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 x y \, \exists z \, \forall u \left[ \, u \in z \leftrightarrow ( \, u \in x \; \vee \; u = y \, ) \, \right] $; $ \mathsf{(D)} \; \forall x y \, \exists z \, \forall u \, \left[ \, u \in z \leftrightarrow ( \, u \in x \; \wedge \; u \neq 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 } $.