Today I was reading this paper, which includes discussion of essential undecidability of various weak theories. On page 24, I was surprised to find out that Vaught has showed that set theory with the following two axioms is essentially undecidable:
$$\forall x\exists y\neg(y\in x)$$ $$\forall x,y\exists u\forall z(z\in u\Leftrightarrow(z\in x\lor z\in y))$$
I was surprised as I have found in some recent paper (which I can't remember) that ST is the simplest known set theory which is essentially undecidable, while (arguably) Vaught's theory is simpler. I wasn't able to track down Vaught's work which would talk of this theory.
On the second thought, this theory seems unlikely to be essentially undecidable, because, first, it vacuously has an empty model, and second, less trivially, a finite model with $\in$ being empty relation seems to satisfy the theory as well, thus giving a decidable extension of the theory.
Can anyone provide a reference for where Vaught proves essential undecidability of this theory?
Thanks in advance.
Edit: Since the result seems to be false, the question arises whether the article I mention on the beginning has a typo. Finding a work of Vaught which the paper appears to be quoting might help clarify that, but again - I couldn't find anything.
These are just typos. (The axioms as given do not form an essentially undecidable theory, as they have lots of finite models.)
The story on top of p. 24 is clearly meant to refer to the adjunctive set theory (AS) with axioms $$\exists x\:\forall y\:\neg(y\in x),$$
$$\forall x,y\:\exists u\:\forall z\:\bigl(z\in u\leftrightarrow(z\in x\lor z=y)\bigr).$$ Indeed, this theory extended with the axiom of extensionality was introduced and proved essentially undecidable by Szmielew and Tarski, and essential undecidability of the version here without extensionality was shown in Vaught’s paper On a theorem of Cobham concerning undecidable theories (Proc. Logic, Methodology and Philosophy of Science 1960, pp. 14–25). The theory AS is mutually interpretable with Robinson’s arithmetic Q.
An even weaker essentially undecidable theory, nowadays often called Vaught’s set theory (VS), was introduced in Vaught’s paper Axiomatizability by a schema (JSL 32 (1967), pp. 473–479). Its axioms consist of the schema $$\forall x_0,\dots,x_{n-1}\:\exists u\:\forall z\:\Bigl(z\in u\leftrightarrow\bigvee_{i<n}z=x_i\Bigr)$$ for all $n\ge0$. The theory VS interprets Robinson’s theory R; like R, and unlike AS or Q, it is not finitely axiomatizable. There is a mention of VS on p. 26 of Beklemishev’s paper (again, with a typo: the axioms need to be stated for $n\ge0$, not just $n\ge1$, i.e., including the axiom of the empty set).