The ultraproduct is an operation which combines a set of first-order structures into a single structure, in a way that respects the semantics of the individuals structures. This construction is used to give a purely semantic proof of the Compactness Theorem for first-order logic. It also gives a method for constructing non-standard models of first-order theories such as Peano arithmetic.


This concept has the prerequisites:


  • Define the ultraproduct of a set of first-order structures
  • Know the Fundamental Theorem of Ultraproducts, which shows that the ultraproduct construction is consistent
  • Prove the Fundamental Theorem of Ultraproducts
  • Using the ultraproduct construction, prove the Compactness Theorem of first-order logic.

Core resources (read/watch one of the following)


Notes on Logic (2013)
Lecture notes for a course on first order logic.
Author: Henry Cohn
Additional dependencies:
  • Peano axioms


See also

-No Additional Notes-