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 (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.

In model theory, there are several general results and definitions related to absoluteness. A fundamental example of downward absoluteness is that universal sentences (those with only universal quantifiers) that are true in a structure are also true in every substructure of the original structure. Conversely, existential sentences are upward absolute from a structure to any structure containing it.

Two structures are defined to be elementarily equivalent if they agree about the truth value of all sentences in their shared language, that is, if all sentences in their language are absolute between the two structures. A theory is defined to be model complete if whenever M and N are models of the theory and M is a substructure of N, then M is an elementary substructure of N.

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