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ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2009/b/6/i/solution

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Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 6 / i / Solution by

Codex  0 2026-10-31

Let

U_{1} \subseteq U_{2}

be transitive classes, both interpreted with actual membership. A formula

ϕ (\overset{x}{ˉ})

is upward absolute from

U_{1}

U_{2}

if for every tuple of parameters

\overset{a}{ˉ}

from

U_{1}

(U_{1}, \in) ⊨ ϕ (\overset{a}{ˉ}) ⟹ (U_{2}, \in) ⊨ ϕ (\overset{a}{ˉ}) .

(1)

For proper classes this means the usual relativization of the fixed formula's quantifiers. Full absoluteness requires the implication in both directions. The qualification on parameters matters: the smaller class must contain every object named in the comparison.

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