A pre‑order on complete first‑order theories that compares their relative complexity by testing which theories have saturated ultrapowers with respect to classes of regular ultrafilters: T1 ≤ T2 if every regular ultrafilter that saturates T2 also saturates T1.

Principle

The organizing idea is to measure model‑theoretic complexity by ultraproduct behavior: a theory is 'harder' if it requires stronger ultrafilters to produce saturated ultrapowers, so ultrafilter saturation provides a scale of difficulty.

Demonstration

Concrete stratification: stable theories typically lie low in the Keisler order (many regular ultrafilters saturate them), while certain unstable theories that encode complex combinatorics require special ultrafilters and occupy higher degrees; this has been exhibited by constructing ultrafilters that saturate some theories but not others.

Misapplication

Confusing Keisler order with a linear ranking like Morley rank or assuming it is absolute across set‑theoretic universes; the order depends on which ultrafilters exist and on set theory, so asserting strict linearity or absoluteness is incorrect.

Consequence

Provides a coarse but robust invariant for classification theory: dividing theories into equivalence classes and strata which reflect combinatorial and stability properties, and linking model theory to ultrafilter/set‑theory methods.

Reversal

Reversing the comparison (claiming T1 > T2) simply states that there is some regular ultrafilter saturating T1 but not T2; the reversal emphasizes that complexity is relative and sensitive to ultrafilter choice.

Boundary

Defined for complete theories and formulated using regular ultrafilters and saturation of ultrapowers; its exact form can vary with the class of ultrafilters considered and can be sensitive to set‑theoretic hypotheses (e.g. existence of particular kinds of ultrafilters).

Semantic Tension

Tension between Keisler order and other hierarchies (stability, simplicity, NIP): some properties track Keisler levels but none coincide perfectly, producing competing ways to judge theory complexity.

Synthesis

The Keisler Order organizes first‑order theories by how demanding they are of ultrafilters to produce saturated ultrapowers: it is a relative, ultrafilter‑sensitive preorder that bridges combinatorial model properties and set‑theoretic ultrapower behavior.