Stochastic Events

Get ready for a long explanation! For everyone’s reference, the joke (supplied by @awesomepus​) was:

Q: What did the mathematician say when he encountered the paradoxes of naive set theory?A: tfw no ZF

You probably already know the ‘tfw no gf’ (that feel when no girlfriend) meme, which dates to 2010. I’m assuming you’re asking about the ZF part.

Mathematically, ZF is a reference to Zermelo-Fraenkel set theory, which is a set of axioms commonly accepted by mathematicians as the foundation of modern mathematics. As you probably know if you’ve taken geometry, axioms are super important: they are basic assumptions we make about the world we’re working in, and they have serious implications for what we can and can’t do in that world. 

For example, if you don’t assume the Parallel Postulate (that consecutive interior angle measures between two parallel lines and a transversal sum to 180°, or twice the size of a right angle), you can’t prove the Triangle Angle Sum Theorem (that the sum of the angle measures in any triangle is also 180°). It’s not that the Triangle Angle Sum Theorem theorem is not true without the Parallel Postulate — simply that it is unprovable, or put differently, neither true nor false, without that Postulate. Asking whether the Triangle Angle Sum Theorem is true without the Parallel Postulate is really a meaningless question, mathematically. But we understand that, in Euclidean geometry (not in curved geometries), both the postulate and the theorem are “true” in the sense that we have good reason to believe them (e.g., measuring lots of angles in physical parallel lines and triangles). Clearly, the axioms we choose are important.

Now, in the late 19th and early 20th century, mathematicians and logicians were interested in understanding the underpinnings of the basic structures we use in math — sets, or “collections,” being one of them, and arithmetic being another. In short, they were trying to come up with an axiomatic set theory. Cantor and Frege were doing a lot of this work, and made good progress using everyday language. They said that a set is any definable collection of elements, where “definable” means to provide a comprehension (a term you’re familiar with if you program in Python), or rule by which the set is constructed.

But along came Bertrand Russell. He pointed out a big problem in Cantor and Frege’s work, which is now called Russell’s paradox. Essentially, he made the following argument:

Y’all are saying any definable collection is a set. Well, how about this set: R, the set of all sets not contained within themselves. This is, according to you, a valid set, because I gave that comprehension. Now, R is not contained within itself, naturally: if it is contained within itself, then it being an element is a violation of my construction of R in the first place. But R must be contained within itself: if it’s not an element of itself, then it is a set that does not contain itself, and therefore it is an element of itself. So we have that R ∈ R and also R ∉ R. This is a contradiction! Obviously, your theory is seriously messed up.

This paradox is inherently a part of Cantor and Frege’s set theory — it shows that their system was inconsistent (with itself). As Qiaochu Yuan explains over at Quora, the problem is exactly what Russell pointed out: unrestricted comprehension — the idea that you can get away with defining any set you like simply by giving a comprehension. Zermelo and Fraenkel then came along and offered up a system of axioms that formalizes Cantor and Frege’s work logically, and restricts comprehension. This is called Zermelo-Fraenkel set theory (or ZF), and it is consistent (with itself). Cantor and Frege’s work was then retroactively called naive set theory, because it was, of course, pretty childish:

There are two more things worth knowing about axiomatic systems in mathematics. First, some people combine Zermelo-Fraenkel set theory with the Axiom of Choice¹, resulting in a set theory called ZFC. This is widely used as a standard by mathematicians today. Second, Gödel proved in 1931 that no system of axioms for arithmetic can be both consistent and complete — in every consistent axiomatization, there are “true” statements that are unprovable. Or put another way: in every consistent axiomatic system, there are statements which you can neither prove nor disprove.For example, in ZF, the Axiom of Choice is unprovable — you can’t prove it from the axioms in ZF. And in both ZF and ZFC, the continuum hypothesis² is unprovable.³ Gödel’s result is called the incompleteness theorem, and it’s a little depressing, because it means you can’t have any good logical basis for all of mathematics (but don’t tell anyone that, or we might all be out of a job). Luckily, ZF or ZFC has been good enough for virtually all of the mathematics we as a species have done so far!

The joke is that, when confronted with Russell’s paradox in naive set theory, the mathematician despairs, and wishes he could use Zermelo-Fraenkel set theory instead — ‘that feel when no ZF.’

I thought the joke was incredibly funny, specifically because of the reference to ‘tfw no gf’ and the implication that mathematicians romanticize ZF (which we totally do). I’ve definitely borrowed the joke to impress friends and faculty in the math department…a sort of fringe benefit of having a math blog.

– CJH