On the closure of forcing

Dmytro Taranovsky dmytro at mit.edu
Wed Nov 16 14:14:32 EST 2022

Set-theoretical forcing ordinarily gives models that are in a sense close to the ground model M.  And yet, every set is generic (i.e. can be added by forcing) over some iterate of any iterable model with a Woodin cardinal.  In this posting, I will discuss how this is possible, along with the closure properties and limitations of forcing.

Forcing and a generic multiverse can be viewed as a modal logic, i.e. a logic about possibilities.  Compared to weaker modal logics, the closure of forcing in a sense matches the closure of the ground model M (or the part of M that is above the used poset P), so M can almost act as if the generic set G is really there.  Every true statement in the generic extension (even with parameters from the ground model) is forced by some condition, and the forcing relation is definable in the ground model.  And by using boolean-valued models, generic sets can be avoided entirely.  Despite this, with forcing, we can effectively act as if V is countable, and rearrange sets subject to the closure properties of the model, hence the power of forcing over the theory of the uncountable.

Over fine-structural models below Woodin cardinals (apparently even allowing *partial* extenders for many Woodin cardinals), the covering lemmas imply a closeness of M[G] to M, with the exceptions to closeness captured by systems of indiscernibles over M, the simplest of which is a Prikry sequence.  (However, forcing can add arbitrary information "out of sight"; for example, zero sharp (0^#) is constructible from some reals Cohen generic over the minimal transitive model of ZFC.)

However, Woodin cardinals make the systems of indiscernibles go wild (or in a sense, universal for the model) and in turn qualitatively change the possibilities for forcing:  For iterable currently well-understood models, one can force any Sigma^V_2 statement not prohibited by the closure properties of the model.

A large cardinal axiom is an assertion of symmetry:  For example, through elementary embeddings, a measurable cardinal kappa can be moved to any ordinal above kappa with sufficient reflection properties.  For a model M, iterability roughly asserts that these symmetries (though operating on M) also work for transformations outside of M; for example, in the above, kappa can be moved above On^M.  Thus, in a specific sense, iterability amounts to a genuineness of the large cardinal symmetries of M.

Stronger large cardinal properties allow iterates to be chosen to capture more of V, including for example knowledge of V-cofinalities of all ordinals (see my question "Complexity of L[cf]" https://mathoverflow.net/questions/279724/ and Sy-David Friedman's paper "Trapping cofinality" http://www.logic.univie.ac.at/~sdf/papers/trapping.cof.pdf ).

Measurable cardinals lead to an extender algebra (for details, see for example, "The extender algebra and Sigma^2_1-absoluteness" https://arxiv.org/abs/1005.4193 by Ilijas Farah), and in turn to an infinitary logic on which sets are consistent with the large cardinal symmetries/embeddings.  (Sets in M are consistent, but a set of integers outside of M might for example encode an ordinal moved by an extender in the algebra.)  Stronger large cardinal properties give stronger logics.  And assuming iterability, any set G can be made consistent by iterating M until the large cardinal structure is sufficiently symmetric with respect to G.

The special thing about a Woodin cardinal delta in a model M is that the complexity of the large cardinal structure below delta matches the complexity of M above delta, making it rich enough for genericity.  This manifests through the delta-cc condition of the extender algebra below delta (which makes it a complete boolean algebra), and in turn delta-cc genericity of all sets consistent with the algebra.  To be generic over an iterate of M, a set G must have enough look-alikes in M, roughly corresponding to questions about G that M can ask, and iteration can create a scaffold that allows the traces of G to be combined, allowing G to be forced.

However, there are different contexts to being Woodin, and different types of forcing give a different transition point from "M[G] must be close to M" to "G can be arbitrary for an iterate M' of M".  The more closure we require (both in the forcing notion and in the model), the more difficult it is to get G generic, and conversely, the more meaningful the genericity is.  For example:
* Unlike set forcing, general class forcing has few closure properties.  It is open whether 0^# is generic by tame forcing over some inner model M without 0^#.  However, because we have enough definability, such an M would have an iterated satisfaction relation for L (and hence cannot be L).
* Ord-cc class forcing has good closure properties, but still much weaker than set forcing.   If K^M is small enough, its sharp is not Ord-cc generic over M since by coiterating the possibilities for the sharp, M would identify it in the multiverse.  The transition to get V Ord-cc generic happens at "Ord is Woodin", which corresponds to the Stable Core (see "Capturing the Universe" http://www.logic.univie.ac.at/~sdf/papers/capture.pdf by Sy-David Friedman).  The transition at "Ord is Woodin" also applies to various theories with enriched closure of proper classes, with higher degree of closure corresponding to higher strength of "Ord is Woodin".
* An iterate of M_1 (the minimal iterable inner model with a Woodin cardinal) can get a desired set generic, but it can only do so once, and is limited in the kind of testing it can do on the generic; in a sense, its generic multiverse is not very closed.
* An iterate of M_omega can get a set X generic while still having M_omega(X) to test it (i.e. M_omega constructed above X); this symmetry is related to having a derived model of AD.
* An iterable inner model with a proper class of Woodin cardinals has an iterate over which every set in V is generic ( https://mathoverflow.net/questions/356597/inner-models-with-all-sets-generic ).  However, to also get Ord-cc genericity of V itself, we apparently need an iterable inner model with "Ord is Woodin limit of Woodin cardinals".
* A generalization of Ord-cc genericity of V is to also have that for every inaccessible delta, V_delta is generic by a delta-cc set subforcing.  I suspect that this corresponds to an iterate of a (conjectured) iterable inner model with indiscernible Woodin cardinals.
* Assuming CH, every iterable model M with a measurable Woodin cardinal is Sigma^2_1 correct since every subset of omega_1 is generic over some iterate of M for a forcing with certain closure properties.  (However, assuming iterability and CH, and analogously to (Sigma^1_1)^HYP = Pi^1_1, I think that for the minimal iterable inner model M with indiscernible Woodin cardinals, (Sigma^2_1)^M = (Pi^2_1)^V.)
* At the level of Woodin cardinals, generic absoluteness results are connected to universally Baire (uB) sets and (Delta^2_1)^uB.  We do not yet know what lies beyond.

Dmytro Taranovsky

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