This is a continuation of my earlier set theory post. In this post, I’ll describe the next three axioms of ZF and construct the ordinal numbers.

1. The previous axioms

As review, here are the natural descriptions of the five axioms we covered in the previous post.

Axiom 1 (Extensionality). Two sets are equal if they have the same elements.

Axiom 2 (Empty Set Exists). There exists an empty set $\varnothing$ which contains no elements

Axiom 3 (Pairing). Given two elements $x$ and $y$, there exists a set $\{x,y\}$ containing only those two elements. (It is permissible to have $x=y$, meaning that if $x$ is a set then so is $\{x\}$.)

Axiom 4 (Union). Given a set $a$, we can create $\cup a$, the union of the …

Back in high school, I sometimes wondered what all the big deal about ZFC and the Axiom of Choice was, but I never really understood what I read in the corresponding Wikipedia page. In this post, I’ll try to explain what axiomatic set theory is trying to do in a way accessible to those with just a high school background.

1. Motivation

What we’re going to try to lay out something like a “machine code” for math: a way of making math completely rigorous, to the point where it can be verified by a machine. This would make sure that our foundation on which we do our high-level theorem proving is sound. As we’ll see in just a moment, this is actually a lot harder to do than it sounds – there are some traps if we try to play too loosely with our definitions.

First of all …

This is an expanded version of an answer I gave to a question that came up while I was assisting the 2014-2015 WOOT class. It struck me as an unusually good way to motivate higher math using stuff that people notice in high school but for some reason decide to not think about.

In high school precalculus, you’ll often be asked to find the roots of some polynomial with integer coefficients. For instance, $$x^3 - x^2 - x - 15 = (x-3)(x^2+2x+5)$$ has roots $3$, $1+2i$, $-1-2i$. Or as another example, $$x^3-3x^2-2x+2=(x+1)(x^2-4x+\mathrm{2\; \dots }$$

This post briefly outlines the process of setting up a dual boot OSX and Arch Linux on a Mac Mini. This is mostly for my reference in the likely event that I will be doing anything similar in some years, so it assumes some competence; fortunately, the Arch Wiki’s Beginner’s Guide probably fills in any gaps I left out. Obligatory Disclaimer: Use at your own risk or not at all.

This is almost the same as any other installation of Arch Linux, with a few changes that took some hours of frustration to figure out because of the EFI booter. My method is to create the partitions in Disk Utility, install rEFInd, and then install the grub bootloader into /dev/sda1.

Setup done in OSX

1. First, install rEFInd. This worked out of the box for me, and makes it possible to boot via USB.
2. Create a Arch Linux …

Updated version of generic advice post: Platitudes v3.

I think this is an important question to answer, not the least of reasons being that understanding how to learn is extremely useful both for teaching and learning.The least of reasons is that people ask me this all the time and I should properly prepare a single generic response.

About a year agoIt’s only been a year? I could have sworn it was two or three., I posted my thoughts on what the most important things were in math contest training. Now that I’m done with the IMO I felt I should probably revisit what I had written.

It looks like the main point of my post a year ago was mainly to debunk the idea that specific resources are important. Someone else phrased this pretty well in the replies to the thread

The issue is many people …

I always wondered whether I could generate olympiad geometry problems by simply drawing lines and circles at random until three lines looked concurrent, four points looked concyclic, et cetera. From extensive experience you certainly get the feeling that this ought to be the case – there are tons and tons of problems out there but most of them have relatively simple statements, not involving more than a handful of points. Often I think, “I bet I could have stumbled upon this result just by drawing things at random”.

So one night, I decided to join the tangency point of $A$-mixtilinear circle with the orthocenter of a triangle $ABC.$ You can guess about how well that went. Nothing came up after two hours of messing around randomly.

Surprisingly, though, I found almost by accident that the following modification has had significant success:

1. First …

I was in Boston over this weekend for the 2014 Harvard-MIT Math Tournament. Before the contest on Friday, I sat in a few of the undergraduate math classes. They were pretty nice; I was actually able to learn some higher math that just by sitting in, despite the fact that I didn’t have the necessary background. I also got the feeling that the lectures moved somewhat slowly, which is probably how I managed to follow what was happening.

Anyways, I promised a sampler, so attached (at the end) are the notes I took during the classes. As I mentioned, I figured out what was happening in the first two lectures but not the third (so the notes for that one might be total gibberish). Hopefully they’re somewhat interesting though :D

HMMT 2014 Sampler

I always scan copies of letters into my computer before I send them out. So I had a bunch of large PDF’s sitting around hogging my Dropbox space.

One day I found this blog post which claimed that simply running (in Bash) the commands

$ pdf2ps original.pdf temp.ps
$ ps2pdf temp.ps new.pdf

would decrease the file size. (The two commands are part of GhostScript, which I had installed on my Linux boxes anyways.) I couldn’t resist trying it – and miraculously, it worked. It generally decreases my scans by a factor of 10 (from 20MB to 2MB or so).

I have no clue why this works, although it probably has something to do with the fact that the PDF’s are scanned pages . Anyone care to enlighten me?

This is a reflection of a talk I gave today. Hopefully these reflections (a) help me give better talks, and (b) help out some others.

Today I was worked from 6PM-8PM with the Intermediate group at the Berkeley Math Circle, middle school students maybe one or two standard deviations above the average honors student. My talk today was “All you have to do is construct a parallelogram!”. Here is a link to the handout problems and their solutions. (Obviously I only went over a very proper subset of the problems during the lecture.)

Background

Some background information: I had actually given an abridged version of the lecture to the honors geometry class at my Horner Junior High (discussing only 1,2,4,10). It had gone, as far as I could tell, very well. The HJH students audibly reacted as I completed the (short) solutions to their problems, meaning they …

So I guess I can resume blogging now, seeing that I’m done with college applications (at last!). I’m not sure what I plan to blog about in general, but I figured I might as well put this domain name to good use :) I also realized that writing things out helped me clarify my thinking a lot (actually Qiaochu Yuan recommended this for math in particular), so I’ll be trying to do that more often this $2014 = 2 \cdot 19 \cdot 53$ and onwards.

Onto the actual content, anyways. In this post I’ll talk about the inspiration and development for one of my afternoon projects, which I’ve named wintermelon for no good reason.

A while back Jacob Steinhardt recommended to the SPARC alumni list that we check our email at most twice a day. I was able to …