Absoluteness
In mathematical logic, a formula is said to be
absolute if it has the same truth value in each of some class of structures. Theorems about absoluteness typically establish relationships between the absoluteness of formulas and their syntactic form. There are two weaker forms of partial absoluteness. If the truth of a formula in each substructure
N of a structure
M follows from its truth in
M, the formula is
downward absolute. If the truth of a formula in a structure
N implies its truth in each structure
M extending
N, the formula is
upward absolute. Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results and definitions are motivated by absoluteness. In set theory, the issue of which properties of sets are absolute is well studied. The
Shoenfield absoluteness theorem, due to Joseph Shoenfield, establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological consequences.