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From: "Prof. Peter Johnstone"
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Subject: Re: Can you spot a flaw in this argument?
Date: Fri, 26 Jun 2009 09:32:47 +0100 (BST)
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Dear Gregory,
I think the problem resides in what you mean by "no internal structure".
If by this you mean that the *set* of names has no internal structure,
then it's obviously impossible -- to require the set to have decidable
equality is to impose internal structure on it. But it's still possible
for the names themselves to be atomic: you can identify the set of names
with the set of natural numbers (which has lots of internal structure,
including decidable equality) without identifying the individual names
with von Neumann-style natural numbers (or Russell-style cardinals,
or ...)
Peter Johnstone
On Thu, 25 Jun 2009, Meredith Gregory wrote:
> All,
>
> This is an argument about effective theories and so i would love to hear
> from the topos-theory crowd. i'll state the argument in lowly computational
> terms, using two distinct proposals for foundational models of computing:
> the lambda calculus and the ?-calculus. Both the lambda calculus and the
> ?-calculus suffer a dependence on a theory of names (aka variables). Both
> require two things of whatever theory of names is provided to them:
>
> - at least countably infinitely many names
> - an effective equality on names
>
> Now, to this recent proposals[1] add a third constraint, namely
>
> - names are atomic -- they have no internal structure
...
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