The Continuum Hypothesis - Logic at Harvard

infinite cardinals: Assuming GCH, if κ λ then κ λ = λ + ; if cf(κ) λ κthen κ λ = κ + ; and if λ < cf(κ) then κ λ = κ.Very little progress was made on CH and GCH. In fact, in the early eraof set theory the only other piece of progress beyond Cantor’s result that2 κ > κ (and the trivial result that if κ λ then 2 κ 2 λ ) was König’s resultthat cf(2 κ ) > κ. The explanation for the lack of progress was provided bythe independence results in set theory:Theorem 1.1 (Gödel, 1938). Assume that ZFC is consistent. Then ZFC+CH and ZFC+GCH are consistent.To prove this Gödel invented the method of inner models—he showed thatCH and GCH held in the minimal inner model L of ZFC. Cohen thencomplemented this result:Theorem 1.2 (Cohen, 1963). Assume that ZFC is consistent. Then ZFC+¬CH and ZFC+¬GCH are consistent.He did this by inventing the method of outer models and showing that CHfailedin a generic extension V B ofV. The combined results ofGödel andCohenthus demonstrate that assuming the consistency of ZFC, it is in principleimpossible to settle either CH or GCH in ZFC.In the Fall of 1963 Easton completed the picture by showing that forinfinite regular cardinals κ the only constraints on the function κ ↦→ 2 κ thatare provable in ZFC are the trivial constraint and the results of Cantor andKönig:Theorem 1.3 (Easton, 1963). Assume that ZFC is consistent. Suppose Fis a (definable class) function defined on infinite regular cardinals such that(1) if κ λ then F(κ) F(λ),(2) F(κ) > κ, and(3) cf(F(κ)) > κ.Then ZFC+“For all infinite regular cardinals κ, 2 κ = F(κ)” is consistent.Thus, set theoristshadpushedthecardinalarithmeticofregularcardinalsas far as it could be pushed within the confines of ZFC.5