Friedman's $\mathsf{WD}$ is not parameter-free sequential

Abstract

Harvey Friedman's $\mathsf{WD}$ is a weak set theory given by the following non-logical axioms: $(\mathsf{W})\forall xy\exists z\forall u\left[u\in z\leftrightarrow (u\in x\vee u=y)\right]$; $(\mathsf{D})\forall xy\exists z\forall u\left[u\in z\leftrightarrow (u\in x\wedge u\ne y)\right]$. We answer a question raised by Albert Visser which asks whether $\mathsf{WD}$ is parameter-free sequential. Let $\mathsf{WD}+\mathsf{EXT}$ denote the theory we obtain by extending $\mathsf{WD}$ with the axiom of extensionality. We show that $\mathsf{WD}+\mathsf{EXT}$, and hence also $\mathsf{WD}$, is not parameter-free sequential by using forcing to construct a model ${\mathcal{V}}^{\star }$ of $\mathsf{WD}+\mathsf{EXT}$ where $\left({\mathcal{V}}^{\star },a\right)\simeq \left({\mathcal{V}}^{\star },b\right)$ for any two elements $a,b$ of ${\mathcal{V}}^{\star }$.