[lisper]

All of that is a red herring. The basic structure of Cantor's proof applies equally to finite (what you call "self-contained") lists. Consider:

Theorem: there are an infinite number of the integers.

Proof: Assume to the contrary that there are a finite number of integers. Then it is possible to construct a finite list L that contains all of the integers. Find the largest integer in L and add one. The result will be larger (by one) than the largest integer in L, and hence larger than all of the integers in L, and hence not in L, contradicting our assumption that L contains all of the integers. Hence no finite list can contain all of the integers. QED.

You are saying, "But if you give me any integer, I can produce a finite list containing that integer." That's true. But if you don't see at this point why that is irrelevant then you are beyond my ability to help.

[chriswarbo]

> finite (what you call "self-contained") lists.

Wow, I thought it was pretty clear when I said that self-contained == not a function. I don't know how you've interpreted that to mean 'is finite'...

The thing that must be finite isn't our list, it's the representation/program/description of our list. Here's a finite representation/program/description of an infinite list:

    The list beginning with zero, where each successive element is one more than its predecessor

That represents/computes/describes an infinite list (containing all of the naturals in ascending order), but it itself is not infinite (it only contains 92 characters) and is not a list (it is a program/description/whatever).

That has nothing to do with being "self-contained". All I meant by "self-contained" is, as I've said, "not a function".

The program above is not a function; it has no inputs; it has no free variables; etc. hence it is a finite, self-contained program.

The list it describes/computes/represents is also not a function (it's a list); it has no inputs; it has no free variables; etc. hence it is an infinite, self-contained list.

If you like, we could flesh out examples elsewhere on these axes:

- An example of an infinite, self-contained program is: "the list containing the number zero, followed by the number one, followed by ..." and so on ad infinitum. That's self-contained, since it's not a function (it has no inputs), but it's infinite (of course, to represent it in a Hacker News comment I've had to use a finite description at a meta-level (the '...'), but such is the nature of these things!)

- An example of an infinite, not-self-contained program is: "given a number N, return the list containing N, followed by N+1, followed by N+2, followed ...". It's not self-contained because it is a function: it has an input (N). Again, we have to use a meta-level (the '...') to 'represent the infinite representation' ;)

- An example of a finite, not-self-contained program is diagonalization: "given a list L, return the number whose Nth digit is equal to the Nth digit of the Nth element of L, plus one mod 10." That's finite (it's only 117 characters long), and it's not-self-contained since it's a function (it takes the input 'L'). Note that in this case, we're computing a real number rather than a list, but that's irrelevant to the concepts of "self-contained" (i.e. not a function) and the distinction between representations and objects.

Just for fun, here's another not-self-contained program: "Given a binary tree, return its left-most leaf". That's got nothing to do with numbers, lists, diagonalization, etc. but we can immediately see that it's not self-contained, since it's a function.

The program/object distinction is needed because Cantors proof talks about "after" we've given/written-down/etc. an infinite list. We cannot write down such a list in 'unexpanded' form (e.g. [0, 1, 2, 3, ...and so on), since we would keep going forever; and hence any statement about what happens "after" (like Cantor finding a number that's not in our list) is meaningless, since there's no "after".

On the other hand, it's trivial to write down a finite representation/program/description of this list; that's exactly what I've written above as "The list beginning with zero...".

> The basic structure of Cantor's proof applies equally to finite lists... Theorem: there are an infinite number of the integers. Proof: ...

Yes. It also applies to Euclid's proof that there are infinitely many primes ("given a list of primes, I can write down a prime that's not in that list"). We can use it to prove that there's no "best" choice in rock/paper/scissors: given a choice, I can beat it". It's a very useful technique (as is diagonalization).

> All of that is a red herring

From what? Paying the bills? In that sense, all pure maths (including Cantor's proof) is a 'red herring'. I was pointing out (what I find to be) some interesting aspects of Cantor's proof:

- If we do the choices in the opposite order, we get the opposite result.

- If we interleave the choices (e.g. pick one more list element, pick one more digit, pick one more list element, etc.) then it's a "stalemate".

- If we make the choices concurrently, in a perfect information setting (i.e. each "player" knows their own "program" and their competitor's) then it's a win for whoever has more computing power: since they can simulate their opponent and perform the 'extra step' of add-one-mod-10, or append-number-with-diagonalized-prefix (depending on which "player" they are).

Isn't 'noticing and discussing interesting things about an abstract/formal system' basically the process of mathematics? So why is it a "red herring"? What were you hoping to achieve, other than (hopefully) understanding these aspects of the proof that I've tried to point out (which you can then ignore, or take in another direction, or whatever you like)?

> if you don't see at this point why that is irrelevant then you are beyond my ability to help.

Again, irrelevant to what and help with what? I don't plan on taking these observations any further (e.g. writing a paper), since I would imagine they're pretty clear to those with a mathematical mindset. I was pointing them out here because:

- The HN audience seems to like mathematical ideas (for example, this article!)

- The HN audience isn't all highly trained mathematicians, for whom this might be too trivial to discuss

- The HN audience seems to like computational ideas, which these aspects of the proof are related to (although classical mathematics isn't limited to turing computability; hence why I said a "program"/"description" could use hypercomputation, if we like)

[lisper]

> Wow, I thought it was pretty clear when I said that self-contained == not a function. I don't know how you've interpreted that to mean 'is finite'...

Because your definition doesn't make sense. Everything in mathematics is a function. I was trying to do you the courtesy of interpreting what you said in a way that made sense rather than just pointing out that you don't seem to understand what a function is.

> The program above is not a function; it has no inputs; it has no free variables; etc. hence it is a finite, self-contained program.

Now it sounds like what you're trying to get at is the concept of a "constant function" (https://en.wikipedia.org/wiki/Constant_function). Constant functions are functions, and they can have inputs.

So now it seems that you've just spent all this time and effort to point out that diagonalization is not a constant function. That's true, but I don't think it's a particularly interesting observation.

[chriswarbo]

> Everything in mathematics is a function.

No, everything in mathematics is precisely what it is defined to be by whoever is doing the defining. If you want to introduce extraneous extra concepts into a theory, then that's fine; for example, Russell and Whitehead seemed to enjoy contorting set-based constructions into areas where they're completely unnecessary.

However, care should be taken to remain compatible with the original definitions. In this case, there is a clear distinction between "a foo" and "a function which takes some input and returns a foo which depends on which particular value was provided as input". To introduce functions as a 'universal substrate' such that everything is a function, then complain that every "foo" is a function, then introduce the concept of 'constant function' serves no purpose other than to obfuscate perfectly clear definitions under a mountain of irrelevant one-upmanship.

> So now it seems that you've just spent all this time and effort to point out that diagonalization is not a constant function.

That's what most of the time and effort have been spent on, yes. That's not at all what I was trying to "point out" though. I was simply assuming that this was common knowledge (for those familar with Cantor's proof, at least), and so I defined some terms like "self-contained" precisely so I that I could ignore algorithms like diagonalization.

> That's true, but I don't think it's a particularly interesting observation.

I agree it's not interesting. But I wouldn't even call it an "observation". Again, it's simply a trivial consequence of the definitions I gave, and I gave those definitions precisely so that I can ignore diagonalization, for this reason.

My actual observations are along the lines of "the steps of Cantor's proof can be rearranged in these ways, which give rise to these alternative results". Diagonalization doesn't matter for any of that, but you seem to keep trying to:

- Insert diagonalization into those rearrangements, despite it being a type error to do so (because diagonalization is a function, not a number; modulo whatever 'constant function' axioms would make you feel better)

- Complain that those rearrangements are "wrong" (they're not; they're my definitions, so they're "right" by definition!).

- Restate over and over various ways that my definitions do not correspond to Cantor's proof. Which is the entire point.

- "Prove wrong" those definitions, by showing that they're incompatible with various aspects of Cantor's proof (e.g. the order of steps) and diagonalization (e.g. its type). Which is completely backwards, since those definitions were chose precisely so that Cantor's proof and diagonalization will not work for them.