In this talk we will explore the current debate regarding justification in the context of non-classical set theories. We will present the main arguments of Incurvati (2020) against paraconsistent set theories based on the naive conception of set, which we will present in the form of two maxims. We conclude that his arguments seems convincing and that we can extend his criticizm to paraconsistent set theories based on the iterative conception of set. In particular we show that ANY lattice-valued model which satisfies minimal properties (such as Extensionality and MP) fail to satisfy these maxims. (!)
So we are either left with the option to explore alternative model constructions for set theory or we need to rethink justification! We go for the second option and introduce a new account of justification based on the ideal element theory of Hilbert. We call this the ideal account of justification and argue that this account behaves much better with respect to paraconsistent set theory. Moreover, we claim that this account should provide a solid justification for paraconsistent set theories based on the iterative conception of set. We conclude with philosophical remarks about the future love childs of Hilbert and Da Costa.