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The notion of a continuously variable quantity or number can be seen as a
generalization of a particular, constant quantity or number, and the
properties of such quantities are then similar to, and derived from, the
properties of constants. For example, the continuous, real-valued
functions on a topological space behave like the field of real numbers in
many ways, but instead form a ring.
The algebraic approach to logic, using categories and toposes, permits one
to apply this same idea, and to consider continuously variable sets
(sheaves). In this expository talk, such applications are explained for
non-specialists. Some recent results in topos theorey are then discussed
in this setting, and the new logical completeness theorems for systems of
higher-order logic are presented.
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