The Universe of Discourse

Uncountable sets for seven-year-olds

I was recently talking to a friend whose seven-year old had been reading about the Hilbert Hotel paradoxes. One example: The hotel is completely full when a bus arrives with 53 passengers seeking rooms. Fortunately the hotel has a countably infinite number of rooms, and can easily accomodate 53 more guests even when already full.

My friend mentioned that his kid had been unhappy with the associated discussion of uncountable sets, since the explanation he got involved something about people whose names are infinite strings, and it got confusing. I said yes, that is a bad way to construct the example, because names are not infinite strings, and even one infinite string is hard to get your head around. If you're going to get value out of the hotel metaphor, you waste an opportunity if you abandon it for some weird mathematical abstraction. (“Okay, Tyler, now let !!\mathscr B!! be a projection from a vector bundle onto a compact Hausdorff space…”)

My first attempt on the spur of the moment involved the guests belonging to clubs, which meet in an attached convention center with a countably infinite sequence of meeting rooms. The club idea is good but my original presentation was overcomplicated and after thinking about the issue a little more I sent this email with my ideas for how to explain it to a bright seven-year-old.

Here's how I think it should go. Instead of a separate hotel and convention center, let's just say that during the day the guests vacate their rooms so that clubs can meet in the same rooms. Each club is assigned one guest room that they can use for their meeting between the hours of 10 AM to 4 PM. The guest has to get out of the room while that is happening, unless they happen to be a member of the club that is meeting there, in which case they may stay.

If you're a guest in the hotel, you might be a member of the club that meets in your room, or you might not be a member of the club that meets in your room, in which case you have to leave and go to a meeting of one of your clubs in some other room.

We can paint the guest room doors blue and green: blue, if the guest there is a member of the club that meets in that room during the day, and green if they aren't. Every door is now painted blue or green, but not both.

Now I claim that when we were assigning clubs to rooms, there was a club we missed that has nowhere to meet. It's the Green Doors Club of all the guests who are staying in rooms with green doors.

If we did assign the Green Doors Club a guest room in which to meet, that door would be painted green or blue.

The Green Doors Club isn't meeting in a room with a blue door. The Green Doors Club only admits members who are staying in rooms with green doors. That guest belongs to the club that meets in their room, and it isn't the Green Doors Club because the guest's door is blue.

But the Green Doors Club isn't meeting in a room with a green door. We paint a door green when the guest is not a member of the club that meets in their room, and this guest is a member of the Green Doors Club.

So however we assigned the clubs to the rooms, we must have missed out on assigning a room to the Green Doors Club.

One nice thing about this is that it works for finite hotels too. Say you have a hotel with 4 guests and 4 rooms. Well, obviously you can't assign a room to each club because there are 16 possible clubs and only 4 rooms. But the blue-green argument still works: you can assign any four clubs you want to the four rooms, then paint the doors, then figure out who is in the Green Doors Club, and then observe that, in fact, the Green Doors Club is not one of the four clubs that got a room.

Then you can reassign the clubs to rooms, this time making sure that the Green Doors Club gets a room. But now you have to repaint the doors, and when you do you find out that membership in the Green Doors Club has changed: some new members were admitted, or some former members were expelled, so the club that meets there is no longer the Green Doors Club, it is some other club. (Or if the Green Doors Club is meeting somewhere, you will find that you have painted the doors wrong.)

I think this would probably work. The only thing that's weird about it is that some clubs have an infinite number of members so that it's hard to see how they could all squeeze into the same room. That's okay, not every member attends every meeting of every club they're in, that would be impossible anyway because everyone belongs to multiple clubs.

But one place you could go from there is: what if we only guarantee rooms to clubs with a finite number of members? There are only a countably infinite number of clubs then, so they do all fit into the hotel! Okay, Tyler, but what happens to the Green Door Club then? I said all the finite clubs got rooms, and we know the Green Door Club never gets a room, so what can we conclude?

It's tempting to try to slip in a reference to Groucho Marx, but I think it's unlikely that that will do anything but confuse matters.

[ Previously ]

[ Update: My friend said he tried it and it didn't go over as well as I thought it might. ]

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A Qmail example of dealing with unavoidable race conditions

[ I recently posted about a race condition bug reported by Joe Armstrong and said “this sort of thing is now in the water we swim in, but it wasn't yet [in those days of olde].” This is more about that. ]

I learned a lot by reading everything Dan Bernstein wrote about the design of qmail. A good deal of it is about dealing with potential issues just like Armstrong's. The mail server might crash at any moment, perhaps because someone unplugged the server. In DJB world, it is unacceptable for mail to be lost, ever, and also for the mail queue structures to be corrupted if there was a crash. That sounds obvious, right? Apparently it wasn't; sendmail would do those things.

(I know someone wants to ask what about Postfix? At the time Qmail was released, Postfix was still called ‘VMailer’. The ‘V’ supposedly stood for “Venema” but the joke was that the ‘V’ was actually for “vaporware” because that's what it was.)

A few weeks ago I was explaining one of Qmail's data structures to a junior programmer. Suppose a local user queues an outgoing message that needs to be delivered to 10,000 recipients in different places. Some of the deliveries may succeed immediately. Others will need to be retried, perhaps repeatedly. Eventually (by default, ten days) delivery will time out and a bounce message will be delivered back to the sender, listing the recipients who did not receive the delivery. How does Qmail keep track of this information?

2023 junior programmer wanted to store a JSON structure or something. That is not what Qmail does. If the server crashes halfway through writing a JSON file, it will be corrupt and unreadable. JSON data can be written to a temporary file and the original can be replaced atomically, but suppose you succeed in delivering the message to 9,999 of the 10,000 recipients and the system crashes before you can atomically update the file? Now the deliveries will be re-attempted for those 9,999 recipients and they will get duplicate copies.

Here's what Qmail does instead. The file in the queue directory is in the following format:

    Trecip1@host1■Trecip2@host2■…Trecip10000@host10000■

where ■ represents a zero byte. To 2023 eyes this is strange and uncouth, but to a 20th-century system programmer, it is comfortingly simple.

When Qmail wants to attempt a delivery to recip1346@host1346 it has located that address in the file and seen that it has a T (“to-do”) on the front. If it had been a D (“done”) Qmail would know that delivery to that address had already succeeded, and it would not attempt it again.

If delivery does succeed, Qmail updates the T to a D:

 if (write(fd,"D",1) != 1) { close(fd); break; }
 /* further errors -> double delivery without us knowing about it, oh well */
 close(fd);
 return;

The update of a single byte will be done all at once or not at all. Even writing two bytes is riskier: if the two bytes span a disk block boundary, the power might fail after only one of the modified blocks has been written out. With a single byte nothing like that can happen. Absent a catastrophic hardware failure, the data structure on the disk cannot become corrupted.

Mail can never be lost. The only thing that can go wrong here is if the local system crashes in between the successful delivery and the updating of the byte; in this case the delivery will be attempted again, to that one user.

Addenda

1. I think the data structure could even be updated concurrently by more than one process, although I don't think Qmail actually does this. Can you run multiple instances of qmail-send that share a queue directory? (Even if you could, I can't think of any reason it would be a good idea.)

2. I had thought the update was performed by qmail-remote, but it appears to be done by qmail-send, probably for security partitioning reasons. qmail-local runs as a variable local user, so it mustn't have permission to modify the queue file, or local users would be able to steal email. qmail-remote doesn't have this issue, but it would be foolish to implement the same functionality in two places without a really good reason.

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Puzzling historical artifact in “Programming Erlang”?

Buy Programming Erlang from Bookshop.org (with kickback) (without kickback)

Lately I've been reading Joe Armstrong's book Programming Erlang, and today I was brought up short by this passage from page 208:

Why Spawning and Linking Must Be an Atomic Operation

Once upon a time Erlang had two primitives, spawn and link, and spawn_link(Mod, Func, Args) was defined like this:

spawn_link(Mod, Func, Args) ->
  Pid = spawn(Mod, Func, Args),
  link(Pid),
  Pid.

Then an obscure bug occurred. …

Can you guess the obscure bug? I don't think I'm unusually skilled at concurrent systems programming, and I'm certainly no Joe Armstrong, but I thought the problem was glaringly obvious:

The spawned process died before the link statement was called, so the process died but no error signal was generated.

I scratched my head over this for quite some time. Not over the technical part, but about how famous expert Joe Amstrong could have missed this.

Eventually I decided that it was just that this sort of thing is now in the water we swim in, but it wasn't yet in the primeval times Armstrong was writing about. Sometimes problems are ⸢obvious⸣ because it's thirty years later and everyone has thirty years of experience dealing with those problems.

Another example

I was reminded of a somewhat similar example. Before the WWW came, a sysadmin's view of network server processes was very different than it is now. We thought of them primarily as attack surfaces, and ran as few as possible, as little as possible, and tried hard to prevent anyone from talking to them.

Partly this was because encrypted, authenticated communications protocols were still an open research area. We now have ssh and https layers to build on, but in those days we built on sand. Another reason is that networking itself was pretty new, and we didn't yet have a body of good technique for designing network services and protocols, or for partitioning trust. We didn't know how to write good servers, and the ones that had been written were bad, often very bad. Even thirty years ago, sendmail was notorious and had been a vector for mass security failures, and even something as innocuous-seeming as finger had turned out to have major issues.

When the Web came along, every sysadmin was thrust into a terrifying new world in which users clamored to write network services that could be talked to at all times by random Internet people all over the world. It was quite a change.

[ I wrote more about system race conditions, but decided to postpone it to Monday. Check back then. ]

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Math SE report 2023-09: Sense and reference, Wason tasks, what is a sequence?

Proving there is only one proof?

OP asks:

In mathematics, is it possible to prove that there is only one (shortest) proof of a given theorem (say, in ZFC)?

This was actually from back in July, when there was a fairly substantive answer. But it left out what I thought was a simpler, non-substantive answer: For a given theorem !!T!! it's actually quite simple to prove that there is (or isn't) only one proof of !!T!!: just generate all possible proofs in order by length until you find the shortest proofs of !!T!!, and then stop before you generate anything longer than those. There are difficult and subtle issues in provability theory, but this isn't one of them.

I say “non-substantive” because it doesn't address any of the possibly interesting questions of why a theorem would have only one proof, or multiple proofs, or what those proofs would look like, or anything like that. It just answers the question given: is it possible to prove that there is only one shortest proof.

So depending on what OP was looking for, it might be very unsatisfying. Or it might be hugely enlightening, to discover that this seemingly complicated question actually has a simple answer, just because proofs can be systematically enumerated.

This comes in handy in more interesting contexts. Gödel showed that arithmetic contains a theorem whose shortest proof is at least one million steps long! He did it by constructing an arithmetic formula !!G!! which can be interpreted as saying:

!!G!! cannot be proved in less than one million steps.

If !!G!! is false, it can be proved (in less than one million steps) and our system is inconsistent. So assuming that our axioms are consistent, then !!G!! is true and either:

1. There is no proof of at all of !!G!!, or
2. There are proofs of !!G!! but the shortest one is at least a million steps

Which is it? It can't be (1) because there is a proof of !!G!!: simply generate every single proof of one million steps or fewer, and check at the last line of each one to make sure that it is not !!G!!. So it must be (2).

What counts as a sequence, and how would we know that it isn't deceiving?

This is a philosophical question: What is a sequence, really? And:

if I write down random numbers with no pattern at all except for the fact that it gets larger, is it a viable sequence?

And several other related questions that are actually rather subtle: Is a sequence defined by its elements, or by some external rule? If the former how can you know when a sequence is linear, when you can only hope to examine a finite prefix?

I this is a great question because I think a sequence, properly construed, is both a rule and its elements. The definition says that a sequence of elements of !!S!! is simply a function !!f:\Bbb N\to S!!. This definition is a sort of spherical cow: it's a nice, simple model that captures many of the mathematical essentials of the thing being modeled. It works well for many purposes, but you get into trouble if you forget that it's just a model. It captures the denotation, but not the sense. I wouldn't yak so much about this if it wasn't so often forgotten. But the sense is the interesting part. If you forget about it, you lose the ability to ask questions like

Are sequences !!s_1!! and !!s_2!! the same sequence?

If all you have is the denotation, there's only one way to answer this question:

By definition, yes, if and only if !!s_1!! and !!s_2!! are the same function.

and there is nothing further to say about it. The question is pointless and the answer is useless. Sometimes the meaning is hidden a little deeper. Not this time. If we push down into the denotation, hoping for meaning, we find nothing but more emptiness:

Q: What does it mean to say that !!s_1!! and !!s_2!! are the same function?

A: It means that the sets $$S_1 = \{ \langle i, s_1(i) \rangle \mid i\in \Bbb N\}$$ and $$S_2 = \{ \langle i, s_2(i) \rangle \mid i\in \Bbb N\}$$ have exactly the same elements.

We could keep going down this road, but it goes nowhere and having gotten to the end we would have seen nothing worth seeing.

But we do ask and answer this kind of question all the time. For example:

1. !!S_1(n)!! is the infinite sequence of odd numbers starting at !!1!!
2. !!S_2(n)!! is the infinite sequence of numbers that are the difference between a square and its previous square, starting at !!1^2-0^2!!

Are sequences !!S_1!! and !!S_2!! the same sequence? Yes, yes, of course they are, don't focus on the answer. Focus on the question! What is this question actually asking?

The real essence of the question is not about the denotation, about just the elements. Rather: we're given descriptions of two possible computations, and the question is asking if these two computations will arrive at the same results in each case. That's the real question.

Well, I started this blog article back in October and it's still not ready because I got stuck writing about this question. I think the answer I gave on SE is pretty good, OP asked what is essentially a philosophical question and the backbone of my answer is on the level of philosophy rather than mathematics.

[ Addendum: On review, I am pleasantly surprised that this section of the blog post turned out both coherent and relevant. I really expected it to be neither. A Thanksgiving miracle! ]

Can inequalities be added the way that equations can be added?

OP says:

Suppose you have !!x + y > 6!! and !!x - y > 4!!. Adding the inequalities, the !!y!! terms cancel and you end up with … !!x > 5!!. It is not intuitively obvious to me that this holds true … I can see that you can't subtract inequalities, but is it always okay to add them?

I have a theory that if someone is having trouble with the intuitive meaning of some mathematical property, it's a good idea to turn it into a question about fair allocation of resources, or who has more of some commodity, because human brains are good at monkey tasks like seeing who got cheated when the bananas were shared out.

About ten years ago someone asked for an intuitive explanation of why you could add !!\frac a2!! to both sides of !!\frac a2 < \frac b2!! to get !!\frac a2+\frac a2 < \frac a2 + \frac b2!!. I said:

Say I have half a bag of cookies, that's !!\frac a2!! cookies, and you have half a carton of cookies, that's !!\frac b2!! cookies, and the carton is bigger than the bag, so you have more than me, so that !!\frac a2 < \frac b2!!.

Now a friendly djinn comes along and gives you another half a bag of cookies, !!\frac a2!!. And to be fair he gives me half a bag too, also !!\frac a2!!.

So you had more cookies before, and the djinn gave each of us an extra half a bag. Then who has more now?

I tried something similar this time around:

Say you have two bags of cookies, !!a!! and !!b!!. A friendly baker comes by and offers to trade with you: you will give the baker your bag !!a!! and in return you will get a larger bag !!c!! which contains more cookies. That is, !! a \lt c !!. You like cookies, so you agree.

Then the baker also trades your bag !!b!! for a bigger bag !!d!!.

Is it possible that you might not have more cookies than before you made the trades? … But that's what it would mean if !! a\lt c !! and !! b\lt d !! but not !! a+b \lt c+d !! too.

Someday I'll write up a whole blog article about this idea, that puzzles in arithmetic sometimes become intuitively obvious when you turn them into questions about money or commodities, and that puzzles in logic sometimes become intuitively obvious when you turn them into questions about contract and rule compliance.

I don't remember why I decided to replace the djinn with a baker this time around. The cookies stayed the same though. I like cookies. Here's another cookie example, this time to explain why !!1\div 0.5 = 2!!.

What is the difference between "for all" and "there exists" in set builder notation?

This is the same sort of thing again. OP was was asking about

$$B = \{n \in \mathbb{N} : \forall x \in \mathbb{N} \text{ and } n=2^x\}$$

but attempting to understand this is trying to swallow two pills at once. One pill is the logic part (what role is the !!\forall!! playing) and the other pill is the arithmetic part having to do with powers of !!2!!. If you're trying to understand the logic part and you don't have an instantaneous understanding of powers of !!2!!, it can be helpful to simplify matters by replacing the arithmetic with something you understand intuitively. In place of the relation !!a = 2^b!! I like to use the relation “!!a!! is the mother of !!b!!”, which everyone already knows.

Are infinities included in the closure of the real set !!\overline{\mathbb{R}}!!

This is a good question by the Chip Buchholtz criterion: The answer is much longer than the question was. OP wants to know if the closure of !!\Bbb R!! is just !!\Bbb R!! or if it's some larger set like !![-\infty, \infty]!!. They are running up against the idea that topological closure is not an absolute notion; it only makes sense in the context of an enclosing space.

I tried to draw an analogy between the closure and the complement of a set: Does the complement of the real numbers include the number !!i!!? Well, it depends on the context.

OP preferred someone else's answer, and I did too, saying:

I thought your answer was better because it hit all the important issues more succinctly!

I try to make things very explicit, but the downside of that is that it makes my answers longer, and shorter is generally better than longer. Sometimes it works, and sometimes it doesn't.

Vacuous falsehood - does it exist, and are there examples?

I really liked this question because I learned something from it. It brought me up short: “Huh,” I said. “I never thought about that.” Three people downvoted the question, I have no idea why.

I didn't know what a vacuous falsity would be either but I decided that since the negation of a vacuous truth would be false it was probably the first thing to look at. I pulled out my stock example of vacuous truth, which is:

All my rubies are red.

This is true, because all rubies are red, but vacuously so because I don't own any rubies.

Since this is a vacuous truth, negating it ought to give us a vacuous falsity, if there is such a thing:

I have a ruby that isn't red.

This is indeed false. And not in the way one would expect! A more typical false claim of this type would be:

I have a belt that isn't leather.

This is also false, in rather a different way. It's false, but not vacuously so, because to disprove it you have to get my belts out of the closet and examine them.

Now though I'm not sure I gave the right explanation in my answer. I said:

In the vacuously false case we don't even need to read the second half of the sentence:

there is a ruby in my vault that …

… The irrelevance of the “…is not red” part is mirrored exactly in the irrelevance of the “… are red” part in the vacuously true statement:

all the rubies in my vault are …

But is this the right analogy? I could have gone the other way:

In the vacuously false case we don't even need to read the first half of the sentence:

there is a ruby … that is not red

… The irrelevance of the “… in my vault …” part is mirrored exactly in the irrelevance of the “… are red” part in the vacuously true statement:

all the rubies in my vault are …

Ah well, this article has been drying out on the shelf for a month now, I'm making an editorial decision to publish it without thinking about it any more.

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