Re: [HoTT] A small observation on cumulativity and the failure of initiality

[Thierry Coquand <Thierry...@cse.gu.se>]

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 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 <homotopyt...@googlegroups.com> on behalf of Thomas Streicher <stre...@mathematik.tu-darmstadt.de>
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













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