Models of Set Theory

Set Theory · Axiom Academy

Understanding what it means for a structure to satisfy the axioms of set theory A model of set theory consists of a pair (M, ∈ᴹ) where M is a collection and ∈ᴹ is a binary relation on M that satisfies all the ZFC axioms. Think of M as a "universe of sets" and ∈ᴹ as the "membership relation" within that universe. The superscript M reminds us that membership might be interpreted differently in different models. 2. Standard vs Non-Standard Models The "intended model" V is what we think of when we do ordinary set theory - the collection of all sets built up from the empty set using the standard membership relation ∈. But there can be other models that look very different! A non-standard model might contain "sets" that behave in unexpected ways, yet still satisfy all the ZFC axioms. An inner model is a model M such that M ⊆ V and M satisfies all ZFC axioms. These are "subuniverses" contained within our standard universe. An outer model is a model N such that V ⊆ N. These are larger universes that contain everything we can construct in V, plus potentially more. Inner models are particularly important because they allow us to prove consistency results: if ZFC is consistent, then so is ZFC + certain additional axioms. A model M is transitive if whenever x ∈ M and y ∈ x, then y ∈ M. In other words, M contains all the elements of its elements.

This is the written version of the interactive lesson above. See the full Set Theory course.