Is there a natural, non-trivial example of a TCA (total combinatory algebra, cf. pca) with a natural notion of an oracle?
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$\begingroup$ Bob, could you provide a little more background on TCA's, or a link where we can find out more? Is this post relevant: mathoverflow.net/a/57489/1946? $\endgroup$– Joel David HamkinsCommented Dec 9, 2014 at 1:36
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$\begingroup$ A reference on TCAs: janwillemklop.nl/Jan_Willem_Klop/Bibliography_files/… $\endgroup$– Joel David HamkinsCommented Dec 9, 2014 at 2:56
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$\begingroup$ Joel, those are some great references. I don't know of any standard reference. Pca's are used in realizability, so any reference work including realizability would have to talk about them. They're in Beeson's book, and likely van Oosten's "Realizability." I'd think they'd be in any good book on the lambda-calculus too. $\endgroup$– Robert LubarskyCommented Dec 9, 2014 at 17:00
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$\begingroup$ I'm surprised that Andrej's suggestion of just defining "divergence" as 42 (with the Turing functions, K_1) actually works. So that's not a bad example, but noticeably unnatural, even if just in that one spot. $\endgroup$– Robert LubarskyCommented Dec 9, 2014 at 17:01
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