TITLE: What practical applications does set theory have?
QUESTION [76 upvotes]: I am a non-mathematician. I'm reading up on set theory. It's fascinating, but I wonder if it's found any 'real-world' applications yet. For instance, in high school when we were learning the properties of i, a lot of the kids wondered what it was used for. The teacher responded that it was used to describe the properties of electricity in circuits. So is there a similar practical app of set theory? Something we wouldn't be able to do or build without set theory?

Edit: Actually, I'm asking about the practicality of the knowledge of the properties of infinite sets, and their cardinality. I'm reading Peter Suber's [A Crash Course in the Mathematics Of Infinite Sets][1] ([Wayback Machine](https://web.archive.org/web/20110703003113/https://earlham.edu/~peters/writing/infapp.htm)). The properties of infinite sets seem unintuitive, but of course, the proofs show that they are true. 
My guess is that whoever came up with the square root of -1 did so many years before it 'escaped' from mathematics and found a practical use. Before then perhaps people thought it was clever, but not necessarily useful or even 'true'. So then, if you need to understand electricity, and you can do it best by using i, then even someone who thinks it's silly to have a square root of negative -1 would have to grudgingly admit that there's some 'reality' to it, despite its unintuitiveness, because electricity behaves as if it 'exists'.
Seeing as how there was so much resistance to infinite sets at the beginning, even among mathematicians, I wonder: has the math of infinite sets been 'proven worthwhile' by having a practical application outside of mathematics, so that no one can say it's just some imaginative games?

REPLY [6 votes]: I recently gave a talk about how rank-into-rank cardinals and the cardinals around the $n$-huge level could be used to construct new public key cryptosystems.
The basic idea is to select some finite subalgebra $(X,*)$ of the quotient algebra of rank-into-rank embeddings $\mathcal{E}_{\lambda}/\equiv^{\gamma}$. Since $(X,*)$ is finite, the algebra $(X,*)$ is always computable. One then extrapolates from $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ a ternary self-distributive operation $t^{\bullet}$ on the set $X$. The operation $t^{\bullet}$ is typically of the form $t^{\bullet}(x,y)=T(x,y)*z$ for some operation $T$ that satisfies $x*T(y,z)=T(x*y,x*z)$. Then from the algebra $(X,t^{\bullet})$, one extrapolates a new algebra $(\Diamond(X,t^{\bullet}),t^{\sharp})$ ($t^{\sharp}$ is ternary) which I call a functional endomorphic Laver table. Functional endomorphic Laver tables may be used as platforms for several public key cryptosystems such as the Ko-Lee key exchange and the Kalka-Teicher key exchange.
Since such cryptosystems are very new, nobody knows anything about the security of these cryptosystems or even about the efficiency of such cryptosystems (set theorists, get to work). However, if these cryptosystems are secure against classical computation, then these cryptosystems will likely remain secure against attacks from adversaries with quantum computers as well (quantum algorithms currently use very few specialized techniques that have nothing to do with large cardinals). Depending on how you count, there are about five other different kinds of public key cryptosystems which are not known to be vulnerable to attacks from quantum computers.