On Tuesday, March 19, 2019 at 9:16:57 AM UTC-5, Bruno Marchal wrote:
> On 18 Mar 2019, at 20:33, Philip Thrift <cloud...@gmail.com <javascript:>> 
> wrote:
> Might have something of interest ...
> https://mathoverflow.net/questions/325702/connection-between-provability-logic-gl-and-geometry
> <https://www.google.com/url?q=https%3A%2F%2Fmathoverflow.net%2Fquestions%2F325702%2Fconnection-between-provability-logic-gl-and-geometry&sa=D&sntz=1&usg=AFQjCNF4OHbDm6tYoYD1Y55XxgK8qFi0TQ>
> ...
> There seems to be an existing literature on topological semantics and 
> provability logics. Thomas Icard has slides on this 
> <http://logic.berkeley.edu/colloquium/IcardSlides.pdf>, and the Stanford 
> Encyclopedia of Philosophy has a bit on the topic 
> <https://plato.stanford.edu/entries/logic-provability/#TopoSemaForProvLogi>. 
> I'm not sure how related this is to the specific topological/geometric 
> intuitions you give here, but it might be of interest more broadly. – Noah 
> Schweber <https://mathoverflow.net/users/8133/noah-schweber> 31 mins ago 
> <https://mathoverflow.net/questions/325702/connection-between-provability-logic-gl-and-geometry#comment813108_325702>
> I know the work of Leo Esakia, (and of Blok) notably his Russian paper (*) 
> and it is indeed quite interesting. It provides interesting topological 
> semantics for variants of K4 and GL (called G here). Icard has extended 
> such semantics for the logic GLP, which is a polymodal logic axiomatising 
> the provability logic on sequence of stronger and stronger theories. At 
> each level n we have a box [n] obeying to G, and with the two axioms:
> [n]p -> [n+1]p
> <n>p -> [n+1]<n>p
> for all n, Beklemishev has shown that this is complete for the 
> arithmetical interpretation extended on this succession of theories. I use 
> this in some more detailed exposition. 
> In the work of Esakia, the consistency <>p becomes a special sort of 
> abstract derivative. The idea of using topological model cale from the work 
> of McKinsey and Tarski, in the spirit of the algebraic semantics of Helena 
> Rasiowa and Roman Sikorski “The mathematics of Metamathematics” (an 
> excellent work on algebraic semantics). 
> The topological space, in the semantics of Esakia, are not Hausdorff 
> spaces, and are rather peculiar from a topologist standpoint, but 
> nevertheless, that work is quite interesting. It is a good complement of 
> the more traditional Kripke or Krpke-inspired semantics (which I use much 
> more).
> Unfortunately, although this will have interesting application for the 
> general theology of “growing machine”, the topology is not related directly 
> to the quantum “geometry” of the “material” variant of G (the Z and X 
> logics I have shown to exist here). 
> But thanks for this link. I missed those slides by Icard, and some 
> references therein.
> Esakia died in 2010, and was responsible for the great interest in 
> provability logic in Georgia (Europa).
> Bruno
> (*) Esakia, L. (1981). Diagonal constructions, Löb’s formula and Cantor’s 
> scattered space (Russian). In Studies in logic and semantics, pages 
> 128–143. Metsniereba, Tbilisi.
> - pt
There are now three links supplied in the MathOverflow post:

(SEP article)  

*Topological Semantics for Provability Logics*

*The Topological Structure of Asynchronous Computability*

- pt

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