Re: Paper and slides on indefiniteness of CH

Dear Pen,

On Sat, 27 Sep 2014, Penelope Maddy wrote:

Dear Sy,

I fear that height actualism is not dead; surely there must be even a few Platonists out there, and for such people (they are not “nuts”!) I’d have to work a lot harder to make sense of the HP. Is the Height Actualism Club large enough to make that worth the effort? It would help a lot to know how the height actualists treat proper classes: are they all first-order definable? And how do they feel about “collections of proper classes”; do they regard that as nonsense?

I have no strong commitment to height actualism, but I did once think about proper classes as something other than what looks like just another few ranks in the hierarchy — something more like extensions of properties, so that they could be self-membered, for example.  My goal was to understand some of Reinhardt’s arguments this way, but it didn’t work for that job, so I left it behind.

So you generated IMH first, then developed the HP from it? Where did IMH come from?

I launched the (strictly mathematical) Internal Consistency Programme. A first-order statement is “internally consistent” if it holds in an inner model (assuming the existence of inner models with large cardinals). To be “internally consistent” is stronger than to be just plain old consistent, so new methods are needed to show that consistent statements are internally consistent (sometimes they are not) and there’s also a new notion of “internal consistency strength” (measured by large cardinals) that can differ from the usual notion of consistency strength. All of this work was of course about what first-order statements can hold in inner models so it was an obvious question to ask if one could “maximise” what is internally consistent. That is exactly the inner model hypothesis.

I see.  Thank you.

Can you remind us briefly why you withdrew your endorsement of IMH?

Because it only takes maximality in width into account and fails to consider maximality in height!

It this the problem of IMH implying there are no inaccessibles?

Yes, exactly!

We’re now out of my depth, though, so I hope we might hear others on this. E.g., it seems the countable models and the literal thickenings (as opposed to imaginary ‘thickenings’) have both dropped out of the picture.  ??

No, otherwise it wouldn’t be the Hyperuniverse Programme! (Recall that the Hyperuniverse is the collection of countable transitive models of ZFC.)

An important step in the HP for facilitating the math is the “Reduction to the Hyperuniverse”. Recall that we have reduced the discussion of “thickenings” of V to a magic theory in a logic called “V-logic” which lives in a slight “lengthening” V^+ of V, a model of KP with V as an element. In other words, the IMH (for example) is not first-order in V but it becomes first-order in V^+. But now that we’re first-order we can apply Loewenheim-Skolem to V^+! This gives a countable v and v^+ with the same first-order properties as V and V^+. What this means is that if we want to know if a first-order property follows from the IMH it suffices to show that it holds just in the countable v‘s whose associated v^+‘s see that v obeys the IMH. The move from V to v doesn’t change anything except now our “thickenings” of v with quotes are now real thickenings of v without quotes! So we can discard the v^+‘s with their magic theories and just talk boldly and directly about real thickenings of countable transitive models of ZFC. Fantasy has become reality.

In summary the moves are as follows: To handle the “thickenings” needed to make sense if the IMH we create a slight lengthening V^+ of V to make the IMH first-order, then apply Loewenheim-Skolem to reduce the problem of deriving first-order properties from the IMH to a study of countable transitive models together with their real thickenings. So in the end we get rid of “thickenings” altogether and can work the math on countable transitive models of ZFC, nice clean math inside the Hyperuniverse!

The above applies not just to the IMH but also to other HP-criteria.

I’m glad you asked this question!


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