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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 (mathematics) In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object .
Continuum hypothesis. In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: "There is no set whose cardinality is strictly between that of the integers and the real numbers." Or equivalently:
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was ...
Forcing (computability) Forcing in computability theory is a modification of Paul Cohen's original set-theoretic technique of forcing to deal with computability concerns. Conceptually the two techniques are quite similar: in both one attempts to build generic objects (intuitively objects that are somehow 'typical') by meeting dense sets.
In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω 2 × ω to {0,1} and p < q if p ⊇ q. This poset satisfies the countable chain condition. Forcing with this poset adds ω 2 distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum ...
Iterated forcing. In mathematics, iterated forcing is a method for constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by Solovay and Tennenbaum ( 1971) in their construction of a model of set theory with no Suslin tree. They also showed that iterated forcing can ...
In set theory, a standard model for a theory is a model for where the membership relation is the same as the membership relation of the set theoretical universe (restricted to the domain of ). In other words, is a substructure of . A standard model that satisfies the additional transitivity condition that implies is a standard transitive model ...