Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 1 / i

An inner model of ZFC is a transitive class containing every ordinal and satisfying all its axioms. Write this class as $A$ with its inherited membership relation. Satisfaction is interpreted by restricting all quantifiers to $A$. In a first-order formulation this is a schema for a definable class, possibly with fixed parameters. The class may equal the whole universe. Transitivity means $x\in y\in A\Rightarrow x\in A$; containing all ordinals rules out treating an arbitrary transitive set model as an inner model.

A strongly inaccessible cardinal is an uncountable cardinal $\lambda$ that is both regular and a strong limit:Regularity excludes expressing $\lambda$ as the supremum of a shorter increasing sequence, while the strong-limit requirement concerns all smaller power sets. Merely being an uncountable regular limit cardinal is the weaker notion of a weakly inaccessible cardinal.

A first-order formula $\phi (x_1,\dots ,x_n)$ is an absolute formula for membership structures $A$ and $B$ if, for every tuple $\vec{a}\in (A\cap B{)}^n$,The same elements interpret the free variables in both structures; bound variables range over their respective domains. Thus absoluteness requires both directions, rather than only preservation of truth from the smaller structure to the larger one.