard part of the time scale: there we can “measure” time with watches in standard numbers. Whereas authentic being is traversing our life-span in the nonstandard numeration, which is beyond measure, and where, in a sense, as soon as we are born we are already dying .
Recently a new, fascinating twist to this old conundrum appeared: Cris Calude’s Lexicon. Here is succinctly how it applies. Zeno’s paradox, and its possible “solutions”, must be somehow explicitly stated in a communicable language. This implies some linear sequence of symbols; the set of allowed symbols constituting the pertinent alphabet. Any finite sequence can be unambiguously coded in binary (or decimal) and thus corresponds exactly to some rational number. This paper, for example, corresponds to the rational number “w”. On the other hand, real numbers are infinite sequences of digits (in whatever chosen code or base). Question: Is there a real number that with certainty contains the word w (i.e.: exactly this paper)? Answer, Yes, and furthermore, there exists a real number that contains every possible “word”. That is, that contains everything that can be explicitly stated, coded, communicated. Here is how that number is constructed, in binary: simply add one after the other every possible binary sequence of 1,2,3,4,... bits:
0,1,00,01,10,11,000,001,011,111,110,100, 010,101, 0000,0001,...
all the way to infinity. By construction, absolutely everything that can be explicitly stated is represented, at least once, in this sequence.
Now it can be shown that this special real number not only contains, by construction, every possible finite linear sequence, say William Shakespeare’s complete works, but also that it contains every possible linear sequence infinitely many times! This is easily proved. Again, call some sequence, say this paper, w. Now construct these sequences:
w000 w0000 .................
all the way to infinity. Since by construction, each of them is already on our specially constructed binary real number, all of these “words” or binary sequences must also appear, at least once. But in each of them w appears, hence w appears infinitely many times. And this is the case for every possible w, QED.
It has been shown in 1998 by Calude and Zamfirescu that there exist real
numbers that present this remarkable property independent of the employed code or alphabet (binary, decimal, or, for instance, all the symbols on a computer keyboard). These are the Lexicons. Thus a Lexicon contains infinitely many times anything imaginable and not imaginable, everything ever written, or that will ever be written, any description of anything, of any phenomenon, real or imaginary, etc., etc.
So what is the relation of this surprising result with Zeno’s paradox? We humans are limited and mortal. We can only name, we can only put our fingers on, we can only actually exhibit, the rational (finite) numbers. But underlying all our mathematics, and Zeno’s paradox, are the real numbers. These, at least the immense majority of them, contain potentially everything, and infinitely many times. What we can actually exhibit is only a vanishing small subset of the underlying potentially possible number set. Now, we may call some “fact”corresponding to some rational numbera novelty. But this is just a Zeno-type illusion: everything is already contained (or expressed) infinitely many times in the infinite set of the reals almost all being Lexicons.
But what is a fact? Something that may have happened and that is amenable to a communicable description (measurement). For instance: the result of the sense- perception by which we observe Achilles catching the tortoise. Thus, something that corresponds to some finite sequence, to some rational number. But this “fact”, this sequence, this change, movement, kinêsis, is infinitely many times recorded in infinitely many real numbers ... if they exist, of course. Then it follows from the admittance of the existence of the real numbers, of Zeno’s infinite divisibility axiom of space and time, that there can never be any novelty. Everything is, always, and nothing ever becomes: Parmenides was right after all!
“The thing that has been, it is that which shall be; and that which is done is that which shall be done : and there is no new thing under the sun” ; so says the Ecclesiastes 1.9 (composed after 250 BC). Everything, the theory tells us, is right there under our nose, so to speak, potentially, but inaccessible. Whereas anything that actually happens, happened, or will happen, is an illusion, since it has been “there”, already, always. A final question remains, however: has anybody anytime laid his fingers on such a Lexicon? Surprisingly, yes, Greg Chaitin’s marvellous and mysterious real number View attachment 1703
is a Lexicon. But that is another story.
After twenty-five centuries Zeno’s paradox* is still with us: if we admit the existence of the real numbers we run into trouble; we deny it, and we find a different set of equally intractable problems since now mathematics, and thus physics describing “change”, become problematic. We are left with admiration for those early Greek philosophers, who unveiled the fundamental limits of human reason (and of mathematics ).
* Zeno's paradox see: https://iep.utm.edu/zeno-par/