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

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 .

Bo Diddley's cover of "Sixteen Tons" sounds very much like one of my favorites, "Can't Judge A Book By Its Cover". It's interesting to compare. Thinking on that it suddenly occured to me that his name might have been a play on “diddley bow”, which is a sort of homemade one-stringed zither. The player uses a bottle as a bridge for the string, and changes the pitch by sliding the bottle up and down. When you hear about blues artists whose first guitars were homemade, this is often what was meant: it wasn't a six-string guitar, it was a diddley bow. But it's not clear that Bo Diddley play his name on the diddley bow. "Diddly" also means something insignificant or of little value, and might have been a disparaging nickname he received in his youth. (It also appears in the phrase "diddly squat"). Maybe that's also the source of the name of the diddley bow.did