Decidable Fragments of the Simple Theory of Types with Infinity and NF

Abstract

We identify complete fragments of the simple theory of types with infinity ($\mathrm{TSTI}$) and Quine’s new foundations ($\mathrm{NF}$) set theory. We show that $\mathrm{TSTI}$ decides every sentence $\varphi$ in the language of type theory that is in one of the following forms:

(A) $\varphi =\forall x_1^{r_1}\cdots \forall x_k^{r_k}\exists y_1^{s_1}\cdots \exists y_l^{s_l}\theta$ where the superscripts denote the types of the variables, $s_1>\cdots >s_l$, and $\theta$ is quantifier-free,

(B) $\varphi =\forall x_1^{r_1}\cdots \forall x_k^{r_k}\exists y_1^s\cdots \exists y_l^s\theta$ where the superscripts denote the types of the variables and $\theta$ is quantifier-free.

This shows that $\mathrm{NF}$ decides every stratified sentence $\varphi$ in the language of set theory that is in one of the following forms:

(A${}^{\prime }$) $\varphi =\forall x_1\cdots \forall x_k\exists y_1\cdots \exists y_l\theta$ where $\theta$ is quantifier-free and $\varphi$ admits a stratification that assigns distinct values to all of the variables $y_1,\dots ,y_l$,

(B${}^{\prime }$) $\varphi =\forall x_1\cdots \forall x_k\exists y_1\cdots \exists y_l\theta$ where $\theta$ is quantifier-free and $\varphi$ admits a stratification that assigns the same value to all of the variables $y_1,\dots ,y_l$.