Now that my time in Part III is over, I feel justified in releasing my essay, which is on the subject of Non-standard Analysis. It was supervised by Dr Thomas Forster (to whom I owe many thanks for exposing me to such an interesting subject, and for agreeing to supervise the essay).

Why jargon is a really useful thing to have and use.

A quick overview of the definition of the mathematical concept of finitistic reducibility.

Most recent exposition: an article on Tennenbaum’s Theorem. Comments welcome. The proof is cribbed from Dr Thomas Forster, but his notes only sketched the fairly crucial last step, on account of the notes not yet being complete.

I’ve written a blurb about what a modular machine is (namely, another Turing-equivalent form of computing machine), and how a Turing machine may be simulated in one. (In fact, that blurb now contains an overview of how we may use modular machines to produce a group with insoluble word problem, and how to use them to embed a recursively presented group into a finitely presented one.)

A modular machine is like a slightly more complicated version of a Turing machine, but it has the advantage that it is easier to embed a modular machine into a group than it is to embed a Turing machine directly into a group. We can use this embedding to show that there is a group with unsolvable word problem: solving the word problem would correspond to determining whether a certain Turing machine halted.

So you’ve heard that the Axiom of Choice is magical and special and unprovable and independent of set theory, and you’re here to work out what that means.

Recall the Monty Hall problem: the host, Monty Hall, shows you three doors, named A, B and C. You are assured that behind one of the doors is a car, and behind the two others there is a goat each. You want the car. You pick a door, and Monty Hall opens one of the two doors you didn’t pick that he knows contains a goat. He offers you the chance to switch guesses from the door you first picked to the one remaining door. Should you switch or stick?

I’ve been trying to learn Clojure through Exercism, a programming exercises tool. It took me an hour to get Hello, World! up and running, so I thought I’d document how it’s done. I’m using Leiningen on Mac OS 10.11.4.

Answering the question, “Why does a continued fraction containing only 1, subtraction, and division result in one of two complex numbers?”.

Another short post to point out my new article on the Friedberg-Muchnik theorem, a theorem from computability theory. It uses what is known officially as a finite injury priority method, and the proof is cribbed entirely from Dr Thomas Forster.

Just a post to draw attention to my new article about representable functors and their links to adjoint functors. It’s very short, but it gives a reason for being interested in representable functors: they are basically “those with left adjoints”, up to minor quibbles.

I’m clearing out my desktop again, and found this document on the multiplicativity of the determinant, which I wrote in 2014. It might as well be up here.

I should note that this document contains no motivation of any kind. It is simply an exercise in symbol-shunting, and it has no clever ideas in it.

Another short post to highlight the existence of an article about the Monadicity Theorems, in which I prove one direction of both the Crude and Precise versions. Comments and corrections would be very much appreciated, because there is an awful lot of work involved in proving those theorems. It would be good to know of any parts where the argument is unclear, unmotivated, too long-winded, or wrong.

Just a post to draw attention to my new article about the General Adjoint Functor Theorem. It’s a motivation of the GAFT and its proof. I’ve never seen it motivated in this way, and it’s actually quite a natural theorem. I haven’t managed to motivate the Special Adjoint Functor Theorem at all, although I’m told that it’s natural if you know Stone-Cech compactification.

In the Part III Topics in Set Theory course, we have used forcing to show the consistency of the Continuum Hypothesis, and we are about to show the consistency of its negation. I don’t really grok forcing at the moment, so I thought I would go through an example.

As an exercise in understanding the definitions involved, I find the Eilenberg-Moore category of a certain functor.

It has been proposed to me that if one is to play the National Lottery, one should be sure to select one’s own numbers instead of allowing the machine to select them for you. This is not an optimal strategy.

Here I explain proof by contradiction so that anyone who has ever done a sudoku and seen algebra may understand it.

In the summer of 2015, I worked through Awodey’s Category Theory, and I produced a large collection of posts as I tried to understand its contents. These posts are probably not of much interest to anyone who is just looking for something to read, so they’re siloed off.

*In which I am a wizard.*

Sometimes as a student, the work piles up and I start to think “I’ll never finish this”. It becomes easy to think that there’s no point in working because the work will never be over. When that happens to me, I imagine that my course is magic/alchemy/something with flashy special effects. I’m going through the Wizardry Academy, and I’ll graduate able to manipulate the four elements. Even if I’m not the best in the year at it, I’m still able to *manipulate the elements*, and if I work at it, I’ll be able to manipulate them better and in flashier ways - that’s not something most people can do!