In the Dunn book on algebraic logic, there is a chapter on something called representation theorems. This was not something I’d come across before and it was not really explained. The first question is: what is a representation theorem? The answer is that it seems to be a canonical way of mapping the structure in question into a set-theoretic structure which contains operations on the set structure that correspond to the operations on the original structure.

The next question is: why are these interesting? I’m not really sure. The set-theoretic structures have the benefit of being extensional. That could be an epistemological benefit if the original structures aren’t obviously extensional. Some of the structures are a little arcane though. For example, the representation of fusion, relevant implication, and the ternary accessibility relation are rather involved. I’m not going to include them; I don’t have the reference handy. It isn’t simplicity that is the aim of the mapping, per se.

The project of showing that all of math can be done in set theory was something that I thought was more of an early 20th century phenomenon. This seems to be reflected in the things cited. They are all before 1950. Nothing from the set theory is used to reveal anything in the original structures, so it isn’t a case of translating from the original language, such as lattice theory, to set theory, finding something neat, then translating the result back into the original language. Some of the representations are technically quite nice but I’m unsure why one would be interested in them in the first place outside of the desire to show that the original structures can be found in the universe of sets.

Generally, Dunn’s book is quite good, but it is kind of frustrating that some of the more heavy duty technical parts of the book, such as the representation chapter and subsequent representation theorems, are not motivated. More importantly, the idea and point of a representation theorem are not explained at all. They haven’t gotten much clearer as the book goes on. [Edit: The comments clear things up a lot.]

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