In model theory, several basic results and definitions are motivated by absoluteness.
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.
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.
Certain properties are absolute to all transitive models of set theory, including the following (see Jech (2003 sec.
The paradox can be resolved by noting that countability is not absolute to submodels of a particular model of ZFC.
It is possible that a set X is countable in a model of set theory but uncountable in a submodel containing X, because the submodel may contain no bijection between X and ω, while the definition of countability is the existence of such a bijection.
The Löwenheim–Skolem theorem, when applied to ZFC, shows that this situation does occur.
One consequence of Shoenfield's theorem relates to the axiom of choice.
In particular, this includes any of their consequences that can be expressed in the (first-order) language of Peano arithmetic.
Shoenfield's theorem also shows that there are limits to the independence results that can be obtained by forcing.
In particular, any sentence of Peano arithmetic is absolute to transitive models of set theory with the same ordinals.
There are certain large cardinals that cannot exist in the constructible universe (L) of any model of set theory.
Nevertheless, the constructible universe contains all the ordinal numbers that the original model of set theory contains.
This "paradox" can be resolved by noting that the defining properties of some large cardinals are not absolute to submodels.