[Haskell-cafe] References for topological arguments of programs?
Stuart A. Kurtz
stuart at cs.uchicago.edu
Mon Dec 10 14:43:43 UTC 2018
I've not been following this, but (a -> Bool) -> a is essentially a choice function, which figured in Ernst Zermelo's proof of the well-ordering theorem.
> On Dec 10, 2018, at 6:38 AM, Siddharth Bhat <siddu.druid at gmail.com> wrote:
> I was recently intrigued by this style of argument on haskell cafe:
> One can write a function
> Eq a => ((a -> Bool) -> a) -> [a]
> that enumerates the elements of the set. Because we have universal quantification, this list can not be infinite. Which makes sense, topologically: These so-called searchable sets are topologically compact, and the Eq constraint means the space is discrete. Compact subsets of a discrete space are finite.
> I've seen arguments like these "in the wild" during Scott topology construction and in some other weird places (hyperfunctions), but I've never seen a systematic treatment of this.
> I'd love to have a reference (papers / textbook preferred) to self learn this stuff!
> Sending this from my phone, please excuse any typos!
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