I conjecture that every regular frame becomes a Polish space instead of just metrizable after collapsing sufficiently many cardinals (I have a couple proof ideas and I think we would be more satisfied if frames become completely metrizable instead of just metrizable). I know that forcing does not add or take away compactness; every compact space remains compact after forcing (the forcing extension adds enough points to a compact space to keep it compact), and forcing cannot take a non-compact space and make it compact. Since sufficient forcing preserves compactness and should make everything completely metrizable, there is not much room to have a non-trivial purely topological virtual large cardinal (unless we consider forcings that do not collapse so many cardinals or other generalizations).

Perhaps, one should add “every regular frame becomes a Polish space in some larger and better universe” to the multiverse axioms.

]]>Over at https://cs.stackexchange.com/questions/82230 I asked a similar question (before discovering your post today) and it was suggested to interpret d/dx (f(x)) as D(lambda x.f(x))(x). The problem with that is, that the expression with the lambda contains a free variable x, while it is not clear that the expression d/dx (f(x)) does so too. (I could of course write d/dx (f(5)), but it is usually considered to be something different from D(lambda x.f(x))(5))

Anyway, thanks for highlighting some of the subtleties over at MO. I find it extremely difficult to discuss this with people, since most consider the problem trivial or useless. I find it quite interesting from the perspective of (computer) formalizing the differential calculus as used by physicist and engineers.

]]>This is a very interesting idea and I thank you for the comment.

I believe that this idea has some resonance with the consideration of virtual large cardinals. Victoria Gitman has worked on this and spoken about it at our seminars. The idea is that one has large-cardinal like embedings $j:M\to N$, but the embeddings $j$ have only a possible existence in that they exist in a forcing extension, not necessarily in $V$. You are proposing a similar thing with frames and virtual topological spaces.

I encourage you to pursue this further!

]]>Have you ever considered how, in the multiverse, moving from one universe to a larger and better universe may affect topological spaces (or uniform spaces, or proximity spaces for that matter)? I hear set theorists commonly talk about adding reals by forcing, but I do not recall hearing anything about how forcing extensions may add points to a frame (recall that a frame is a complete Heyting algebra, and frames model the open sets in a topological space, so you should think about frames as point-free topological spaces. For example, think of the set of all branches of height omega_1 in an Aronszajn tree as a point-free topological space where the basic open sets are all of those imaginary cofinal branches that contain a certain point.).

I personally think that a multiverse or generic multiverse philosophy will help clarify some issues that many may have with general topology and point-free topology (even to those who don’t adhere to such a multiverse philosophy) including the following:

1. Many people have trouble with the idea of point-free topology. How can a space not have any points in it? Conversely, people who are comfortable with point-free topology may be unfamiliar with forcing or skeptical about forcing.

2. It may be unclear what the correct axioms for a “good” topological space are (by “good”, I informally mean the topological spaces that satisfy enough separation axioms and which are reasonably well-behaved). It is easy to convince yourself that every metric space is a “good” topological space, but it is also clear that non-T1 spaces are not “good” topological spaces. Where do we draw the line?

3. Set theoretic topology has little in common with other branches of topology such as algebraic topology and the theory of manifolds.

To clarify these mathematical issues, let me propose the following characteristics or completely regular frames/spaces that we would like to see in a multiverse or generic multiverse.

i. A completely regular frame is a “completely regular space” whose points do not live necessarily in the universe V but whose points live in some larger universe.

ii. Passing from V to a larger universe sometimes adds points to a topological space and to frames but it never removes points from a topological space or from frames (for example, random forcing adds points to the space of all real numbers). In other words, topological spaces are objects which may have more points in better universes.

iii. Passing from a smaller universe to a larger universe does not add any new open sets to a frame or topological space unless these new open sets are simply unions of old open sets (the open sets become larger though when going to a larger universe since the open sets get more points).

iv. Every regular frame becomes a separable metrizable space in some larger universe.

v. Frame homomorphisms in V functorially produce continuous functions between topological spaces in larger universes.

vi. Uniform frames become metric spaces in larger universes.

If L is a frame, then the points in the frame L are the frame homomorphisms from L to the trivial Boolean algebra 2. Passing to a larger universe will sometimes add such homomorphisms and hence add points to the frame. Collapsing |L| to omega will make the frame into a metric space. Forcing therefore takes very bad spaces and frames (as long as they are regular) and turns them into good spaces that mathematicians can easily imagine (the formalization works better in the context of point-free topology).

From this realization of frames as metric spaces in forcing extensions that collapse cardinals, we have the answers to the three possible objections that I had originally listed.

1. Frames are simply topological spaces whose points live in forcing extensions. There is therefore no reason for any skepticism about point-free topology if one is comfortable with forcing extensions. Similarly, topologists would be more at ease with forcing if they simply consider forcing extensions to be the universes where all the points in point-free topology live.

2. Since regular spaces become metrizable after collapsing cardinals, one should consider regularity (or complete regularity) to be the line between a “bad” topological space and a “good” topological space (I consider complete regularity rather than regularity to be the line between a bad and a good topological space because completely regular spaces are precisely the uniformizable space and precisely the spaces with a compatible proximity. The problem with regularity is that regular spaces only become good in forcing extensions but they are just ok in the ground model).

3. Collapsing cardinals takes bad spaces and makes them nice so that one can use these various techniques to investigate these spaces.

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