This is the edited transcript of a talk introducing finite factored sets. For most readers, it will probably be the best starting point for learning about factored sets.

Video:

(Lightly edited) slides: https://intelligence.org/files/Factored-Set-Slides.pdf

1. Short Combinatorics Talk

1m. $Some Context$

Scott: So I want to start with some context. For people who are not already familiar with my work:

My main motivation is to reduce existential risk.
I try to do this by trying to figure out how to align advanced artificial intelligence.
I try to do this by trying to become less confused about intelligence and optimization and agency and various things in that cluster.
My main strategy here is to develop a theory of agents that are embedded in the environment that they're optimizing. I think there are a lot of open hard problems around doing this.
This leads me to do a bunch of weird math and philosophy. This talk is going to be an example of some weird math and philosophy.

For people who are already familiar with my work, I just want to say that according to my personal aesthetics, the subject of this talk is about as exciting as Logical Induction, which is to say I'm really excited about it. And I'm really excited about this audience; I'm excited to give this talk right now.

1t. $Factoring the Talk$

This talk can be split into 2 parts:

Part 1: a short pure-math combinatorics talk.

I suspect that if I were better, I would instead be giving a short pure-math category theory talk; but I'm trained as a combinatorialist, so I'm giving a combinatorics talk upfront.

Part 2: a more applied and philosophical main talk.

This talk can also be split into 4 parts differentiated by color: $Motivation$ , $Table of Contents$ , $Main Body$ , and $Examples$ . Combining these gives us 8 parts (some of which are not contiguous):

	Part 1: Short Talk	Part 2: The Main Talk
$Motivation$	1m. Some Context	2m. The Pearlian Paradigm
$ToC$	1t. Factoring the Talk	2t. We Can Do Better
$Body$	1b. Set Partitions, etc.	2b. Time and Orthogonality, etc.
$Examples$	1e. Enumerating Factorizations	2e. Game of Life, etc.

1b. $Set Partitions$

All right. Here's some background math:

A partition of a set $S$ is a set $X$ of non-empty subsets of $S$ , called parts, such that for each $s \in S$ there exists a unique part in $X$ that contains $s$ .
Basically, a pa

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