The Lowenheim-Skolem theorems state that if a first-order theory has an infinite model, then it has models of every infinite cardinality. It implies that certain foundational theories in mathematics, such as Peano arithmetic and Zermelo-Frankl set theory, must have nonstandard models.
This concept has the prerequisites:
- Know the statements of the upward and downward L-S theorems.
- Prove the upward and downward L-S theorems.
- Know what Skolem's Paradox refers to and why it is paradoxical.
- Know why the L-S theorems imply nonstandard models of the Peano Axioms.
Core resources (read/watch one of the following)
→ Notes on Logic (2013)
Lecture notes for a course on first order logic.
→ A Course in Mathematical Logic
A graduate textbook in mathematical logic.
- Section 5.1, "Basic ideas of model theory," pages 161-168
- Section 5.2, "The Lowenheim-Skolem Theorems," pages 168-173
→ A Mathematical Introduction to Logic
An undergraduate textbook in mathematical logic, with proofs.
Location: Section 2.6, "Models of theories," subsection "Size of models," pages 151-155
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