Colossal Gardner, ch. 28

The section on Infinity ends with Surreal Numbers. Surreal Numbers were developed by John Conway (The Game of Life), and named by Donald Knuth.

(All rights belong to their owners. Images used here for review purposes only. John “Horned” Conway.)

Surreal numbers are created by following set rules, and can define all integers, all integral fractions, all irrationals, all of Cantor’s transfinite numbers, the reciprocal of Cantor’s numbers, and infinite classes of “weird numbers”, like:
(w + 1)^(1/3) – PI/w
Where w is omega, Cantor’s first infinite ordinal.

We start with two sets, a left set L, and a right set R. No member of L is equal to or greater than any member of R. Then, there is a number {L|R} that is the “simplest number” (as Conway defines it) in between. With the empty sets for L and R, { | } gives us the definition of 0. Everything else follows from plugging back in newly created numbers into the L and R sets. {0|0} is not a number, but {0| } defines 1, and { |0} defines -1. More info can be found in the wiki article.

The rest of the chapter covers Conway’s extension of the surreal number process to create what he calls “games.” A Conway “game” is constructed in a similar, but more general way. If L and R are any two sets of games, there is a game {L|R}. Some games correspond to numbers, others don’t. They all, however, are built on the empty set. More details can be found in the wiki article on Combinatorial Game Theory.

(Cutcake puzzle.)

The above illustration is for the game cutcake. You have a 4×7 sheet cake, in Conway’s notation of value 0. This means that the second player wins regardless of who goes first. Last person to be able to make a cut along the lines wins. Left’s move is to break a piece of the cake into 2 parts along any horizontal lattice line; Right’s is to break a piece along any vertical line. It looks as if the vertical breaker, who has twice as many options as their opponent, would have the advantage, but not if they go first. Assume that the vertical breaker goes first and breaks along the line indicated by the arrow, what is player 2’s response to win?




The Two Sides of Yes

(All rights belong to their owners.Images used for review purposes only.)

Rick Wakeman: The Two Sides of Yes
I asked for a Rick Wakeman / Yes CD for my birthday, and this is the one I got. It’s not really Yes music, though. Instead, Rick wrote all new arrangements for 13 songs, and rerecorded them as instrumental tracks, which he performed more or less solo (some sound like multi-track recordings). Half are just piano pieces and the other half have multiple layers of synths on them. If you’re familiar with Yes’ discography, then you already know every song here. If you are a Wakeman fan, the reason for getting this 2-CD set is to listen to his versions of the songs, now that he’s had a lot of practice playing them. To me, they sound too much like elevator music. They’re fine, but lack the genius synth expressions and energy of the original songs. They do work well as background sounds when you’re busy concentrating on something else.

Hachette 3D Wood Puzzles Series Update

Back in March, I wrote a blog entry on the latest 3D wood puzzles series from Hachette. At the time, there were to be 18 puzzles in the series, coming out one every two weeks. I didn’t have much interest in the first few volumes, and I only bought numbers one and nine. In June, I’d written that the last few volumes looked the most interesting and that I’d start buying them as they came out, beginning with the one in mid-August. So, what happened?

Well, it turns out that these types of serialized kits are not all that popular with the bookstore operators. It’s one thing if the series is for Mountains of Japan, or Important Figures in Japan, because these things are just thin 20-page magazines. But, the Build Your Own 3D Printer, and Build Astro Boy kits are 52+ volume boxes that can be up to 2″ thick each. Between DeAgostini and Hachette, there can be up to 20 kits ongoing at any given month, and the stores don’t want to dedicate half their shelves to these kits by having 2-3 copies of every volume in inventory in any one store. Initially, though, the three stores near me, Kinokuniya, Maruzen and Junkudo, did at least carry extra copies of the newest volume with each printing, and I could hold the 3D puzzles boxes in my hands to decide if I wanted to buy one or not.

But right around August, all three stores stopped carrying surplus copies of most of the ongoing series, including the 3D wood puzzles, and only kept the first 1-3 volumes of the newest series as they came out (like with the Robie second series). What this means is that even if any of the stores here do get in one of the 3D puzzle volumes, it’s only for holding for customers that place pre-orders. Otherwise, the only way to get any of the latest puzzles is either to order them off Amazon (and not know how big the puzzle is until you get it in your hands) at $20 USD a shot (including shipping and tax), or preorder the entire 18-box series through one of the stores, and get every single one of the kits, including the volumes that I don’t want for a total of something around $360.

Many of the puzzles are actually kind of small, and are overpriced at the $17 USD cover price (plus tax), and the only reason I had any interest in the ones I did was that they looked challenging in the photos in the volume 1 booklet, and not because I was really committed to obtaining things just simply because they were wooden 3D puzzles. Anyway, the point is that I haven’t written about this series in a long time because I haven’t seen any of the newer kits in the stores here in Kagoshima in the last 3 months. And now you know.

Colossal Gardner, ch. 27

I’m not sure I would consider Fractal Music to be part of Infinity, but that’s how Martin chose to categorize it in this book. On the other hand, the concept mixes two of my favorite things – fractals and synthesizers – so I’m willing to be forgiving.

Gardner starts out by discussing music’s relationship to the other arts within Plato’s thought that all art imitates nature. This may be true for paintings and written fiction, but in what way does someone singing opera imitate a bird call or burbling water? He then introduces a discovery by Minnesotan physicist Richard F. Voss, who joined IBM’s Watson Research Center. This discovery involves spectral density, AKA the power spectrum, and autocorrelation. Mandelbrot, who developed the field of fractals and also worked at Watson, suggested an easy way to understand these concepts. Say you record a sound and play it back at different speeds – you’d expect something like a violin to sound very different when played slower. But, there is a class of sounds where, if you change the frequency, all you have to do is adjust the volume to make it sound exactly the same again. He called these sounds “scaling noises.”

(All rights belong to their owners. Images used here for review purposes only.)

The simplest of the scaling noises is “white noise,” or Johnson noise, which is a random collection of all frequencies. It’s what you get when you listen to a detuned radio. The correlation, which measures how fluctuations at any given moment are related to previous fluctuations, for white noise is 0, except right at the beginning, when it is 1. You can use spinners (above) for generating a “white tune,” where there’s no correlation between any two notes. The note names are for a piano keyboard, and you can pick “fa” for middle C. The sizes of the segments for the notes can be completely arbitrary. With the above spinner, the notes have been weighted to give precedence to those around fa. A second spinner can be used to assign the lengths of the notes as 1/8th beats, 1/4 beats, 1/2 beats or a full beat. Alternatively, you can use dice, random number tables, or the digits of an irrational number (such as PI or e).

The next step is to use Brownian noise, which mimics the random thermal movements of small particles in a liquid. To create Brownian music, add one more spinner divided into segments marked 0, +1, -1, +2 and -2. So, you start by spinning the Do, Re, Me spinner for the first note, then use the offset spinner to step to the subsequent notes based on the first one (up one or two notes, or down one or two notes each time). The duration of the notes could also be controlled by the offset spinner to make it Brownian. You may note that Brownian music is going to have a higher correlation factor.

If you want to find a halfway point between White and Brownian music, you can go stochastic. In this case, you set rules for changing to a note based on the previous 3 or 4 notes. One way to do this is to analyze Bach’s music and look at how often a particular note follows another triplet. If certain transitions never happen in Bach’s music, we can introduce rejection rules to prevent our using them. The result would be Bach-like at the beginning, but eventually it would still become random.

So, what Voss did was to find a compromise between White and Brownian music by selecting a scaling noise halfway between. This is called 1/f in spectral terminology. Music based on 1/f is mildly correlated, and it repeats over both short and long runs. 1/f noise is sometimes referred to as flicker noise, and Mandelbrot found that it’s very widespread in nature. The annual flood levels of the Nile is a 1/f fluctuation. From here, Gardner goes into a brief excursion into fractals, and how they relate the Peano curve to the Koch snowflake, and Bill Gosper’s flowsnake. Other 1/f fluctuations include variations in sunspots, the wobble of the Earth’s axis, membrane currents in the nervous systems of animals, and the uncertainties in time measured by an atomic clock. So yeah, we’re now getting to the idea of music imitating nature, if we use a 1/f spectrum. Most classical, jazz, and rock is 1/f.

(Comparing white, 1/f and Brownian noise.)

Voss devised a way to simulate a computer program for generating 1/f music. Pick 8 notes and a scale of 16 tones, and write 0-7 on a chart as below. Write the numbers out in binary form and label the columns “Red,” “Blue,” and “Green.” Take three 6-sided dice, one of each color. Roll all three to get a number from 3 to 18 for your starting tone. Now, look at the chart, and see that when you go from 000 to 001, only the bit for the red die changes. So, pick up the red die, leaving the blue and green as-is. Roll the red die, and the total of the 3 dice gives you your next tone. Going from 001 to 010, both the red and green bits change, so roll those dice, and continue until you have your full song, 8 notes at a time. Gardner comments that this process is not exactly 1/f, but it’s close enough for jazz. You can also use 4 dice for a range of 21 tones. Note lengths are generally fixed, but that’s tweakable with a similar chart and dice system. Gardner adds that 1/f music is fractal, in that if you compare a short note string to a much longer passage, they are similar. “The tune never forgets where it has been.” He also comments that a melody can be changed to something new by playing it backwards, to turn it upside-down, or both, and they still retain their 1/f spectral densities. He specifically mentions Mozart’s Mirror Canon as an example of the style (which was not actually written by Mozart).

(Dice chart for making music.)

Now, the synthesizer part that I mentioned above is not directly described in this chapter, but there is a lead-in with “mountain music.” The idea is that mountain ranges look fractal or random, so you go out, photograph a mountain range, and then use the skyline to translate the heights into notes. Villa Lobos did this using the mountain skylines around Rio de Janeiro, and Sergei Prokofiev did the same for Sergei Eisenstein’s film “Alexander Nevsky” (1938). Terahiko Terada (1878-1935), a Japanese physicist, discussed doing something similar, imprinting skylines or people’s profiles on records to create new sounds. And, this is where things get fun with synths. If you digitize said skyline, or someone’s profile, and normalize that for a 5V control signal, you could use it for sound envelopes in a synth, either replacing the ADSR, or looping it as an audio waveform, or feeding it into a VCO to control pitch or filtering. The possibilities are endless for experimental music – all you need is a video camera, facial recognition software, and a breakout board… Anyway, Gardner returns to mountain music in chapter 47.

Sidenote: As I was writing this blog entry, I happened by the Scientific American magazine website, and found an article that had just been posted a few minutes earlier about Betty Shannon, wife of computer pioneer Claude Shannon. In the article, the author mentioned that one of Betty’s technical papers was on “Composing Music by a Stochastic Process.”


New Video. Ok. Go. Obsess.

Direct youtube link

I love these guys. I love these videos.

Direct youtube link

Supertask Wednesday

Supertask answer:

This is a false proof that Cantor was mistaken about Aleph-Null and C being different orders of infinity.

Integers – Decimal Fractions
1    .1
2    .2
3    .3
10    .01
11    .11
12    .21
100    .001
101    .101

The left side is an endless list of integers in order, the right side matches them with a number that reverses the order of the digits and puts a decimal point in front of them. Since the list on the left goes to infinity, it should eventually include every possible combination of digits. If so, then the list on the right will eventually contain every real number between 0 and 1. The real numbers form a set of size C. Since this has been put in one-to-one correspondence with the integers, an Aleph-Null set, the two must be equivalent. What’s the flaw?

No matter how long the list of integers is, no number with an Aleph-Null quantity of digits will ever appear on each side. Therefore, no irrational decimal fraction will appear on the right. The list on the right is simply a subset of the integral fractions between 0 and 1. To get 1/3, you need 33333… with an Aleph-Null number of digits. As Gardner says, this just proves that the counting numbers can be matched with a subset of integral fractions.

My note: Infinities are tricky to play with, but they can be fun when you have proper supervision from a trained, licensed instructor. Buy a box today at the Dollar Store, and see what I mean!

Colossal Gardner, ch. 26

Supertasks starts out with a recap of Cantor’s Aleph-Null concept, focusing on the properties of the empty set. The basic idea being that in set theory, the empty set is a subset of every other set, including itself (to get the number of members of a set, you take 2^n, and 2^0=1). Then we get the question, “The n elements of any finite set obviously cannot be put into one-to-one correspondence with its subsets because there are always more than n subsets. Is this also true of infinite sets?” The answer is “yes” and the proof is a reductio ad absurdum. “Assume that all elements of N, a set with any number of members, finite or infinite, are matched one-to-one with all of N’s subsets. Each matching defines a coloring of the elements:

1. An element is paired with a subset that includes that element. Let us call all such pairings blue.
2. An element is paired with a subset that does not include that element. We call all such elements red.”

You might be able to see where this is going. All of the red elements are a subset of the initial set N. Can we attach this subset to a blue element? No, because the blue elements are already matched. Can we attach it to a red element? No, because the red element would then be included in its own subset and would already be colored blue. Conclusion? No set, even if it’s infinite, can be put into one-to-one correspondence with its subsets. “If n is a transfinite number, then 2^n – by definition it is the number of subsets of n – must be a higher order of infinity than n.” (“Transfinite” was coined by Cantor to refer to infinite numbers that are not “absolutely infinite”, and are in the Aleph Null category.)

Gardner gives a few more examples of infinity, such as Cantor’s matching of all the points on a square to all the points of a line segment, which can easily be extended to a cube. He then asks the question, “Is there a set in mathematics that corresponds to 2^C?” The answer is yes – It’s the set of all real functions of x, including the set of all real one-valued functions. Geometrically, it’s the set of all curves that can be drawn on a plane, or finite portion of a plane (even if it’s the size of a postage stamp).

Finally, we come to Supertasks. This word was coined by James F. Thomson, creator of Thomson’s lamp. The idea of supertasks is to perform a “countably infinite sequence of operations that occur sequentially within a finite interval of time.” It’s a variation on the Zeno paradoxes. In the case of the lamp, start by pressing a switch to turn on a lamp for half a minute. Turn the lamp off for a quarter of a minute. Then, on for an 1/8th, off for a 1/16th, etc. The sum of this halving series is 1 (1/2 + 1/4 + 1/8 + 1/16 +…) At the end of one minute you’ll have pushed the switch Aleph-Null times. Will the lamp be on or off?

With Zeno’s constant speed runner, who has to cover half the distance first, then half the remaining distance, then half of that, at the end of one minute he does reach his goal. Can we say the lamp made its last move at the end of one minute? No, because the lamp must either be on or off, and it’s the same as saying that there is a last integer that is either even or odd. Another supertask involves an infinity machine that calculates and prints the value of PI. Each digit is printed in half the time taken for the preceding one. Plus, the digits are printed on an idealized tape of finite length but with the digits having half the width of the preceding one. Because both the time and width series converge to the same limit, in theory you could expect the machine to print out the full value of PI on a piece of tape. Except that there is no final digit for PI, and the task is self-contradictory.

Gardner also mentions Max Black’s marble machines. You have two machines, the first transfers a marble from tray A to tray B in one minute, and rests one minute as the second machine transfers the marble back to tray A. Then the times get cut in half over and over until you just get a gray blur. After 4 minutes, each machine has made Aleph-Null transfers. Which tray is the marble in?

At the moment, the difference between these supertasks and Zeno is that his runner moves continuously, but the supertasks involve discrete steps. Adolph Grunbaum suggested that Zeno’s runner could have a staccato motion of Aleph-Null steps. When he moves, he’s twice as fast as his analog version, but the overall average speed is the same. This would turn Zeno’s runner into a supertask and we could ask “which leg crosses the finish line first?”

Then, we get the puzzle. This is a false proof that Cantor was mistaken about Aleph-Null and C being different orders of infinity.

Integers – Decimal Fractions
1    .1
2    .2
3    .3
10    .01
11    .11
12    .12
100    .001
101    .101

The left side is an endless list of integers in order, the right side matches them with a number that reverses the order of the digits and puts a decimal point in front of them. Since the list on the left goes to infinity, it should eventually include every possible combination of digits. If so, then the list on the right will eventually contain every real number between 0 and 1. The real numbers form a set of size C. Since this has been put in one-to-one correspondence with the integers, an Aleph-Null set, the two must be equivalent. What’s the flaw?


Modern Magic Comments

I received Professor Lewis Hoffmann’s book of Modern Magic as a birthday gift a year ago, and I’ve been slowly working my way through it ever since. Hoffmann’s real name was Angelo John Lewis, and he was born in 1839, and died in 1919. He was a lawyer by profession, and took on a stage name because he felt that having a reputation for lying to people as a magician might affect his work in the courts (not an issue this days). He was one of the first magicians to sit down and really describe the operations of various magic tricks in English in his first book, Modern Magic, which was released in 1876, and then reprinted in 1877. You can find a list of his other books on magic at the Genii magazine site.

Modern Magic is a great collection of many of the illusions commonly presented up to the 1870’s, with illustrations of prepared objects, and hand movements for various passes. A lot of these tricks can still be seen today, and you don’t have to worry about the illustrations being too muddy, because you can buy updated versions of the mechanical tricks in magic shops. The patter Hoffmann suggests to go with the tricks is out of date and can be ignored, but his suggestions for grouping illusions together so as to make doing them easier or more impressive is still useful. His comments on the importance of the psychological element of magic are also still relevant. The main take away is that projecting the image of a magician is at least as important as actually being able to do the tricks without looking at your hands as you do them.

I’ve got a few comments I want to add here. Hoffmann mentions several other magicians by name, including Edme-Gilles Guyot (1706-1786), as being an expert on the cups and balls, with his book Mathematical Recreations (1769). Ozanam (1640-1718), as having written an earlier book on magic of the same title in 1694. Robert-Houdin (1805-1871) for his developmental work in using electricity and magnetism in magic, and especially for the Light and Heavy Chest. Herrmann (1844-1896), for several illusions. And, F. A. Gandon (d’ Antoine Francois Gandon) for his 1849 treatise on how Houdin’s Second Sight mentalist trick was implemented.

After finishing Modern Magic, I started searching out the books he mentions, as well as anything else I could find. It’s difficult locating public domain copies of either version of Mathematical Recreations, or Gandon’s book. All three were originally written in French, and only one was translated commercially into English. If you know French, then it’s ok.

Along the way, I made multiple stops at the Gutenberg Project. Initially, I wanted to see what they had on Houdin, but that pulled up Harry Houdini’s Unmasking Robert-Houdin (1906). Harry starts out saying that he’d been a great fan of Robert’s as a child, even taking the stage name “Houdin + i” to mean “like Houdin.” He’d wanted to write a tribute book about Robert’s life and magic, but after doing a fair amount research, came to realize that Houdin himself was something of a big huckster, claiming the invention of many illusions that had been created long before him by other magicians. What I found most interesting was that the tricks given in the wiki article on Robert (Second Sight, Ethereal Suspension and the Marvelous Orange Tree) are all the ones that Harry debunks. Granted, Robert made them famous in his act, but they weren’t original to him. Which is made even more interesting when you take into account the fact that his mechanic, Le Grand, pirated Robert’s own designs and was arrested for making duplicates that were sold to others, including Herrmann.

Another point that attracted my attention was Hoffmann’s mention of a Japanese troupe that had toured London, and the explanations of the Japanese Obedient Ball and Paper Butterfly Tricks. I’d never really heard much about Japanese magicians or jugglers touring outside of Japan prior to the 1890’s, and I was curious as to who they were. Hoffmann didn’t give specifics, but I knew that it had to have happened prior to the 1876 printing of Modern Magic. I couldn’t turn up anything immediately useful in a Goggle search, so I instead looked for “Japanese magicians.” This led me to the Japanese wiki article, which lists about 150 names, most of whom were born in the 1900’s. So, I took a brute force approach and clicked on every single name, eventually uncovering the one person born early enough, Namigorou Sumidagawa (1830-?) to qualify for participating in the troupe.

Following up on additional leads, it seems that an American gymnast of the name Richard Risley Carlisle, and billing himself as Professor Risley, left the circus he’d been with to move to Japan in 1864 and became the first western professional acrobat there. He started selling milk and ice to make money, then went back to performing a couple years later. He became the manager of the Imperial Japanese Troupe, which had several trained professional Japanese acrobats and magicians, one of whom was Namigorou. Carlisle took the Troupe to the Paris Exposition in 1866, and then San Francisco, New York, London and back to Paris in 1867. After touring Europe and the rest of the world for 2 years, they returned to Japan in 1869. According to the Japanese wiki article, Namigorou presented the Paper Butterfly Trick to the San Francisco Academy of Music in 1867. Fred Schodt, one of America’s premiere writers on Japanese culture, published in 2012 the book Professor Risley and the Imperial Japanese Troupe, which has a more-detailed description of the troupe, plus poster reprints and illustrations. I also found a historical blog entry for their appearance in Birmingham, England, in 1867. I assume then that this is what Hoffmann was referring to regarding the appearance of the Paper Butterfly and Obedient Ball tricks in London.

Next, electricity. The principles behind electricity, and the development of electromagnets were well-established before 1860, and Maxwell published his Treatise on Electricity and Magnetism in 1873. Even so, the use of electromagnets was already prevalent in stage magic to the point where they were integral parts of seances and mindreading acts (solenoids were used to tap the underside of tables for “contacting the spirits”), and apparently-wireless floating electric bells and buzzers were also common. Hoffmann gives descriptions for how to make sulfuric acid/zinc batteries, wrapped copper coils and transformers. He highly recommends the reader to get a copy of Intensity coils, how made and how used, by Dyer. Most of the copies I can find on the net were originally published around 1889, which would have been after Modern Magic came out. I assume Dyer may have reprinted this tract several times over the years. In the section on Robert-Houdin’s Light and Heavy Chest, Hoffmann goes into great detail on its history. Essentially, the chest has an iron plate in the bottom, and is brought out on a heavy cart. The cart has an electromagnet concealed within it, and when turned on, even the strongest person can’t lift the chest off the cart.

I just find the entire concept of electricity used in this way so long ago absolutely fascinating. Houdin also had wires from a step-up transformer brought up to the handle of the chest, and an assistant backstage would open and close the battery contacts quickly to mimic an AC input voltage to the transformer, shocking the victim with high-voltage, low-amperage current so that their muscles would seize up and they couldn’t let go of the handle until Houdin magically “gave them permission” to do so.

Finally, japanning. Japanese lacquer work was introduced to Europe in the 1600’s. The lacquer was in high demand, but could only be produced from the sap of the toxicodendron vernicifluum tree, which didn’t grow outside of Asia. The word “japanning” originated in the 1600’s to refer to a European-made heavy black lacquer that was more like simple enamel paint. Hoffmann describes many of the stage tricks as being made of tin and “japanned” as necessary to hide the appearance of the object’s interior. I’d seen this word used before, many years ago, and it wasn’t until now that I finally looked it up to find out what it meant.

Anyway, Modern Magic is a good reference book if you’re into these kinds of things. Recommended.

Wednesday Aleph-Null Answers

Using J. B. Rhine’s ESP card symbols, are there any symbols that can be drawn an Aleph-One number of times on a sheet of paper, assuming lines of no thickness, with no overlapping or intersections of the lines? (They don’t have to be the same size, but they must have similar shapes.)

Only the cross symbol cannot be replicated an Aleph-One number of times without overlap.

Colossal Gardner, ch. 25

Aleph-Null and Aleph-One
Martin starts out by mentioning a question raised by Paul J. Cohen (29) in 1963. Is there an order of infinity higher than the number of integers but lower than the number of points on a line? To answer this, Martin goes into an explanation of Georg Cantor’s discovery of the infinity of higher levels of infinity. The base level, the number of integers, was called Aleph Null. We include in this set any selection of numbers that can be counted (not that anyone is actually going to try counting them), such as all prime numbers, all natural numbers, all even, odd, positive or negative numbers, all rational fractions, and so on. Yes, some of these sets can include values that get much bigger, much faster than some of the other sets, but the important distinction is that they can be counted.

Example with prime numbers:
1 (1), 2 (2), 3 (3), 5 (4), 7 (5), 11 (6), 13 (7), 17 (8), 19 (9), etc.

Aleph-Null numbers are said to be “countable” or “denumerable”. This brings us to a paradox, in that with infinite numbers, “they can be put into correspondence with a subset of themselves,” or with their “subsets.” One example was developed by logician Charles Sanders Pierce. Start with the fractions 0/1 and 1/0 (ignoring the divide by zero issue for the moment). Sum the two numerators and then the two denominators to get the new fraction 1/1 and place it between the original pair. Repeat this process with each pair of adjacent fractions to get the new fractions that go between them.

0/1 . 1/0
0/1 . 1/1 . 1/0
0/1 . 1/2 . 1/1 . 2/1 . 1/0
0/1 . 1/3 . 1/2 . 2/3 . 1/1 . 3/2 . 2/1 . 3/1 . 1/0

As Gardner mentions, as this series continues, every rational number will appear once and only once. Reducible fractions like 10/20 never show up. If you want to count the fractions at any given step, you can take them in their order of appearance. Any two fractions equally distant from the center, 1/1, are reciprocals of each other. And, any two adjacent pairs ab and c/d have the following equalities: bc – ad = 1; c/d – a/b = 1/bd. Also, as Gardner says, this series is closely related to Farey numbers. Pierce’s process creates an infinite number of countable fractions, as a subset of all numbers.

(All rights belong to their owners. Images used here for review purposes only. Example of creating subsets of a set of three elements.)

The next step is to show that there’s another set with a higher order of infinity than Aleph-Null. The above figure uses three objects – a watch, a key and a ring. Below each, you lay out a row of 3 cards, face down (black) or face up (white) to represent subsets. A white card shows that the above object is in the subset, while black indicates that it is not. The first subset consists of the original set. The next three rows are subsets of 2 objects each, the following three are subsets of one object each, and the bottom row is the empty, or null, subset that doesn’t have any objects. “For any set of n elements, the number of subsets is 2^n.”

Apply this procedure to an Aleph-Null set, which is countable, but infinite. The above figure attempts to show that this new set is countable, with each row represented by the white and black cards extending to infinity. They can be listed in any order, and numbered 1, 2, 3, etc. “If we continue forming such rows, will the list eventually catch all the subsets? No – because there is an infinite number of ways to produce a subset that cannot be on the list.” We can do this by taking every card on the diagonal line and flip it white-for-black, making a new subset that hadn’t existed before, even though the original set was infinite. This set is not countable, showing that the assumption was wrong. So, “the set of all subsets of an Aleph-Null set is a set with the cardinal number 2 raised to the power of Aleph-Null.” As a side effect, Cantor’s diagonal proof shows that the set of real numbers (rationals and irrationals) is also uncountable. Again, using the above figure, we’ll assume a line segment with end points at 0 and 1. Assign binary values to the decimal portion of the segment, and list them on the rows of subsets, and you get the same result as before.

Cantor then showed that the subsets of Aleph-Null, the real numbers and the total points on a line segment all have the same number of elements. He gave this cardinal number the name “C,” the “power of the continuum,” and said that this is Aleph-One, the first infinity greater than Aleph-Null.

Gardner continues to talk about the importance of the Cantor sets in geometry, using the above figure. Say you have an infinite plane tessellated with hexagons. “Is the total number of vertices Aleph-One or Aleph-Null?” The answer is – Aleph-Null, because you can count them along a spiral path. But, the number of circles of one-inch radius placed on a sheet of paper is Aleph-One, because inside any small square near the center of the page are an Aleph-One number of points that are the centers of their own 1-inch circles.

Puzzle for today:
Using J. B. Rhine’s ESP card symbols, are there any symbols that can be drawn an Aleph-One number of times on a sheet of paper, assuming lines of no thickness, with no overlapping or intersections of the lines? (They don’t have to be the same size, but they must have similar shapes.)

To come back to Cohen… Cantor thought there would be an infinite hierarchy of Alephs, obtained by raising the previous one to the power of 2, and that there are no other Alephs in between. There’s no Ultimate Aleph, either. He tried, but could not prove that there’s no Aleph between Aleph-Null and C. Kurt Godel showed in 1938 that the so-called Cantor conjecture could be assumed to be true. And here we get Cohen, who proved in 1963 the opposite could also be assumed – that C is not Aleph-One. There can be at least one Aleph between Aleph-Null and C, although no one has any idea how to specify it. All he did was demonstrate that the continuum hypothesis is unprovable within standard set theory.