Unfoundedness in Mathematical Logic and Model Theory

Unfoundedness is a concept used in mathematical logic and model theory to describe the property of a statement or a formula being unprovable within a given system. In other words, it is a statement that cannot be deduced from the axioms of the system.

For example, the statement "this sentence is false" is unfounded in classical propositional logic, because it cannot be proved or disproved within the system. Similarly, the continuum hypothesis is unfounded in Zermelo-Fraenkel set theory, because it cannot be proven within the system.

Unfoundedness is an important concept in model theory, because it allows us to distinguish between statements that are true but unprovable, and statements that are false but unprovable. In other words, unfoundedness provides a way to identify statements that are not provable within a given system, but may still be true.