Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 24 / 4 / a

A standard membership model of set theory has the actual membership relation:When “standard model” is used with the transitive-model convention, one also requires $M$ to be transitive: $x\in y\in M$ implies $x\in M$. Under a convention where standard means only actual membership, this extra requirement defines a transitive model. The set-theoretic absoluteness arguments below use transitive models of ZFC, so that bounded quantification and ordinals have their intended meaning.

The Mostowski collapse theorem states that if $R$ is a well-founded extensional relation on a set $X$, then there is a unique transitive set $M$ and a unique bijection $\pi :X\to M$ such thatExtensionality means that distinct elements of $X$ have distinct predecessor sets. The collapse is characterized by the recursionWell-founded recursion defines it, extensionality makes it injective, and its range is transitive. Uniqueness follows by well-founded induction. For a class version, the relation must also be set-like; its predecessor sets must be sets.

An absolute formula has the same truth value in the structures under comparison for parameters common to them. For transitive models $M\subseteq N$ and $a_1,\dots ,a_n\in M$, the requirement isOne specifies the relevant class of models, for example transitive models of ZFC. A formula is upward absolute when the forward implication holds and downward absolute when the reverse implication holds. Bounded formulas in set theory are absolute between transitive membership structures: all their quantifiers range over the same elements of their parameter sets.