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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
etc.

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!

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
etc.

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?

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.

Aleph-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.

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.

There’s a coffee shop just across the street from City Hall in Kagoshima that I see all the time when I go to and from the English school. They used to roast their own coffee, and I went one time a few years back to check it out. The hand-ground and poured coffee was good, but they were charging something like 400 ($3.70 USD) yen for one small cup, with no free refills, so I didn’t go back.

Then, a few weeks ago, a couple of my students told me about this “suugaku cafe” (mathematics), where the owner was a recreational math enthusiast. The cafe serves as a coffee shop during the day, and then as a cram school for junior high students in the evening. My students suggested I check it out, but they didn’t know the name of the place. So I went online and did a google search on “suugaku cafe kagoshima”. The results showed it was near my school, and the next day I had a little free time and was in the area, so I swung by. I was surprised to see that it was the former coffee roaster, Sandeco, and that they still had the same sign as before. Underneath the sign, it says “mathematics cafe and cram school.” So, maybe it always was math-oriented and I just didn’t realize it then (but I don’t think so. I think they may have changed ownership.)

In general, they look like a regular cafe, serving curry rice with sweet Kagoshima pork, and Shirokuma shaved ice desserts. This lunch set of the curry rice, onion soup, salad, and a small cup of hot coffee after the meal was 750 yen ($7 USD). A bit more than I want to pay for lunch on a regular basis, but it was good, at least.

The only thing that hints at the math aspect of the place is this row of text books, and the white case of drawers in the lower right corner of the nook. (Plus, advertising in the menu for their mascot, Suuga-kuma, (a play on Suugaku (math) and kuma (bear)).) The drawers hold 12 different math puzzles on 3×5 cards, and are divided up into 1st year through 3rd year junior high-level difficulties, 4 cards per level.

I asked the waitress how the system worked, and she explained it, saying that the easiest puzzle was in the upper left drawer, and the hardest one, which she thought was really hard, was in the lower right drawer. (So, difficulty goes from top to bottom, left to right.) I decided to try my luck with the easiest one. What I got was 2(x-1) = 3(x+2) + (x-4), solve for x.

It took me more time to figure out the instructions than it did to do the actual problem. I showed all my steps just in case that’s what the rules required. While I was at the cafe, there were another 9-10 customers, who were also there to do the problems. The rules allow two puzzles per customer per order, so I grabbed the second puzzle for 1st year students. I messed up on one step, showing that it’s better for me to not do these things in pen. I did correct my mistake, though.

Sandeco loves its coffee and math.

There’s kind of a window display in the hallway leading from the door to the main seating area, and in the display were coffee-themed Halloween decorations.

The prizes for completing the puzzles correctly are little paper stickers featuring Suuga-kuma (a white caricature of a bear, wearing a textbook for a professor’s cap) and some kind of joke saying. The one on the left says “eating meals, taking baths, and sleeping are good for you.” There are a total of 50 seals, but only 12 puzzle cards, so I don’t know how the cafe selects the seals you get at any given time, if they’re related to the puzzle, or if they’re totally random.

It’s not worth going back every week just to collect the little seals, but I might consider getting one of the shaved ice desserts on a Saturday if I have a 1-hour break between lessons.

(Secret – I did sneak a look at the hardest problem just to find out how hard it is. It’s pretty simple, but I’m having trouble understanding the Japanese instructions. It’s asking for dy/dx of a simple equation, but I’m not sure if this is supposed to be differentiation or integration. I’m guessing differentiation. Keep in mind, this is a junior high-level problem, and in the U.S. in the 1970’s, I didn’t get into differentiation until I got into college.)

Finally got some news out of Gakken on their Otona no Kagaku (Adult Science) magazine/book/kit series.

Their facebook page announced near midnight on Tuesday that their tiny letterpress printer kit will come out mid-December, for 3,500 yen (not including tax). There’s nothing on this yet on the official website, and the Amazon page is just a blank stub, showing a Dec. 15 release date.

The machine translation of the Facebook announcement reads:
“I’ve been waiting for you for a long time!
The Latest Edition of an adult science magazine, a bath, “small typographical printing machine” on sale on December 15th!
A Letterpress Printing machine called ” Texaco You can create a business card, a message card, etc. The price is 3500 Yen (excluding tax). Thank you for your patronage!
By the way, this is the actual chintakureifu.”

We are now into the section on Infinity, with the chapter on Infinite Regress. The idea here is that you take increasingly many increasingly smaller slices of something on into infinity. Gardner gives examples both from real life, as well as math. These include the plays Tiny Alice, (1964) by Edward Albee and Six Characters in Search of an Author (Luigi Pirandello), and stories like The Town in the Library in the Town in the Library (E. Nesbit), Point Counter Point (Aldous Huxley), The Sorrow of Search (Dunsany), and The Notebook (Norman Mailer). Plus, we also have M. C. Escher’s Drawing Hands.

(All rights belong to their owners. Images used here for review purposes only. A graphic proof for the impossibility of cubing the cube.)

On the math side, both positive and negative numbers disappear into infinity, and every infinite series is an infinite regress. Above is part of a proof for whether it is possible to cube the cube, which is related to squaring the square. In the latter problem, you want to create a tiling of “an integral square only using other integral squares”. Is it possible to do this with cubes? The answer is no.

(Droste ad using regression.)

Gardner also mentions processes that are repeated on smaller and smaller segments of a regular polygon to create something with an infinite perimeter but finite area, including the Koch snowflake.

And the cross-stitch curve (I drew this one.)

Puzzle:
For the cross-stitch curve, how long is the final perimeter, and how large is the enclosed area?