# Algebraic Set Theory

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## Why "algebraic set theory"?

There are two reasons for referring to this research as "algebraic
set theory":
The first reason is that the models of set theory that are
produced by these methods are algebras for an abstractly presented
"theory", in a precise, technical sense known to category theorists
as a monad. The notion of an algebra for a monad subsumes and
generalizes that of a model for a conventional algebraic theory, such
as groups, rings, modules, etc. Indeed, the first significant work
in this style on the applications of category theory to the study of
set theory was the monograph __Algebraic Set Theory__ (Cambridge, 1995)
by André Joyal and Ieke Moerdijk.
The second reason is that we believe the locution "algebraic
logic" should properly refer to categorical logic rather than just
the logic of Boole and his modern proponents, since categorical logic
subsumes such lattice theoretic methods and not the other way around.
Hence the term "algebraic set theory" rather than "categorical set
theory". This is in keeping with the use of "algebraic" to mean,
essentially, "functorial" in modern algebraic topology, algebraic
geometry, etc.

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