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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Bob, could you provide a little more background on TCA's, or a link where we can find out more? Is this post relevant: http://mathoverflow.net/a/57489/1946? – Joel David Hamkins Dec 09 '14 at 01:36
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A reference on TCAs: http://janwillemklop.nl/Jan_Willem_Klop/Bibliography_files/89.extendingpca.pdf – Joel David Hamkins Dec 09 '14 at 02:56
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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. – Robert Lubarsky Dec 09 '14 at 17:00
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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. – Robert Lubarsky Dec 09 '14 at 17:01