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:
- semantics of first-order logic (The ultraproduct is a way of combining structures in first-order logic.)
- equivalence relations (The ultraproduct is defined using an equivalence relation.)
- ultrafilters (The ultraproduct is defined in terms of ultrafilters.)
- structural induction (The proof of the Fundamental Theorem of Ultraproducts requires structural induction.)
- countable sets (The Compactness Theorem is stated in terms of finite sets.)
- 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.
Location: Section 15, "Ultrafilters," pages 36-39
- Peano axioms
→ A Course in Mathematical Logic
A graduate textbook in mathematical logic.
Location: Section 5.3, "Ultraproducts," pages 174-184
- Refer to Section 5.1 for basic definitions and notation.
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