Paradoxes, paradoxically

So I was able – fortunately – to wake up early and to do some legit reading, despite having only a handful of sleep hours (4-ish?). That’s a definite positive. Right now, I’m about 30 minutes away from a forced obligation (that’s a definite negative), but I wanted to use the 30 minutes I have to still do something constructive. Rather than spend this time wracking my brain with really difficult, hard-to-understand reading that would leave me mentally exhausted for the aforementioned obligation, I decided to come here and write a little exposition regarding something mathematical.

In particular, I’m going to talk about the so-called Richard’s Paradox (see here).

Of course, the fact that I’m avoiding theoretical math to postpone mental exhaustion while using the time to come here and talk about theoretical math is a bit of a paradox as well, so I’ll basically be expositing, paradoxically, about paradoxes.

You have no idea how much I crack myself up.

The ideology that birthed Richard’s paradox is intimately tied to the idea of metamathematics, that is, the study of metatheories – theories about mathematical theories – using mathematical ideas and quantification. I’m not going to get too deeply involved in the discussion on that particular topic; the interested reader, of course, can scope out more here.

To begin, we let \mathbb{N} denote the set of nonzero positive integers (aka, the natural numbers) and we investigate the collection of all “formal English language statements of finite length” which define a number n of \mathbb{N}. For example, The first prime number, The smallest perfect number, and The cube of the first odd number larger than five are such statements, as they verbally describe the numbers 2, 6, and 73=343, respectively. On the other hand, statements like The number larger than all other numbers and Scotland is a place I’d like to visit fail to make the list due to the fact that the first doesn’t describe a number in \mathbb{N} and the second doesn’t describe a number at all. Let \mathcal{A}_n denote the collection of all so-called qualifying statements, that is, statements that do describe elements n\in\mathbb{N}.

Note, first, that the collection \mathcal{A}_n is infinite due to the fact that the statements The ith natural number is a qualifying statement for all i=1,2,\ldots. It’s also countable: Only a countable number of words exist in the English language, and each statement in \mathcal{A}_n consists of a finite union of these countably many words. This fact, along with obvious language considerations, says that \mathcal{A}_n can actually be given an ordering.

Indeed, consider a two-part ordering: First, organize the statements in \mathcal{A}_n by length so that the shortest statements appear first, and then organize statements of the same length by standard lexicographical (dictionary) ordering. The result is an ordered version of the countably infinite collection \mathcal{A}_n which we’ll again denote by \mathcal{A}_n.

As of now, almost nothing has been done.

Note, here, that there are two distinct possibilities.

  1. One possibility the i^{th} element in \mathcal{A}_n actually does describe the i^{th} element of \mathbb{N}. For example, if the nineteenth element of \mathcal{A}_n is the statement The largest prime less than 20, then the nineteenth element of \mathcal{A}_n describes precisely the number 19. We don’t really care about this possibility.

  2. The second possibility is the other case. In particular, it’s perfectly possible – and in fact, perhaps even probable – that the i^{th} element of \mathcal{A}_n fails to describe the i^{th} natural number. An example here is that the seventy-third element of \mathcal{A}_n is One more than the largest prime number smaller that 20. This statement obviously describes the element 19+1=20\in\mathbb{N}, and the “index” in \mathcal{A}_n doesn’t match the “index” in \mathbb{N}.

We call the numbers in case 2 Richardian: A number n\in\mathbb{N} is said to be Richardian if n does not have the property assigned to it by the mapping \mathcal{A}_n\to\mathbb{N}. We’ll denote the latter mapping \varphi hereon.

At this point, it’s probably not clear why all this notation has been introduced and what this has to do with paradoxes. Coincidentally, the Richardian condition is precisely the point! Note, in particular, that Is Richardian is an English statement that unambiguously describes a property possessed by some natural numbers. Hence, Is Richardian is an element of \mathcal{A}_n. Suppose this statement is the x^{th} element of \mathcal{A}_n. Now, let’s take a second to examine the number x\in\mathbb{N} and consider again two possibilities.

  1. Suppose x is Richardian. Then by definition of the Richardian condition, x fails to have the property defined by \varphi^{-1}(x)\in\mathcal{A}_n, namely the statement “x is Richardian”. This contradicts the assumption first made about x.

  2. Suppose to the contrary that x fails to be Richardian. In particular, then, x satisfies whatever statement in \mathcal{A}_n ascribed to it by \varphi. By assumption, the x^{th} element of \mathcal{A}_n is the statement Is Richardian, whereby it follows that x is Richardian.

The conclusion? An element n\in\mathbb{N} is Richardian if and only if it fails to be Richardian, whereby it follows that the Richardian condition is a paradox. There are a couple questions we can ask here, and because I’m bored waiting for my aforementioned obligation to begin, it may be fruitful to contemplate those momentarily.

First of all, I think it’s worthwhile to ponder what happened? From what piece of the construction above does this paradox originate? There are many intelligible answers to this question, and my take on it is this:

The map \varphi is clearly a bijection: For every i, \varphi assigns to the i^{th} element of \mathcal{A}_n the natural number i\in\mathbb{N}, and thus the function \psi:\mathbb{N}\to\mathcal{A}_n which selects for each natural number j the j^{th} element of \mathcal{A}_n is obviously \varphi^{-1}. This says several things about the collections \mathbb{N} and \mathcal{A}_n, though the most important from a strictly set theoretic standpoint is that the two collections are indistinguishable as sets: Hence, if all arithmetic, etc., properties of the two structures are forgotten, \mathcal{A}_n is \mathbb{N}. They’re one in the same. Everything we say about \mathcal{A}_n as a set is true about \mathbb{N} and vice versa, and so – in terms of sets and elements of sets – each statement in \mathcal{A}_n is a nonzero positive integer.

But think about what \varphi is doing, then: It’s using “natural numbers” to “verbally describe” natural numbers, and not necessarily in a way that’s “linear”. What I mean is: We’re treating the number 1,349 as an adjective, and we’re saying that some integer n – maybe n=1349, maybe not – “is” 1,349 (the adjective). In the case that n\neq 1349, you’re essentially being told that a red car is blue or that a tall, fat man is short and skinny.

If that’s not a reason to stop and scratch your head, I surely don’t know what is.

The second valid question is why? Why does anyone care about this? I mean, really, what do these math guys do, anyway? They just sit around and think of things that aren’t possibly true just to write books and teach classes that nobody can understand? What the hell, math guys?!

This question – though not particularly unreasonable – is one that I can’t really answer accurately. The biggest complaint lots of educated people have about theoretical mathematics is that it’s thinking for thinking’s sake, and in lots of instances, that’s completely entirely 100% true. Given that, the idea of metamathematics to those people has to almost seem pretentious, like the douchebag Harvard socialite in Good Will Hunting whose entire reason for existing in academia is to tell other people that he exists in academia.

As someone who’s been in academia at various levels for a pretty significant period of time, I can honestly say I’ve never encountered a mathematician who was (quite) that guy. More than anything, I’d say metamathematics is the approach a few people have taken to expand the realm of human understanding. It’s been my experience that mathematicians – the good ones – the ones who haven’t been trampled into submission by the bureaucracy that is organized academia – the ones who didn’t decide to take the easy route just because it’s easy – are, at their core, genuinely curious. And the really good ones – the guys like Terence Tao and Grigori Perelman and the other two dozen or so Fields Medalists – are the ones who possess all of those traits, along with a fearlessness that motivates them to tackle problems that nearly every other person would consider unconquerable.

To those guys, questions about questions about questions probably don’t seem stupid or far-fetched or unnecessary: They probably seem more like air and food and water and shelter.I imagine that those guys live to do theorems about theorems about postulates about axioms about the theory of theories. Why? Because why not, that’s why, and because who knows what we can learn by axiomatizing what we know about what we already know?

As for what I know? A solid three hours have passed since I started this thing. My oft-mentioned obligation has come and passed, as has my nightly baby-routine duties, etc. etc., and now it’s time to do some reading and maybe look at something that doesn’t say “WordPress” at the top.

Until next time….

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