Truth as a homeomorphism group

#1

I was given a riddle in topology. For increased interest, it is clumsily reset in a prison.

There is a prison run by a crazed and tyrannical warden who plays increasingly complicates games with the prisoners’ souls. The prisoners, of course, have been trying to figure out some way of getting out for years. It started simple with ideas like digging a tunnel, but each simple idea spawned a dozen problems. Every solution brings even more… The problems have become increasingly abstract, and at this point the prisoners have developed a full mathematical theory of the problem of escape.

A brilliant prisoner has just released a new proof proving the ultimate problem of escape is homeomorphic to a new, possibly accessible problem. Reaction among the prisoners is muted, as few understand even the statement of the problem…

Define a homeomorphism group which represents the concept of “truth”. What is the space and what is the topology?

How do they escape?

1 Like
#2

I’ve been thinking about it a bit, but I’m very out of practice with topology.

True things are continuous functions, so perhaps statements in general are the functions? They are functions on what? Other statements, used as variables? Then what topology could make true statements continuous and others not??

If statements are functions and the space itself, is it it’s own function space? Does a function space from a set to itself necessarily have greater cardinality?

#3

I don’t have a lot to add at this time, but this is an interesting idea. I
I’m reminded of Gödel numbering, which might provide some additional context. Gödel’s work in general seems like a good launchpad for exploration of this stuff.

#4

Also see here.

1 Like
#5

Very interesting and relevant! Here is a crazy category theory extension: topos.

It also got me started on some other things I’ve been looking for:

A set which is closed under powerset!! A Gronthendieck Universe

I’ve started ascending Cantor’s attic…