Sacks property
In mathematical set theory, the Sacks property holds between two models of Zermelo–Fraenkel set theory if they are not "too dissimilar" in the following sense. For formula_1 and formula_2 transitive models of set theory, formula_2 is said to have the Sacks property over formula_1 if and only if for every function formula_5 mapping formula_6 to formula_7 such that formula_8 diverges to infinity, and every function formula_9 mapping formula_6 to formula_6 there is a tree formula_12 such that for every formula_13 the formula_14 level of formula_15 has cardinality at most formula_16 and formula_17 is a branch of formula_15 . The Sacks property is used to control the value of certain cardinal invariants in forcing arguments. It is named for Gerald Enoch Sacks. A forcing notion is said to have the Sacks property if and only if the forcing extension has the Sacks property over the ground model. Examples include Sacks forcing and Silver forcing. Shelah proved that when proper forcings with the Sacks property are iterated using countable supports, the resulting forcing notion will have the Sacks property as well. The Sacks property is equivalent to the conjunction of the Laver property and the formula_19-bounding property. Sacks property