Journal of Symbolic Logic

A Spector-Gandy Theorem for $\mathrm{cPC}_d(\mathbb{A})$ Classes

Shaughan Lavine

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Let $\mathfrak{U}$ be an admissible structure. A $\mathrm{cPC}_d(\mathfrak{U})$ class is the class of all models of a sentence of the form $\neg\exists\bar{K} \bigwedge \Phi$, where $\bar{K}$ is an $\mathfrak{U}$-r.e. set of relation symbols and $\phi$ is an $\mathfrak{U}$-r.e. set of formulas of $\mathscr{L}_{\infty\omega}$ that are in $\mathfrak{U}$. The main theorem is a generalization of the following: Let $\mathfrak{U}$ be a pure countable resolvable admissible structure such that $\mathfrak{U}$ is not $\Sigma$-elementarily embedded in $\mathrm{HYP}(\mathfrak{U})$. Then a class $\mathbf{K}$ of countable structures whose universes are sets of urelements is a $\mathrm{cPC}_d(\mathfrak{U})$ class if and only if for some $\Sigma$ formula $\sigma$ (with parameters from $\mathfrak{U}$), $\mathfrak{M}$ is in $\mathbf{K}$ if and only if $\mathfrak{M}$ is a countable structure with universe a set of urelements and $(\mathrm{HYP}_\mathfrak{U}(\mathfrak{M}), \mathfrak{U}, \mathfrak{M}) \models \sigma$, where $\mathrm{HYP}_\mathfrak{U}(\mathfrak{M})$, the smallest admissible set above $\mathfrak{M}$ relative to $\mathfrak{U}$, is a generalization of $\mathrm{HYP}$ to structures with similarity type $\Sigma$ over $\mathfrak{U}$ that is defined in this article. Here we just note that when $\mathrm{L}_\alpha$ is admissible, $\mathrm{HYP}_{\mathrm{L}\alpha}(\mathfrak{M})$ is $\mathrm{L}_\beta(\mathfrak{M})$ for the least $\beta \geq \alpha$ such that $\mathrm{L}_\beta(\mathfrak{M})$ is admissible, and so, in particular, that $\mathrm{HYP}_{\mathbb{HF}}(\mathfrak{M})$ is just $\mathrm{HYP}(\mathfrak{M})$ in the usual sense when $\mathfrak{M}$ has a finite similarity type. The definition of $\mathrm{HYP}_\mathfrak{U}(\mathfrak{M})$ is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem.

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J. Symbolic Logic, Volume 57, Issue 2 (1992), 478-500.

First available in Project Euclid: 6 July 2007

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Lavine, Shaughan. A Spector-Gandy Theorem for $\mathrm{cPC}_d(\mathbb{A})$ Classes. J. Symbolic Logic 57 (1992), no. 2, 478--500.

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