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To prove his result, Cohen developed the method of forcing, which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds, and constructs another model which contains more sets than the original, in a way that CH does not hold in the new model.
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) [1] was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal.
Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original
Minimal model (set theory) In set theory, a branch of mathematics, the minimal model is the minimal standard model of ZFC . The minimal model was introduced by Shepherdson ( 1951, 1952, 1953) and rediscovered by Cohen (1963) . The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows from the ...
A choice function (also called selector or selection) is a function f, defined on a collection X of nonempty sets, such that for every set A in X, f (A) is an element of A. With this concept, the axiom can be stated: Axiom— For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X to an ...
Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Every infinite set which can be enumerated by natural numbers is the ...
The existence of such a set is consistent, for example given in Cohen's first model. Surprisingly, however, being an infinite Dedekind-finite set is not enough to rule out a group structure, as it is consistent that there are infinite Dedekind-finite sets with Dedekind-finite power sets.
Another example is the set of proper and bounded open intervals of real numbers with rational endpoints. ZF+AC ω suffices to prove that the union of countably many countable sets is countable. These statements are not equivalent: Cohen's First Model supplies an example where countable unions of countable sets are countable, but where AC ω ...