[bibtex key=Hamkins2012:TheSet-TheoreticalMultiverse]
The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.
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Very thought-provoking! I’m glad to see that you emphasized the ‘algebraic’ nature of ‘models of set theory’. For whatever it’s worth, I think that that’s proper. However, just because there exist exotic ‘subalgebras’ of V (deeming V to be a very generalized ‘algebra’) does not mean that the ‘exotic subalgebras’ constitute
the ‘proper subalgebras’ of V (I define a proper submodel of V to be any model–but no–taking your multiverse view how can one even come to an adequate conception of such, given that, according to you, the cumulative knowledge of the multiverse is necessary for a ‘true knowledge’ of ‘set’). Therein lies the problem. If a concept is perfectly elastic, what good is it?
It seems to me that the multitude of models of set theory, constructed (so to speak) by any one of a number of means has so stretched the meaning that
it has no coherent meaning at all, or at the very least, has no real usefulness for mathematics proper (for example, how would one go about defining a ‘generic’ Dedekind Cut?). Perhaps one could say that the models (of whatever kind) of ZFC, NF, Morse-Kelly, Godel-Bernays, etc. are simply the precise formulations of Cantor’s original definition of set (“By an ‘aggregate’ we are to understand any collection [of any kind whatsovever–my comment] into a whole M of definite and separate objects m of our intuition or our thought.”) and if that is the case, then certainly the multiverse view you espouse is unproblematic. But then which type of collection is to be adequate for the formulation of mathematics (how much classical mathematics could be formulated, say, in L)? What collection of models will now, in your view, be deemed as ‘standard’ (perhaps a synonym for proper)? What is to be the criteria?
I wish to correct a term that might make my previous comment somewhat confusing. Replace the term ‘proper’ in ‘proper subalgebra’, ‘proper model’, and ‘proper’ in the phrase “perhaps a synonym of proper” with the term ‘preferred’. The question then becomes, in essence, what are the criteria for distinguishing the preferred submodels of the set-theoretic universe from the exotic? Both the preferred submodels and the exotic submodels certainly exist, but only the preferrred submodels directly contribute to an understanding of the term ‘set’ and help determine its meaning. For my part, L would have to be the basis for constructing the preferred submodels of V as it is the minimal submodel of V (it would constitute, so to speak, the ‘skeletal structure’ of V) and the ‘preferred’ forcing extensions of L would have to add ‘all possible’ subsets A of a set X to P(X) just short of inconsistency so that the cardinal structure of L holds in all the preferred forcing extensions (that way the universal quantifier (A) in (For some B)(A)( If A is a subset of X then A is a member of B) would make P(X) absolute). Perhaps one could redefine P(X) via some inductive definition where the induction operator (call it P) could be applied to X to get more and more ‘new’ subsets of X in such a way that the closure ordinal would be an Aleph. Just speculation, of course, but a possible start ( in fact, one might characterize V as L plus forcing extensions since the ZFC -provable principle of forcing are exactly those in the modal theory S4.2 )….
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Hi Joel,
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.
Joseph,
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!
Now that you have mentioned it, I am thinking about whether one can get anything similar to virtual large cardinals from frames. The most obvious place to look for such large cardinals is the notions of compactness, complete metrizability, or the Baire category theorem, but these notions do not seem to produce new or existing large cardinal axioms.
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.