Re: circular Godel Phi

MoeBlee wrote:

On Jul 27, 7:26 am, Aatu Koskensilta <aatu.koskensi...@xxxxxx> wrote:
MoeBlee <jazzm...@xxxxxxxxxxx> writes:
If I recall correctly, Enderton DEFINES 'recursive' as
'representable', so the representation lemma is thereby bypassed
(i.e., the substantive result is still unproven).

Interesting. But surely he does prove something to the effect that all
(Turing, register machine, whatever) computable functions are
representable?

Two points to be clear: I'm talking about the first edition; the
second edition might have more on this. And, I'm speaking off the cuff
without the book in front of me, so hope I'm not mistating the case.
That said, I think what we're talking about is the the circle: mu-
recursive/Turing computable/representable, right? As best I recall,
I've not seen a textbook that completes that circle, but rather only
mu-recursive/Turing computable (and certain others such as Markov
computable, register machine, Post computable, lambda definable, etc.)
but not also representable. I'd welcome to be shown incorrect on
this.

And I don't recall exactly about Shoenfield, but I don't remember
seeing proof of the representation lemma there either.

I have Shoenfield with me but I'm too lazy to check. My favourite
treatment of definability, representability, and all that, is in
Smorynski's _Logical Number Theory I_.

That one is on my "wish list". I'll move it higher up the list now. So
you say it has the proof of the "reperesentability lemma" (i.e., the
part Godel himself shrugs off in his famous incompleteness paper)?

Representability is also in Mendelson.

--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.
.

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