I was going to start with “it has been a while since we had one of these” but then I saw that the previous one started with “Wow, has it really been 7 months since I did one of these?” and that was in December 2023. Maybe I can make it a more regular thing. Anyway blogging itself has become a more regular thing of late so there is quite a lot to report just from last month. This article was written by an opaque and incomprehensible stoachastic process, namely me. . I have retracted this description.In 2007 I described Naomi Wolf as “very smart” now mentions that "Johnny Doc” was released from prison on the orders of Donald J. Trump.My article from January about crooked policitians and crab cakes In 2022 I published , which ended with “This article, which I wrote at 3:00 in the morning, is filled with many errors, including some that I would not have made if it had been daytime. Please disbelieve what you have read, and await a correction.” I have decided that it is uncorrectable, and retracted the whole thing.an article about the etymology of a purported Dutch phrase Dan Luu pointed out an important omission from .my July article following up on Steph Curry's 2015–16 crushing of the 3PFG record In 2021 I argued for . The coming of LLM technology has rendered my suggestion completely imprctical.an interesting replacement for the usual engineering interview process

Lately I've been reading Orwell's collected work from 1945–1950, which was being deaccessioned from the library. It opens with a short but fascinating essay on , published 19 October 1945. It seems clear that Orwell's thinking about this development led directly to his novel , which was written in late 1948.“You and the Atom Bomb”Nineteen Eighty-Four Orwell's initial focus is: This is important, he says, because: Obviously, the atom bomb is in the first category. Then we come to the part that is clearly the road to : Nineteen Eighty-Four By 1945 the three super-states were taking shape. Orwell writes about how in 1941 it had appeared that the Axis powers would win the war (he mentions James Burnham's ) so that Germany (not Russia) might control Europe and Asia, while Japan (not China) might control East Asia, but he said this was just a “miscalculation”:The Managerial Revolution The frozen and oppressive political milieu of is clearly laid out:Nineteen Eighty-Four was published in 1949, and Orwell died the following year after a long battle with tuberculosis. His essay on courtesy of the Orwell Foundation and Orwell's estate.Nineteen Eighty-Four“You and the Atom Bomb” is available online The question that is of most urgent interest to all of us, namely: “How difficult are these things to manufacture?” Ages in which the dominant weapon is expensive or difficult to make will tend to be ages of despotism, whereas when the dominant weapon is cheap and simple, the common people have a chance. … A complex weapon makes the strong stronger, while a simple weapon — so long as there is no answer to it — gives claws to the weak. We have before us the prospect of two or three monstrous super-states, each possessed of a weapon by which millions of people can be wiped out in a few seconds, dividing the world between them. It has been rather hastily assumed that this means bigger and bloodier wars, and perhaps an actual end to the machine civilisation. But suppose — and really this is the likeliest development — that the surviving great nations make a tacit agreement never to use the atomic bomb against one another? Suppose they only use it, or the threat of it, against people who are unable to retaliate? In that case we are back where we were before, the only difference being that power is concentrated in still fewer hands and that the outlook for subject peoples and oppressed classes is still more hopeless. More and more obviously the surface of the earth is being parcelled off into three great empires …. The haggling as to where the frontiers are to be drawn is still going on, and will continue for some years, and the third of the three super-states — East Asia, dominated by China — is still potential rather than actual. Mr H.G. Wells and others have been warning us that man is in danger of destroying himself with his own weapons…. Nevertheless, looking at the world as a whole, the drift for many decades has been not towards anarchy but towards the reimposition of slavery. We may be heading not for general breakdown but for an epoch as horribly stable as the slave empires of antiquity. … The kind of world-view, the kind of beliefs, and the social structure that would probably prevail in a state which was at once and in a permanent state of “cold war” with its neighbours.unconquerable

The bookshelf by my elbow, the one that I can reach without getting up, has seven books on it, not necessarily the ones I look in the most, but the ones whose emanations I most hope will infuse me as I write. Roget's Thesaurus (4th edition) The one I actually refer to most often is the Harper and Row Roget's Thesaurus. I thought I had acquired this in my teens, but the note on the flyleaf says 1989. This is the fourth edition. I was very excited to get the eighth edition, which I thought I might like better, and for some time I kept them next to each other so that I could look up the same things in both, and compare. my conclusion was that while the eighth edition had more stuff in it, it wasn't stuff I needed. And it is really fat. So I have retired it to a farther shelf and will eventually get rid of it. The thesaurus is a book that is widely misunderstood. It is not, as many people mockingly imagine, just a compendium of synonyms, and its correct and intended use is not to replace common words with more impressive-sounding ones. just as the correct use of a screwdriver is not to scrape the veneer off of an expensive cabinet. “Thesaurus” means “storehouse" or “treasure room”. Roget's idea, similar to that of John Wilkins before him, was to classify everything in the world into a hierarchy, in this case a hierarchy with a thousand divisions. At the top level the divisions are grouped into "Abstract concepts", "Space”, “Physics”, “Matter”, “Sensation” and so on. Then under “abstract concepts” there are subclasses, of which subclass VI is “Time”, subdivided into five smaller sections: A. Absolute time At the next level down, section (1)(VI)(B) is divided into: §116. Priority Roget's idea is that if you are thinking or writing about time, and specifically about how it goes by, you will leaf through those sections for inspiration, not to find a more pompous way of expressing something you have already written, but to refine your own idea of what it is you wanted to express. Perhaps you are trying to say that one event followed immediately after another. You might look at “§117 posteriority (later time)” which mentions “ensue”, “consequence”, “aftermath”, and “subsequent” — not synonyms, but related aspects of similar concepts, worth more or less consideration depending on what you are trying to emphasize. §117 will also suggest common phrases like “step into the shoes of” — not a synonym by any means, but a related idea. This is probably not what you wanted in this case, but it in another it might be just the thing, and in any case it might give you a bright idea. If nothing in section 117 seems suitable, it is right next to “§116 priority”, and you might discover that instead of saying that the second event followed immediately after the first, you would rather say that the first immediately preceded the second. Or perhaps you realize, looking at “§118 Simultaneity”, that what you really want to say is that the two event were not quite simultaneous. Or perhaps, finding your way to “§131 Earliness” and “§132 Lateness” you realize that your meaning would be more clearly expressed if you said that the second event was a little tardy, or that the first event was premature. Looking through the index for “immediately” you will see that the index distinguishes several senses of “immediate”: are you trying to suggest intantaneity, or continuity, or haste, or promptness, or punctuality? And in this way the book helps you refine your understanding of what you were trying to say. One can use the thesaurus for more concrete tasks. Perhaps I am trying to remember a word, but I can't quite put my finger on it. I know it it is not “coexisting”, but is something like it. I can look up “coexisting” in the index, and it will take me to “§118 Simultaneity” where I find “contemporaneous”… aha, that's what I was looking for! The really important thing about the thesaurus is this large-scale organizing principle, which puts related ideas near one another. Note that none of this works for someone who doesn't know what the words actually mean. All that person can do with the thesaurus is to replace one wrong word with another one, more or less at random. Effective tool use requires skill and training, and careful thought. An online version would be more convenient, but again, it wouldn't have the same stuff and I am very attached to the one I have. My banishment of the 8th edition left a lot of space on the shelf, some of which I have filled with an anthology of the prose of Sir Thomas Browne. I think this will be healthful and inspiring for me, especially if I remember to take it up and thumb through it from time to time. The Prose of Sir Thomas Browne One recurring theme on this blog since the very earliest days has been the writers of the English Baroque period. In 2008 I wrote: [Browne] is witty, and learned, and wise, and humane, and to read his books is to feel that you are in the company of this witty, learned, wise, humane man, one of the best men that the English Renaissance has to offer, and that you are profiting thereby. His work was also a favorite of Jorge Luis Borges', in case you consider that a recommendation. Browne has shown up here a number of times, although not so much as he should have, because I started the blog the year after I was on my big Thomas Browne kick. One reason I have put this book next to my elbow is that I hope it will spark a new Browne kick. (I wrote in 2006 “I'm sure I will return someday”, and it is long past time for that return.) My favorite book by Browne is his Pseudodoxia Epidemica, which is a compilation of stuff that people in 1646 believed that Browne thought was probably wrong. I wrote about that in come detail in 2008 although I didn't get around to publishing it until 2020. And somehow the other three articles I was writing about this have never seen the light of day. One is about his discussion of whether John the Baptist actually ate locusts or whether they were locust beans or something else. Browne is firmly on the side of it being actual locusts, as am I. My unpublished article says: Chester Brown's version of the gospels makes it clear that John was a crazy old bug-gobbler. Panels from Yummy Fur #17, page 15, by Chester Brown. Also Sir Thomas comes up in connection with whether snails have eyes in their horns — a rare example where he was wrong, and for a dumb reason: If we concede they have two eyes, we must alse grant, they have no lesse than four… And therefore if they have two eyes, they have also four, which will be monstrous, and beyond the affirmation of any. Browne seems to be noping out of the very idea of four-eyed snails, and therefore that they must have none at all. In a later edition of the book, he changed his mind, which is to his credit. He had a thoughtful and well-informed opinion about whether Pythagoras forbade his followers from eating beans, supposedly because he thought they contained the souls of the dead. (Browne says the former is true, but not the latter.) I have trouble connecting with the thinkers of the Middle Ages. Their thinking seems to me to be frightened, so overcautious, so cramped and circumscribed, I can't read it without sadness for the way that medieval Christianity strangled the human spirit for so long. But in the early Renaissance there is a flowering of a joyfully brave willingness to try to understand the world, and to follow any inquiry, no matter how extravagant or ridiculous. The whole idea of God has transformed, changed from something constricting to something empowering. The world before belonged to God, and humans were in it only grudgingly and on promise of good behavior. But when the Renaissance started, the world became a beautiful gift, in which humans had been placed to honor God by admiring and marveling at his creation. This admiration and marvel, the willingness to follow any path to understanding, is how I want to be about knowledge and how I hope I am. Reading Browne, I always feel like he and I would have gotten along well, and that that is one of the best parts of myself. Boccaccio's Decameron The story of the Decameron is this: It is 1348, and Venice is devastated by Black Plague. Nothing can be done, despair is everywhere, and there are not enough left living to bury the dead. So ten young people, still healthy, decide to turn their backs on suffering and quit town. They take provisions and servants, retire to the country, and try to forget the horrors they have seen. There they spend the time feasting, walking in the gardens, playing chess, and, once a day, for ten days, they meet, choose a theme, and then each of them tells a story on the theme. I explained this once to a friend who said “That sounds cool, when was it written?” I said “In 1348!” It is one of the two great works of classical Italian literature, the other of course being Dante. Dante is solidly medieval, hierarchical, doctrinaire, and obsessed with a God who is supposedly loving but doesn't seem to know how to show it. That was in 1308 or so, and then, only a few decades later, we have the Decameron which could not be more different. It is about people, doing people things in the real world, eating, drinking, singing, arguing, and making love. God is present, but not oppressive. He has sent a terrible plague for who knows what reason, but rather than submit to it the characters of the Decameron try to take practical steps to make the best of it. There is a story in the Decameron for every mood, usually more than one. Some are sad, some romantic, some funny and salacious. Dioneo is exempt from following the daily theme and usually has a story that is more or less dirty. My favorite story is probably the one about the cross-dressing English princess, or perhaps the one about how young Caterina wanted to sleep on the balcony so that she could hear the nightingale, which I find very sweet. But the funniest one is about the abbess who is called out of her cell one night to berate a nun for having her lover stay over, and who doesn't realize that in her hurry she has put her own lover's trousers on her head instead of her wimple. I have several different Decamerons, but this copy is the Cormac Ó Cuilleanáin translation, which has made several previous appearances here: On the word “squillions”. Following up a chance encounter in the Oxford English Dictionary is what led me to discover the Decameron in the first place The phrase “two-bit huckster” “soup-guzzling pie-muncher” “soup-guzzling pie-muncher” again There's also an unpublished blog article inviting me to look into this passage: Messer Lotto Gualandi gave him a daughter of his, Bartolomea by name, one of the fairest and handsomest young ladies of Pisa — although most of the females from that benighted town look like tarantulas. The J.M. Rigg translation says “spotted lizards”. This is closer to the original Italian, which is lucertole verminare, literally small wormy lizards. I have my doubts about the desirability of living to be a thousand years old, but if I do decide to do it, one reason will certainly be that I will need the time to learn Medieval Italian and translate the Decameron. From Frege to Gödel, edited by van Heijenoort This is a collection of the most important papers in mathematical logic from the time of Frege (who, I have written before, was responsible for kicking the field of logic out of its medieval period into the modern world) to Gödel (who of course spoiled everything). In between these van Heijenoort hits all the most important ideas, starting with Frege's explanation of Begriffsschrift, which is wacky and weird and which didn't catch on except it kind of did and it still underlies half of mathematical logic and which is the prototype for many of the symbols we still use. After this there is Russell's tragic correspondence with Frege in which he pointed out, too late, that Frege's foundational theory didn't work. The book reprints Peano's original description of the Peano numbers, perhaps the most successful single mathematical theory of all time. The book includes Zermelo's proof of Zermelo's theorem that every set can be well-ordered, and Ackermann's discovery of Ackermann's function, which demonstrated the not every computable function is primitive recursive. The book has Russell on type theory and early work by Kolmogorov and Brouwer on the origin of intuitionism. (Heyting is missing.) Van Heijenoort has come up here when I wanted to quote from Schönfinkel's paper about the SKI-calculus, Wiener's paper inventing the ordered pair, and implicitly in probably a dozen other math and logic articles here over the years. The book is on my shelf because I refer to it pretty often, but also because I can usually find something interesting just by thumbing through it. For example, these remarks by Thoralf Skolem about the futility of deriving induction from set-theoretic foundations. Bonus trivia: Van Heijenoort was the personal secretary of Leon Trotsky, and while he was accompanying Trotsky during the latter's exile in Mexico, he was one of Frida Kahlo's lovers. Orbis Sensualium Pictis (English edition), Johannes Comenius I adore this book. My heart swells with love when I think of it. I don't have a blog article about it and there is a story behind that. In 2018 I went to a conference in Cleveland and my hotel was in a building that had formerly been the Cleveland Department of Education. This one has two big murals, one depicting “The Progress of Education”: I planned to write a blog article about these people. It's clear who some of them are. For example, Moses is easy to recognize at lower right, because of the glowing horns, and Confucius is next to him. Some people I was familiar with once they were identified for me: the red-haired guy second from right in the back row is Friedrich Fröbel, who I knew; his “gifts” are a forerunner of the Montessori materials. But in doing the research I got to the bearded hat-wearing dude topmost on the right side and completely fell off the bus, because that is Johann Comenius who is famous because he wrote one of the most marvelous and enchanting books I've ever read, the Orbis Pictus. I have to resist the temptation to say too much, because Orbis Pictus derailed the article about “The Progress of Education”, it then derailed its own article which has been in progress for eight years, and if I let it it will derail this article too, because every time I pick up Orbis Pictus I forget whatever I was doing and I am lost in the pages with a happy and innocent smile on my face. I'm going to precommit to writing only one paragraph about this incredible book. It was the first illustrated children's book published in Europe, in 1658, and it was an immediate hit, being translated from German into English the following year, then into French, Italian, and many other languages. It swept the continent because everyone loved it. Most of the book follows this pattern: there will be an engraved illustration, depicting some aspect of ordinary human activity, such as (I open it up to a random page) “Tame Foul” (that is, “fowl”): Items of interest in the engraving are annotated with numbers, and the facing page explains the illustration, one item at a time: The Cock 1 (which croweth in a morning), hath a comb, 2. In a second column to the right of this is the same text, but in Latin, so that while the reader is learning about tame fowl, they are also learning Latin: Gallus 1. (qui manè cantat) habet Cristam, 2. The prose is limpid, gentle, pithy, and direct. It hits the important points of interest, invites questions, and ends before anyone can get bored. There are pages on anatomy, butchery, feasting, winemaking, various principal virtues, family trees, cities, burials, ships, wells, horology, amphibians. Now I will reluctantly put it down, rather than leaving this article unfinished as I have so many before. The Bible (New International Version, large print) This of course is the cornerstone of Western culture and no well-educated person can be without a knowledge of what is in it. It is full of great wisdom and great stories, and also cruelty, evil lies, and reminders that the world now is in many ways better than it was because people are better. I would like to understand the world I live in, and there is no way to understand 21st-century America without understanding the Bible. The NIV is not the most poetical translation, but it is clear, modern, and accurate. (I got it on the recommendation of Sterling Hanenkampf. Thanks, Sterling!) In former times I had a collection of Bibles but this is the only one that remains. I even got rid of my old King James, since office space is precious and I have had a digital copy on my computer since the early 1990s. I find that most of my articles mentioning the Bible are unpublished for some reason. It comes up a bit in connection with Ploni Almoni, and in passing in many other places. One of the unfinished articles is a series of notes on the theme of Jesus's admonition “Do not put the Lord your God to the test” (Matthew 4:7) and its relationship to a lot of other things like lightning rods, Christian Science (not Christian science), how Larry Wall became a computer programmer, Pikuach nefesh, and the story of the old lady who refused to evacuate from her house during a flood. It'll be epic if I ever finish it, but I probably won't. Another incomplete one is about the incredible story of Samson and Delilah: She asks him flat out: [Judges 16:6] Tell me the secret of your great strength, and how you can be tied up and subdued. Instead of just telling her to fuck off, Samson lies: [16:7] If anyone ties me with seven fresh bowstrings that have not been dried, I'll become as weak as any other man. The Philistines bring her bowstrings and she tries it that night, but Samson snaps the bowstrings as easily as a piece of string snaps when it comes close to a flame. … Then it goes as before! He tells her a different lie, knowing full well that she will betray him, and she does betray him, and he makes a fool of her again! (16:11–12) Okay, that was fun. Let's do it again! (16:13–14) After several repetitions of this, Samson decides that being shaved, blinded and killed will be less exasperating than listening to any more of Delilah's nagging. I read once that the whole point of the book of Judges is that the people in it are all terrible, they are all far from the path of righteousness, and so you definitely shouldn't act like them. I don't know if that interpretation is correct, but it is certainly true that the people in it are all terrible. The Belles Heures of Duc de Berry This book turned up in one of my very first blog articles, on abbreviations in medieval manuscripts, although I didn't know it at the time. In my teens, on a visit to the Metropolitan Museum of Art, I picked up a print of this: Then I carried it with me for the next forty years, eventually framing it and hanging it up, and it is hanging in my house now. Many years after, when I was still on Twitter and Twitter was still fun, I subscribed to a daily feed from the Met, and one day they tweeted this page, or perhaps another page from the same book, stylistically similar enough that I recognized it immediately. They said where it was from: it's the Belles Heures, a “book of hours”, which tells the reader when to pray and how, and which days are sacred to which saints. Very wealthy people had super-fancy ones made from the very best materials, with illustrations by the very best craftsmen. The Duc de Berry was so wealthy that he had more than one, as I found out when I accidentally ordered and received the Tres Riches Heures. But I got the one I wanted eventually. The Duc de Berry book is by Millard Meiss and Elizabeth H. Beatson, and alternates beween the magnificent color plates and prose discussing each one. From the inscription on the page above I had been able to figure out that this was John the Baptist (see previous article), and the authors aren't sure who the other two people are, but they did at least tell me that John was the Duc de Berry's name-saint. (Funny how John keeps popping up, isn't it?) More recently I had another very similar Internet revelation. I've had this framed postcard hanging up for many years: and thanks to a recent Mastodon toot by Cam Larios, I found out that it is from the “Black Hours” of the Morgan Library. #8? The banishment of the very large Roget 8th edition has left enough space on the shelf for an eighth book. I took a quick look around my office to see if there was anything else that wanted to fill that space, but nothing volunteered. (Actually I think Tristan Needham's Visual Complex Analysis might be waving to me from across the room.) Other stuff There are other things in the photo that should not be on this shelf and I don't know why they are: A packet of googly eye stickers Glass and ceramic coasters that I don't use because my coffee cup is always on my electric mug warmer A small audio speaker that might or might not work A boxful of 8mm-helical scan backup tape from the 1990s A pair of old laptop 2.5-inch hard disks that I hope to someday get the data out of A set of Korean playing cards The shelf is like my brain, I guess, full of stuff, and and what's in it doesn't always make sense or go together with the other stuff.

This post is going to be about what infinite ordinal numbers are, and about is in particular. I had a brainwave a while back (18 months now, wow, I have definitely not been blogging enough) and suddenly understood much better than I did before. I have several related ideas here and I am going to try to write one blog post about each of them, instead of one gigantic blog post about all of them together that I never finish. I really like the ordinal numbers. For some reason I was repeatedly exposed to the infinite cardinals as a child and, while they are pleasingly mysterious, they're also somewhat uninteresting because they have no internal structure, they are just bignesses. It's super cool that there is more than one possible bigness of an infinite set, of course, but sets can have all sorts of interesting structure, and looking just at the bigness ignores all that. The ordinals are much more satisfying, and also I feel that they are more like numbers. This post explains how they work and introduces the interesting ordinal . (I wrote an article a while back about how, when your twelve-year-old asks “what is infinity” you should answer as if they had asked “what is ”. Later I found out that Joel Hamkins recommended the same strategy, and I still stand by it.) What we're doing The idea behind the ordinals is that we want to define something like the “natural” numbers , where each number has a successor and there is a less-than relation. But we want to do it in the context of elementary set theory, which is simpler. Extremely simple, in fact. What is set theory? I don't know how intelligible this article will be if you don't already know, but I am going to try to explain it as briefly as possible. People who already know what means can skip to the next section. In set theory, the only kind of object is a “set”, which is like a featureless bag of things, which are called elements. What kind of things? We don't care, that's not part of the model. The only properties a set has are which things are in the bag. It doesn't make sense to ask what color a set is or whather it is a citizen of Belgium; sets don't have colors, they aren't citizens of anywhere, and they don't have any other extrinsic properties. The only kind of question you can ask is about a set is: Is this thing in that set ? When it is, we write , and when it isn't we write . When a set contains the things and , and nothing else, we write it as $$ \{ p, q, r\} $$ so for example but There is one special set called the “empty set” that has nothing in it at all; it's written . The one other piece of set theory you need to know for this article is that if you have two or more sets, you can combine them into a single set that contains everything that the original sets did. This is called the union of the sets. When combining two sets and , we write for their union. For example: $$ \{\text{tea}, \text{coffee}\} \cup \{\text{mango}, \text{octopus}\} = \{ \text{tea}, \text{coffee}, \text{mango}, \text{octopus} \} $$ There is a lot more than that to set theory but that is the basic idea and I think it's enough to get pretty far in this article. To define numbers in the context of elementary set theory means that we want to find sets that we can interpret as numbers, and a way to interpret arithmetic and such as being operations on these sets. We want to show that those sets can be made to behave the way we expect numbers to behave, and that we can prove that the arithmetic operations have the properties that we expect numbers to have. For numbers, it's true that , and we want to be sure that, whatever we decide that means for sets, and whatever sets we've chosen to stand in for and , we should still have . Understanding when we can model a complicated system in terms of a simpler one, and how to do that, is one of the main concerns of mathematics. Set theory is just about the simplest system there is, so mathematics spends a lot of time trying to interpret various complicated systems in terms of set theory. Less-than Numbers have a less-than relation , and elementary set theory has only one relation, , so it makes sense to try to use that for less-than, and see if it works. We’ll say that if and are sets that represent numbers, then means the same as . We want to be transitive. That is if and then we should also have . If we're taking to be synonymous with , then this means that if and are sets that represent numbers, and if and , then we should also have . This is kind of a weird situation. It means that is not a set of fish or carrots, it means that is a set of sets. And it means any element of any of ’s sets is also an element of itself. When this happens we say that the set is transitive, using the word “transitive” analogously to the way we do what we say that is transitive. Transitivity puts fairly strict constraints on what a set can be like. There are lots of sets, but relatively few of them are transitive. Here are some examples of transitive sets, and the numbers they represent: $$ \begin{align*} 0 &= \{\}\\ 1&=\{0\} \\ \end{align*} $$ Since we are using here as just another way to write the empty set , we could have written instead of . They mean exactly the same. But I feel that the nested curly braces quickly get confusing and don't really contribute to understanding. Still, remember that when we write the symbols , and so on, we're not using them in their usual sense of numbers. Rather, we are talking about these particular transitive sets. The next one is: $$ \begin{align*} 2 & = \{0, 1\} \\ \end{align*} $$ Since and the is an abbreviation for the set $$ \{\{\}, \{\{\}\} \}. $$ I hope you can see why I want to avoid the raw curly-brace notation. Continuing, we have: $$ \begin{align*} 3 & = \{0, 1, 2\}\\ 4 & = \{0, 1, 2, 3\}\\ \vdots\\ 9 & = \{0, 1, 2, 3, 4, 5, 6, 7, 8\},\\ \vdots\\ 53 &= \{0, 1, 2, \dots, 52\}\\ \vdots \end{align*} $$ And so on. These sets are all transitive. For example, and and sure enough, also. This isn’t trivial: Not every set of numbers is transitive. For example is not a transitive set because and but . We'll say that an ordinal number is a set that is transitive, and whose elements are also all transitive. All the sets in the list above are examples. There are transitive sets that aren't ordinals, but we're not interested in them in this article, because they aren't number-like in the same way. This identification of numbers as theser particular sets does also make behave like the less-than relation in the way we wanted. For example, we have because , but not vice versa, it's not true that because . Technically this definition has a lot to recommend it. It’s extremely simple, which makes it easy to work with, and many natural theorems are easily proved. For example, when dealing with familiar numbers, it’s always false that , for any . We'd like to able to prove the analogous thing for our synthetic sets-as-numbers. If we can’t (or worse, if we can prove the opposite) then our model is missing something important (or worse, it’s just wrong). Well, by our definition of less-than, simply means , which is false because has no elements, and that's the proof that is false. Successorship Another thing we need from numbers is a successor operation: each number should be followed by another, different one, and it should be possible to calculate which one. This has been recognized since the 19th century as the most important organizing principle that the natural numbers have. It’s is one of the few foundational things that almost every mathematician not only accepts but is happy with. If is some transitive set, we should be able to identify another, different transitive set that we can designate as the successor of , the number that follows in the sequence of numbers. It’s not hard to show that if is transitive then so is $$ T\cup \{T\} $$ See how it works when : the successor of is $$ 2\cup\{2\} = \{0, 1\}\cup\{2\} = \{0,1,2\} = 3 $$ as we would hope. Limits This gets us the numbers, as we wanted, and we could go on from here to explain how and work and so on, but today we are going a different direction. It turns out that if we add one more ingredient we get a lot more than just familiar numbers. There’s one other way of making an ordinal number out of smaller ordinal numbers. If is any family of transitive sets, then their union, the set that contains everything that is in any of them, is an ordinal also. When the family has a largest element , (typically because it’s a finite family) then the union is not anything new, it’s just again. For example . But if the family of transitive sets has no largest element, we do get something new. In particular, the union $$ \omega = 0\cup 1\cup 2\cup\dots $$ is an ordinal number. By constructing the numbers as transitive sets, we got what we wanted: the finite ordinals behave just like numbers. But if we also consider infinite ordinals, we get infinite numbers like that behave, in some ways, like bigger siblings of the numbers. participates very nicely in less-than comparisons and minimum and maximum operations, and somewhat nicely in addition and multiplication. is an ordinal number but not a familiar one. Under our definition of as a synonym for , every finite number is less than ; there's no familiar number that behaves that way. It’s different from finite numbers in another way also: except for , each finite number is a successor of some other finite number and so has a predecessor, whereas is not a successor of anything and has no predecessor. Ordinals like that are not successors are called limit ordinals. Every ordinal has a successor, and is an ordinal set, so it has one, usually written as , which is the next ordinal after . Then there follow , and the union of all of these is the set $$ \{0, 1, 2, \dots, \omega, \omega+1, \omega+2,\dots\} $$ which is called —still an ordinal. After these come , and eventually . Then after a long series of things like $$\omega^2·17 + \omega·39+117$$ comes , then including an infinite ordinal for every polynomial involving , and then the union of all those, . The series continues — it continues forever, we can always find a bigger transitive set — with things like $$ \omega^{\omega^{\omega^{53}·3+11}·2+\omega·19+1}·7 + \omega^{\omega^{17}·143+53}·12 + \omega^{99938}·12712781 + \omega^{99936}·12712781 +\omega+ 2 $$ where it’s like a polynomial in , except that the exponents don’t have to be finite numbers, they can be other super-polynomials in whose exponents don’t have to be finite. And then, after all of these, the limit of this mind-boggling sequence, is the ordinal called $$ \epsilon_0 $$ It’s just gotten too complicated to express with regular mathematical expressions involving . It transpires that this is the smallest ordinal satisfying the property that $$ x = \omega^x $$ This is the thing I have finally been able to get my head around, a little. The next article will explain how.

The ancient Egyptians had a terrible notation for fractions. They had notations for for each , for , but everything else was written as a sum of these, with repeats forbidden, so that for example had to be written as . (Wikipedia) In an older article about Egyptian fractions and the Rhind Mathematical Papyrus, I said: Getting the table of good-quality representations of is not trivial, and requires searching, number theory, and some trial and error. It's not at all clear that . I think I see now where this comes from. , so two of the summands must have denominators divisible by and by respectively. The first thing you should do is consider $$\u5 + \u7 = \frac{12}{35} = \frac{36}{105}.$$ But you don't want , you want , so you multiply by : $$\u{18}\left(\u5 + \u7\right) = \u{90}+\u{126} = \frac 2{105}$$ and there it is. Why pick and rather than, say, and ? I suspect the answer is probably: Ahmes (or someone earlier) tried it both ways and picked the result they liked best. Remember Ahmes compiling a reference table here, so he does these calculations once, writes down the best result, and throws the others away. If you do the same trick with and instead you get . Then you multiply everything by producing $$\u{84} + \u{140} = \frac2{105}$$ which seems a little worse than the other one. Using the and the produces $$\u{75} + \u{175} = \frac2{105}$$ which seems much worse. Of course this only works when the denominator is composite. Here's another approach, which doesn't work too well in this case but might be useful for other examples. Consider that . We want . So $$ \begin{align} \frac2{105} & = \u{35}\cdot\frac23 \\ & = \u{35}\left(\u2+\u6\right) \\ & = \u{70} + \u{210} \end{align} $$ The denominators here are a lot bigger than the first expansion, but they do at least have the advantage of being multiples of . The Egyptians like this because they, like us, often need to multiply numbers by , and whereas a fraction like is hard for them to multiply by , it's trivial to multiply by .