A quite detailed interpretation of type theory in set theory is presented in the paper of Peter Aczel "On Relating Type Theories and Set Theories". On can define directly the interpretation of lambda terms (provided that abstraction is typed), and using set theoretic coding, one can use a global application operation (so that the interpretation is total). One can then checked by induction on derivations that all judgements are valid for this interpretation. Thierry ________________________________ From: homotopyt...@googlegroups.com on behalf of Thomas Streicher Sent: Sunday, October 15, 2017 9:45:30 AM To: Gabriel Scherer Cc: Michael Shulman; Steve Awodey; Dimitris Tsementzis; Homotopy Type Theory Subject: Re: [HoTT] A small observation on cumulativity and the failure of initiality It is certainly tempting to assign meanings to derivations and then to show that different derivations of the same judgement get assigned the same meaning. When writing my thesis in the second half of the 80s I found this too difficult and instead used an a priori partial interpretation function assigning meaning to prejudgements. It was then part of the correctness theorem that all derivable judgements get assigned a meaning. Most people have gone this way at least when they took pains to write down the interpretation function at all. BTW the above allows one to show that assigning a meaning to derivations just depends on the judgement: one jyust has to show that the meaning assigned to a derivation of a judgement J coincides with the meaning assigned to J beforehand. Thomas -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeThe...@googlegroups.com. For more options, visit https://groups.google.com/d/optout.