Intuitionistic Zermelo–Fraenkel set theory ($\mathsf{IZF}$)

Full intuitionistic Zermelo–Fraenkel set theory uses intuitionistic first-order logic, extensionality, pairing, union, infinity, full separation, power set, collection and epsilon induction. Collection is stronger than replacement intuitionistically. This is distinct from predicative constructive ZF, which restricts separation and replaces full power set. The full theory supports a Boolean-valued inner universe over intuitionistic set theory, giving the relative consistency of classical ZF.

Using the double-negation Boolean algebra, form well-founded names whose immediate subnames have truth-value weights. Define valued membership and equality by recursion on the underlying well-founded trees. Logical conjunction and universal quantification use meets; disjunction and existence use Boolean joins. Explicit names for pairs, unions and subsets verify the elementary set axioms, and collection for Boolean-valued names handles unbounded witnesses. The resulting internal valued universe validates classical ZF. This is a negative inner interpretation, not an assertion that the external intuitionistic universe satisfies excluded middle.

A well-founded recursively built set code with a set of immediate subnames and a Boolean coefficient for each. The coefficient records the extent to which the corresponding subname belongs. More generally one may permit repeated coded subnames, joining their coefficients. Recursive valued equality identifies different codes representing the same valued set; external equality of codes is not the interpreted equality.

For a name $a$ and formula $\phi$, form the set of pairs $(u,p)$ with $u$ in the immediate domain of $a$, $p$ a Boolean truth value, and some name $v$ satisfying $p\le a(u)\wedge [[\phi (u,v)]]$. Full separation makes this a set, and intuitionistic collection gives a set of witness names covering all its pairs. A name with these witnesses in its domain collects the entire Boolean value of every existential assertion over $a$, without selecting one witness whose value is top. No choice or maximum principle is used.