Since I started in March, while chasing compactness results in real analysis with my excellent co-author, Maarten McKubre-Jordens (Christchurch), and building up recursive functions towards a paraconsistent version of Godel's theorem, I've been doing a lot of work too on my old favourite: transfinite cardinals.
Maybe ironically, thinking about the size of infinite sets has got me focused on finitude.
I've argued for a few years now that there is an injective function from any set into the singleton {On}, where On = {all the ordinals}. Because of the Burali-Forti paradox, we know that
(i) On is an ordinal, and therefore
(ii) On < On. And therefore
(iii) On < On < ... < On < ... for as long as you want.
So {On} is very structurally rich -- a wellfounded subset of the ordinals that any set can map into, just by dint of a constant function, f(x) = On.
For a long time, I thought this meant that the singleton, {On}, is in fact infinite, appearances of being size 1 notwithstanding. In fact, I was convinced {On}, and indeed the singleton of any inconsistent set, is absolutely infinite, meaning it has the same cardinality as the entire universe, V. After all, an injection from V to {On} -- which exists, by that constant function -- shows that |{On}| is greater than or equal to |V|. I called this the "swelling lemma." Graham Priest suspected that I did because I thought it was, well, swell.
But then I thought some more.
It turns out that, by naive comprehension, every set has an inconsistent subset. (Let A be a set and let A[R] be the set of all members of A, to the degree that the Russell set is self-membered. Are members of A members of A[R] or not? Errr....) Okay, this proves straight away that every set is dedekind infinite, because these inconsistent subsets A[R] are proper subsets. That's not so bad. But by swelling, we also have
|V| < |A[R]| < |A|
which by transitivity implies that |A|=|V|, for any A at all. All sets are the size of the universe? Probably not.
In working (hard) to understand this problem, I've gotten thinking that swelling is the wrong direction for looking at this. We have that any set A at all injects into {On}. Well, then in some sense, |A| is less than |{On}|. That is, instead of thinking that {On} is so big, maybe this means that, no matter what A is, it is rather small.
Finite, even.
Read x < y as "y is bigger than x" and you get swelling. Read it "x is less than y" and you get soothing. I'll spare you the details for now of how to break up the notion of finitude into distinct parts (trans-finitude?) that make sense of this, so that arbitrary sets are (still) not necessarily less than or equal to 1, for instance; all that is in the final paper.
But after thinking for so long about numbers and sets so large, there is something comforting -- beautiful even -- about returning to the notion that everything is so small. Finite, even.
Wait -- everything finite?! How could this be? Well, the idea is that sets only look big, sometimes, because they are inconsistent. That's how, if we're thinking too consistently, we perceive contradictions: as infinity. (The same goes for vagueness: there's no real indeterminacy, only inconsistency. We just weren't expecting that.) So yes, infinity is real, but only as real as contradiction.
Because: the work I'm doing on Godel's theorems at the moment bears out a prediction of Priest, and, inadvertently, of Stewart Shapiro. With a semantically closed arithmetic that represents all its own recursive functions, there is a contradiction at the level of finite numbers.
And once you hit the first inconsistency in the numberline, then, like infinity, they're all inconsistent after that. So it looks like there is some natural number n = |V|. What a world.
What a pleasantly small world. (No wonder there are contradictions, with everything so crowded together.)
Finitism 2011, anyone?
While we head into the New Year, notice that if the universe is finite -- not even countably infinite, but finite--then it shouldn't take long at all to look under every last rock,
and find the Truth, wherever it's hiding.
