Absoluteness
From The Art and Popular Culture Encyclopedia
|
Related e |
|
Featured: |
In mathematical logic, a formula is said to be absolute if it has the same truth value in Template:Clarify span of structures (also called models). 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 (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological consequences. The absoluteness of large cardinal axioms is also studied, with positive and negative results known.
See also
