Re: Paper and slides on indefiniteness of CH

Dear Peter,

Before I forget I should mention now that tomorrow I will be off to a conference, so may not be responding to e-mail promptly for the rest of this week.

Thanks a lot for your messages; they are very helpful for sharpening the arguments that I am making. And I apologise if my description of radical potentialism caused so much confusion! Let me try to clarify it better in this mail.

Your second point is about “desirable properties”; let me address that first. The HP is aimed primarily at what is derivable from the intrinsic feature of maximality, i.e. it is concerned with the maximal iterative conception. But I also mentioned “omniscience”, which I do not see as derivable from maximality, at least no one has presented such an argument and I don’t know of one. If omniscience were to be included as intrinsic to the “concept of set” then Pen would have been right to say that we have changed to a different concept! I only used the word “desirable feature” informally to suggest that I find omniscience desirable, nothing more. I would very much like to hear suggestions about what practice-independent notions like omniscience should be regarded as “desriable”; I have no idea how to formulate that. Actually I am curious to know: Do you see it as a “desirable” feature of the universe of sets? Maybe you don’t want to talk about “desirable features” at all, and I can understand that.

OK, now to radical potentialism: Maybe it would help to talk first about something less radical: Width potentialism. In this any picture of the universe can be thickened, keeping the same ordinals, even to the extent of making ordinals countable. So for any ordinal alpha of V we can imagine how to thicken V to a universe where alpha is countable. So any ordinal is “potentially countable”. But that does not mean that every ordinal *is* countable! There is a big difference between universes that we can imagine (where our aleph_1 becomes countable) and universes we can “produce”. So this “potential countability” does not threaten the truth of the powerset axiom in V!

The standard form of potentialism can be viewed as a process of lengthening as opposed to thickening. Once again, there is no model of ZFC “at the end” because there is no “end”.

Now radical potentialism is in effect a unification of these two forms of potentialism. We allow V to be lengthened and thickened simultaneously. If we were to keep thickening to make every ordinal of V countable then after \text{Ord}(V) steps we are forced to also lengthen to reach a (picture of a) universe that satisfies ZFC. In that universe, the original V looks countable. But then we could repeat the process with this new universe until it is seen to be countable. The potentialist aspect is that we cannot end this process by taking the union of all of our pictures. In fact, whereas in the standard discussion of lengthenings there could be a debate about whether we can arrive at “the end”, if we allow both lengthenings and thickenings, potentialism is the only possibility; actualism is ruled out because the union of our “universes” would not be a model of ZFC and would therefore have to be lengthened further! And again, the “potential countability of V” does not threaten the truth of the axioms of ZFC in V!

Now in powerset and ordinal maximality we are not comparing V to pictures of other universes which see V as countable, even though there are such pictures. We are only looking at lengthenings that have V as a rank-initial segment and thickenings that have the same ordinals as V. From the perspective of a given V, these lengthenings and thickenings are only pictures of course, we are not talking about actual universes of sets, as those would be contained in V. But as I said in my last mail, even a platonist, with his own special V can imagine lengthenings and thickenings. It seems that I have some platonistically-leaning colleagues who discuss the set-generic multiverse surrounding V, which makes no sense if all universes are contained in V. The relevant set-generic extensions can be “pictured” but not “produced”. There are other constructions which take a countable universe and lengthen it, and doing this to V can also be “pictured” by a Platonist.

So set theory has not evaporated, CH is still a good problem. There is a huge wealth of pictures of V and some are “better” than others in the sense that some are better witnesses to maximality than others. The minimal model of ZFC is a terrible witness to maximality. A witness to the IMH does a much better job. In the HP we want to figure out which are the “best” witnesses to maximality. We may conclude that these “best” witnesses to maximality satisfy not CH, or we may conclude otherwise. It is too early to make such a judgment.

Now the next move (please see the outline) is to realise that if the spectrum of pictures is as rich as I describe, allowing V to “look countable” in pictures of larger universes, then in the “maximality test” where V is compared to pictures obtained through thickening and lengthening we might as well carry out this test inside a large picture of V where the orignal universe looks countable and where the lengthenings and thickenings you need actually exist as transitive models of ZFC. The result is that if you want to know if something first order holds in all universes that pass the “maximality test” you can simply assume that the test is taking place in the Hyperuniverse of some background V. (Of course depending on the choice of that backgound V, there may or may not exist universes that pass the maximality test.) This is the reduction of the problem to the Hyperuniverse, where these pictures can actually be realised as transitive models of ZFC.


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