Hi all. Sorry for the long time between posts; for fear of sounding like a cracked record, it's because I am behind and would rather spend my time trying to catch up than writing about trying to catch up. That said, with almost two months passed since my last posting, I figured what's another half an hour and have opted to write a little.
As of tonight, or sometime tomorrow at the latest, I'll actually be on schedule with M381, but this is at the expense of M337, in which I am still a unit behind. I so desperately want to know how it feels to be ahead of schedule, but I am beginning to accept that this is not going to happen. Salvation lies in the fact that the final unit of mathematical logic is not TMA-assessed, and while I am aware there is a question on Godel's incompleteness theorems in the exam, I get the impression that you can still do well without tackling it. I do really want to understand them and the proofs (and to look rather clever when I ask someone to get me 'On Formally Undecidable Propositions in Principia Mathematica and Related Systems' for Christmas - as well as the new Pink Floyd box set :), but unless there's a miracle, I will settle for leisurely working my way through it after the course is done and dusted.
So how have I been finding it recently? Well, I still don't feel the love for number theory, although to say I actively dislike it would be an exaggeration. It is what it is. I'm happy to do the work, and am finding some of it to be quite lucid and clever, but at the same time, I could easily live without it. Logic on the other hand, I'm really getting into. But it is a lot of work. Take the proof in Unit ML 2, that the function p which enumerates the primes is primitive recursive: this involves showing that the characteristic function for Pr the set of prime numbers is primitive recursive, which itself involves showing the remainder function to be primitive recursive, the division function, and the characteristic function of the equality relation. Then we use the 'less than' relation, the exponentiation function, the new rule of 'bounded minimization' and god knows what else, covering around six pages of textbook, give or take detours via examples and problems. The gradual building up of all these pieces to prove the result is very very clever, but equally as heavy-going. The upshot is that ultimately this function is computable from just the basic primitive recursive functions of zero, successor and projection*. How anyone could do this is the first place is beyond me. To quote Ian Dury, there aint half been some clever bastards!
I've taken to developing my own short hand for working through this stuff. Any idea how many time's I've had to write 'primitive recursive' or plain old 'recursive' and the various permutations of these words with function, relation, set, etc? I don't! So I now use a little zig-zaggy squiggle for 'recursive' with a vertical line through it to make it primitive, then I can stick an F on the end for function, R for relation, and so on. I did something similar eventually for URM because although it's an abbreviation, I still felt that the amount I was writing it meant I was wasting a lot of time, so now lolly-pop-like spiral = URM, lolly-pop-like spiral with c on the end is URM-computable, and so on. Also, doesn't writing 'such that' start to get annoying after three years as well? You bet it does, so now it's a long horizontal line with an x in the middle. Sort of like this --x--.
Okay, that's enough for now. I'd really better get back to work. My comment on M337 therefore, must be brief: "It's going fine."
Happy studying!
P.S. On a totally unrelated note, I found out recently, that a book on Japanese volcanic earthquakes that I translated in summer 2009 has been officially released in Japan! It clearly won't have my name in it, and I bet it was heavily edited after I finished with it. But I hope to get a copy soon, as it is something tangible that I have achieved! That and I've not done anything else like it. (A Japanese friend from my Masters course in Bath said she'll pick it up for me and bring it over when she comes to my wedding this summer).
*As with the URM rules I mentioned previously, zero turns n into 0 and successor adds 1 to n. Projection functions basically pick out one from a number of variables in N-whatever, such as the second entry n2 from (n1, n2, n3).
