Question regarding algebra-valued models for set theory

Question

Suppose you have the following algebra $4={1, a, b, 0}$ such that $0≤b≤a≤1$. Now we define the model $V^A$ by transfinite recursion. Conjunction and Disjunction are defined as usually as max and min. , whereas the implication is defined by $\neg (a \wedge \neg b)$ ( where $\neg$ is an intuitionistic negation). Can you come up with a sentence $\sigma$ in the language of set theory such that $[\sigma ] = b$ ?

Answer 1

Without parameters, this is impossible.

The reason is that you're dealing with a homogeneous forcing (there is a dense, homogeneous subposet in your algebra, namely $\{a,b,1 \}$) and for homogeneous forcings we have $$ p \Vdash \phi \iff 1 \Vdash \phi $$ for all $p \neq 0$ in your algebra. Now remember that $$ \| \phi \| = \sup \{ p \mid p \Vdash \phi \} $$ to conclude the claim.