2

Working in first order languge $\mathcal L(\in, W)$, where $W$ is a constant symbol.

Reflection: if $\varphi$ is a formula in $\mathcal L(\in)$, in which $x$ is free, and $\vec{p}$ is the string of all of its parameters, and if $\psi$ is a formula in which $z$ is free and $y$ not free, then all closures of: $$\vec{p} \in W \wedge \exists x (\varphi) \to \forall c\exists x [\varphi\wedge \exists y \in W \ \forall z (z \in y \leftrightarrow z \in x \wedge \psi) ] $$; are axioms.

Now with Extensionality this would interpret all axioms of $\text{ZFC}$ since it would trivially interpret Harvey Friedman $\text{K(W)}$ theory (page 3). Although I'm not sure but I think even Extensionality is interpretable using the above scheme only.

Has there been prior attempts to non-trivially shortly axiomatize ZFC with a single comprehension schema? and had this schema been studied before specially in absence of Extensionality?

Zuhair
  • 4,555

0 Answers0