We introduce and study some variants of a notion of canonical set theoretical truth. By this, we mean truth in a transitive proper class model M of ZFC that is uniquely characterized by some $$\in$$ ∈ -formula. We show that there are interesting statements that hold in all such models, but do not follow from ZFC, such as the ground model axiom and the nonexistence of measurable cardinals. We also study a related concept in which we only require M to (...) be fixed up to elementary equivalence. We show that this theory-canonicity also goes beyond provability in ZFC, but it does not rule out measurable cardinals and it does not fix the size of the continuum. (shrink)

The parameter-free part $$\textbf{PA}_2^*$$ of $$\textbf{PA}_2$$, second order Peano arithmetic, is considered. We make use of a product/iterated Sacks forcing to define an $$\omega $$ -model of $$\textbf{PA}_2^*+ \textbf{CA}(\Sigma ^1_2)$$, in which an example of the full Comprehension schema $$\textbf{CA}$$ fails. Using Cohen’s forcing, we also define an $$\omega $$ -model of $$\textbf{PA}_2^*$$, in which not every set has its complement, and hence the full $$\textbf{CA}$$ fails in a rather elementary way.

By an "inverse iteration" of Sacks forcing over a model of V = L, we produce a model in which the degrees of constructibility of nonconstructible reals have order type ω 1 *.