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From: "Prof. Peter Johnstone"
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Subject: Re: Realizibility and Partial Combinatory Algebras
Date: Thu, 6 Feb 2003 10:44:33 +0000 (GMT)
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On Mon, 3 Feb 2003, Galchin Vasili wrote:
>
> Hello,
>
> I understand (to some degree) full combinatory algebra, but I don't
> understand the motivation behind the definition of a partial combinatory
> algebra. E.g. why do we have Sxy converges/is defined?
I'd like to know the answer to this too. Why does *everyone*, in writing
down the definition of a PCA, include the assumption that Sxy is always
defined? As far as I can see, the only answer is "because everyone else
does so"; the condition is never used in the construction of
realizability toposes, or in establishing any of their properties.
In every case where you need to know that a particular term Sab is
defined, it's easy to find a particular c such that Sabc is provably
defined.
Peter Johnstone