Lowenheim-Skolem theorems


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)


See also

-No Additional Notes-