The German mathematician Gerhard Osius made important contributions to the topos-theoretic account of set theory and logic in the early 1970s.
Peter Johnstone, Topos Theory , Academic Press New York 1977 (Dover reprint 2014). (sections 9.2-3)
G. Osius, The internal and external aspect of logic and set theory in elementary topoi , Cah. Top. Géom. Diff. Cat. 14 no.2 (1973) pp.199-201 . (proceedings colloque Amiens 1973, 3.92 MB)
G. Osius, Categorical Set Theory: A Characterization of the Category of Sets , JPAA 4 (1974) pp.79-119.
G. Osius, The internal and external aspect of logic and set theory in elementary topoi, Cah. Top. Géom. Diff. Cat. 15 no.2 (1974) pp.157-180. (pdf)
G. Osius, Logical and set-theoretical tools in elementary topoi , pp.297-346 in Lawvere, Maurer (eds.), Model Theory and Topoi , LNM 445 Springer Heidelberg 1975.
G. Osius, A note on Kripke-Joyal semantics for the internal language of a topos , pp.349-354 in Lawvere, Maurer (eds.), Model Theory and Topoi , LNM 445 Springer Heidelberg 1975.
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