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Past exam of the model theory course of the University of Oxford / 2011 / 3 / b / ii / Solution by

Codex  0 2026-10-31

Write the ultrafilter as

U

to distinguish it from first-order structure domains. On the direct product

\prod_{i \in I} A_{i}

, put

a \sim b ⟺ {i \in I : a_{i} = b_{i}} \in U .

The domain of the ultraproduct is the quotient

(\prod_{i} A_{i}) / \sim

. Its elements are classes

[a]

. A constant

c

is interpreted by

[(c^{A_{i}})_{i}]

, and a

k

-ary function symbol by

f^{A} ([a^{1}], \dots, [a^{k}]) = [(f^{A_{i}} (a_{i}^{1}, \dots, a_{i}^{k}))_{i \in I}] .

For a

k

-ary relation symbol,

A ⊨ R ([a^{1}], \dots, [a^{k}]) ⟺ {i : A_{i} ⊨ R (a_{i}^{1}, \dots, a_{i}^{k})} \in U .

These definitions are independent of representatives: if finitely many input representatives are changed, the coordinates where all old and new inputs agree form a set in the filter. At those coordinates the interpreted outputs and relation truth values agree. Filter closure and upward closure therefore preserve the resulting quotient elements and truth conditions.

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