Constructive Zermelo-Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-Löf type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice.
Rathjen and Tupailo REF make a close analysis of the interpretation of CZF into Martin-Löf type theory, and the question about what general classes of set-theoretic statements are validated in type theory.
Characterizing the interpretation of set theory in Martin-Löf type theory
Ann. Pure Appl. Log.
Ann. Pure Appl. Log.
Computer Science Mathematics