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Conservative extension by surjective pairing
I hae a question related to your LICS '89 paper which we just had
the pleasure of presenting in our research seminar.
You establish that surjective pairing axioms conservatively extend
lambda-beta conversion, and mention that this question is open for
lambda-beta-eta conversion.
Of course there are lots of other lambda-beta, indeed lambda-beta-eta,
theories where conservative extension HOLDS simply because the theory
consists of the valid equations in some lambda model in which surjective
pairing functions exist, e.g,, D_infinity. Also, I have no doubt there
are other theories besides lambda-beta-eta where conservativity is OPEN.
Is there any theory for which conservative extension by surjective pairing
is know to FAIL? If not, would you conjecture that surjective pairing
axioms yield a conservative extension of EVERY lambda-beta-theory?
Regards, A.