Zermelo–Fraenkel Set Theory

The units on set theory and logic have used ZF set theory without specifying precisely what it is. To investigate which arguments are possible in ZF and which not, you must have a precise description of it. A major question asked during the development of ZF was what system of logic should be used as its framework. Logicians eventually agreed that the framework itself should not depend very much on set-theoretic reasoning. Investigators could then focus on the difficult problems of set theory: there would be little interplay between the framework and the subject under study. During 1920–1940, first-order logic crystallized as a convenient framework for the study of algebraic structures. Applying it does not require use of techniques commonly regarded as set-theoretic. Moreover, the Gödel completeness theorem shows that it encompasses (but doesn't necessarily simulate) many arguments that mathematicians commonly use to prove theorems expressed in a first-order language. Thus, to facilitate investigation of the scope of set theory, it seems appropriate to express it in a first-order language, and restrict it to use logic that is compatible with the first-order framework. When we apply set theory formulated that way, we can highlight the use of its major principles. They're explicitly stated in first-order set-theoretic axioms, and explicitly mentioned in first-order proofs. To be sure, some very elementary parts of set theory are involved in the underlying logical framework, necessary even to formulate those axioms and proofs. But the more powerful set-theoretic principles are displayed conspicuously. ZF is formulated in a first-order theory with minimal apparatus: • countably many variables, which we regard as varying over all sets; • no constants; • no operators; • just two predicates, equality and the binary membership predicate 0. The nonlogical axioms of ZF can be reduced to a small number, as follows. extensionality power set choice separation pair set foundation replacement union infinity The ZF axioms are kept to the minimum number in order to simplify studies of their properties. The list can be pared even further by deriving some axioms from others, but those arguments are uninformative. Each of these axioms is stated below in detail, with some remarks to show how the axioms are used to develop formally the set theory used in the various other units.