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 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 of . 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 and the second doesn’t describe a number at all. Let denote the collection of all so-called qualifying statements, that is, statements that do describe elements .
Note, first, that the collection is infinite due to the fact that the statements The ith natural number is a qualifying statement for all . It’s also countable: Only a countable number of words exist in the English language, and each statement in consists of a finite union of these countably many words. This fact, along with obvious language considerations, says that can actually be given an ordering.
Indeed, consider a two-part ordering: First, organize the statements in 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 which we’ll again denote by .
As of now, almost nothing has been done. Continue reading