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axiomatic set theory : FOLDOC | axiomatic set theory One of several approaches to set theory, consisting of a {formal language} for talking about sets and a collection of axioms describing how they behave. There are many different {axiomatisations} for set theory. Each takes a slightly different approach to the problem of finding a theory that captures as much as possible of the intuitive idea of what a set is, while avoiding the paradoxes that result from accepting all of it, the most famous being Russell's paradox. The main source of trouble in naive set theory is the idea that you can specify a set by saying whether each object in the universe is in the "set" or not. Accordingly, the most important differences between different axiomatisations of set theory concern the restrictions they place on this idea (known as "comprehension"). Zermelo Frnkel set theory, the most commonly used axiomatisation, gets round it by (in effect) saying that you can only use this principle to define subsets of
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