Definition:Relativisation
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Definition
Let $p$ be a well-formed formula of the language of set theory.
Let $A$ be a class.
The relativisation of $p$ to $A$ shall be denoted $p^A$ and shall be defined recursively on the symbols in $p$:
- $x \in y ^A \iff x \in y$
- $\left({\neg p}\right)^A \iff \neg p^A$
- $\left({p \land q}\right)^A \iff \left({p^A \land q^A}\right)$
- $\left({\forall x: P \left({x}\right)}\right)^A \iff \forall x: \left({ x \in A \implies P \left({x}\right)^A}\right)$
Thus, the relativisation of $p$ is simply the well-formed formula achieved when replacing all instances of $\forall x$ with $\forall x \in A$.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 12.4$