Forcing

Forcing is the method given by Paul Cohen. Cohen proved that adding the negation ¬CH of the continuum hypothesis (CH) to the axioms of set theory does not lead to contradiction, unless the axioms themselves are already contradictory. Cohen's work rules out a refutation of ¬CH by ordinary mathematical means. Forcing can be applied to prove (relative) consistency for hypotheses in transfinite arithmetic, infinitary combinatorics, general topology, measure theory, topology of the real line, universal algebra, and model theory. This chapter introduces some formal symbolism for writing about sets: variables for sets, logical signs, quantifiers, membership sign, identity sign, and plus parentheses for punctuation.