This is the third article in a series called Before I Forget, an exposition for the semi-layman of freaky high-level mathematics. Last time we talked about cardinal numbers: numbers which are used for counting the elements in very large sets. This time I’ll use the compactness theorem to introduce an infinity that you can use in ordinary arithmetic to get ordinary results. Remember limits from calculus? Turns out you don’t need ’em to do calculus. We’ll see how this time.
This is the first article that will dive into the details of first-order logic. I’m not going to spend the time here to describe first-order logic: it is a deep subject with a rich vocabulary and lots of little nooks and crannys where faulty reasoning can hide. However, I highly recommend that everyone learn its principles and how to use it. It will change the way you think; it will endow you with the ability to approach certain situations with extreme precision, and to clear away bullshit that many authors throw at you. See the wikipedia page, including the external links section, if you’d like to know more. It’s not technically required to understand this article, but it is required to grok this article.
Sexy mathematical models
I will begin by presenting the difference between a model and a set of sentences. Okay, so we have a sentence of first order logic endowed with one extra symbol: S (successor). We will use this to represent the natural numbers. The existence of the number 0 is expressed by the following sentence: ∃x ∀y S(y) ≠ x (there is an x which is not the successor of any number) (remember we’re in the natural numbers, there is no -1). So when we say 0, we really mean to introduce a new variable and that sentence to identify our 0. The important thing is that we can express it though. And now we may represent any natural number using successor and 0. 1 = S(0), 2 = S(S(0)), 3 = S(S(S(0))), etc.