 Solution

ID: past-exam-of-the-axiomatic-set-theory-course-of-the-university-of-oxford/2011/2/a/solution

Σ_{0}

\exists u \in v

and

\forall u \in v

, with the usual Boolean connectives. A Sigma-one formula in set theory has the form

\exists u_{1} \dots \exists u_{m} θ (u, x),

where

θ

is a bounded formula in set theory; harmless logically equivalent rearrangements are allowed. A Pi-one formula in set theory is defined with a universal unbounded block instead.

φ (x)

for which there are such

Σ_{1}

and

Π_{1}

σ, π

with

ZF ⊢ \forall x (φ \leftrightarrow σ), ZF ⊢ \forall x (φ \leftrightarrow π) .

Equivalently, both

φ

and

\neg φ

have Sigma-one formulas in set theory provably equivalent to them in ZF. These are levels of the Lévy hierarchy, with bounds by sets.

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