Balanced forcing extensions Paul Larson Department of Mathematics - PowerPoint PPT Presentation

Balanced forcing extensions Paul Larson Department of Mathematics Miami University Oxford, Ohio 45056 larsonpb@miamioh.edu November 18, 2018

P.B. Larson Balanced conditions Variations of balance Discontinuous homomor- phisms More forms of joint work with Jindˇ rich Zapletal balance Geometries Questions Models

P.B. Larson Solovay models Balanced conditions Variations of balance Suppose that κ is a strongly inaccessible cardinal, and that Discontinuous homomor- phisms G ⊆ Col ( ω, <κ ) More forms of balance is a V -generic filter. The resulting model Geometries Questions HOD V [ G ] Models V , P ( ω ) is a Solovay model (which we will call W ). We study forcing extensions of the Solovay model which recover forms of the Axiom of Choice.

P.B. Larson Suslin partial orders Balanced conditions Variations of Definition balance A preorder � P , ≤� is Suslin if there is a Polish space X such Discontinuous homomor- that phisms More forms of 1 P is an analytic subset of X ; balance Geometries 2 the ordering ≤ is an analytic subset of X 2 ; Questions 3 the incompatibility relation is an analytic subset of X 2 . Models We are mostly (but not only) interested in the case where P is σ -closed. We do not require ≤ to be antisymmetric, so different elements of P can represent the same condition in the separative quotient.

P.B. Larson Examples Balanced conditions Variations of balance • Countable partial functions from ω ω to 2 Discontinuous homomor- • Countable subsets of P ( ω ) with the finite intersection phisms property, under the relation of generating a larger filter More forms of balance ( P ( ω ) / Fin ) Geometries • Countable partial selectors for a Borel equivalence relation Questions Models • Countable subsets of R which are linearly independent over Q (adds a Hamel basis) • Countable almost disjoint families (of various kinds) • Disjoint pairs ( a , b ) ∈ [ R ] <ω × [ R ] ω , under containment (forces ¬ DC R )

P.B. Larson More examples Balanced conditions Variations of balance • Countable subsets of C which are algebraically Discontinuous independent over Q homomor- phisms • The poset of countable injections from the E -classes to More forms of balance the F -classes, for Borel equivalence relations E and F on Geometries Polish spaces. Questions • The poset of countable linear orders of E -classes, for a Models Borel equivalence relation E on a Polish space. • The poset of countable acyclic subsets of a Borel graph on a Polish space. • The post of countable partial k -colorings of a Borel graph on a Polish space, for k ∈ ω + 1.

P.B. Larson Virtual conditions Balanced conditions Variations of balance Discontinuous A virtual condition for a Suslin partial order P is a pair ( Q , τ ) homomor- phisms such that More forms of • Q is a partial order, balance Geometries • τ is a Q -name for an element of P , and Questions • τ realizes to an equivalent P -condition in every V -generic Models Q -extension. There is a natural notion of equivalence for virtual conditions : realizing to equivalent conditions in all generic extensions.

P.B. Larson Balanced conditions Balanced conditions Variations of balance Given a Suslin forcing P , a balanced condition for P is a pair Discontinuous homomor- ( Q , τ ) such that phisms • ( Q , τ ) is a virtual condition, More forms of balance • for any two V -generic filters G 0 and G 1 for Q existing Geometries respectively in mutually generic extensions V [ H 0 ] and Questions V [ H 1 ], and any two conditions Models p 0 ≤ τ G 0 and p 1 ≤ τ G 1 in V [ H 0 ] and V [ H 1 ] respectively, p 0 and p 1 are compatible in V [ H 0 , H 1 ].

P.B. Larson Balance Balanced conditions Variations of V [ H 0 , H 1 ] balance Discontinuous homomor- p 2 ≤ p 0 , p 1 phisms More forms of balance V [ H 0 ] V [ H 1 ] Geometries Questions p 0 ≤ τ G 0 p 1 ≤ τ G 1 Models V

P.B. Larson Example Balanced conditions Variations of balance Discontinuous For the partial order of countably generated filters, the homomor- phisms balanced pairs are (up to equivalence) the pairs ( Q , τ ) where More forms of Q = Col ( ω, 2 ℵ 0 ) and τ is a Q -name for an enumeration of an balance Geometries ultrafilter on ω in V . Questions Models Sketch : If U is an ultrafilter on ω , the union of two mutually generic filters containing U has the finite intersection property, since any name for a member of either of these filters must have U -many possible members below each condition.

P.B. Larson More examples Balanced conditions • For the partial order of countable partial selectors for a Variations of balance pinned equivalence relation, the balanced pairs are the Discontinuous Col ( ω, 2 ℵ 0 )-names for enumerations of the total selectors. homomor- phisms • For the partial order of countable partial functions from X More forms of balance to 2 (for X a Polish space), the balanced pairs are the Geometries Col ( ω, 2 ℵ 0 )-names for (codes for) total functions from X Questions to 2 in V . Models • For the partial order of countable linearly independent subsets of R over Q , the balanced pairs are the Col ( ω, 2 ℵ 0 )-names for enumerations of Hamel bases. • If E is a Borel equivalence relation, and P is the partial order of countable partial tournaments on the E -classes, the balanced conditions are classified by the total tournaments on the virtual E -classes.

P.B. Larson Even more examples Balanced conditions Variations of balance Discontinuous • For the partial order of disjoint finite/countable pairs of homomor- phisms reals, the balanced conditions are characterized by finite More forms of sets and their complements. balance Geometries • The partial order of countably generated filters disjoint Questions from a given F σ ideal. There may be ultrafilters disjoint Models from the ideal which do not give balanced conditions. • Does the partial order of countable subsets of C algebraically independent over Q have balanced conditions?

P.B. Larson Balanced partial orders Balanced conditions Variations of balance Discontinuous A Suslin partial order is said to be balanced if below each homomor- phisms condition there is a balanced virtual condition. More forms of balance Geometries Balance is not in general absolute between V and its forcing Questions extensions. Models We say that P is cofinally balanced below κ if every partial order in V κ is regularly embedded in one forcing that P is balanced in V κ .

P.B. Larson Henle-Mathias-Woodin for Balanced cofinally balanced partial orders conditions Variations of balance Discontinuous homomor- Theorem phisms If More forms of balance • P is a Suslin order, cofinally balanced below a strongly Geometries inaccessible cardinal κ (in V ), Questions • α is an ordinal, Models • W is a Solovay model for κ and • G ⊆ P is W -generic, then P ( α ) ∩ W [ G ] ⊆ W .

Proof. P.B. Larson Suppose that, in W , some condition p ∈ P forces some Balanced P -name σ to represent a subset of α . In the Levy extension, σ conditions is definable from some z ⊆ ω and an element of V . Variations of balance Discontinuous homomor- Let V [ K ] be a forcing extension of V contained in W such phisms that p , z ∈ V [ K ] and such that P is balanced in V [ K ]. Let More forms of balance ( Q , τ ) ∈ V κ [ K ] be a balanced virtual condition below p . Geometries Questions Fix β < α and suppose there exist Q × Col ( ω, <κ )-names Models ρ 0 , ρ 1 in V [ K ] for conditions below the realization of τ forcing different truth values to the statement ˇ β ∈ σ . Since ( Q , τ ) is balanced, there are mutually generic extensions V [ K ][ H 0 ] and V [ K ][ H 1 ] in which the realizations of ρ 0 and ρ 1 are compatible, giving a contradiction. It follows that, in W , any V [ K ]-generic realization of τ forces that σ G ∈ V [ K ].

P.B. Larson Consequences of balance Balanced conditions Theorem Variations of balance If P is cofinally balanced below a strongly inaccessible cardinal Discontinuous κ , W is a Solovay model for κ and G ⊆ P is W -generic, then homomor- phisms the following hold in W [ G ] : More forms of balance • every wellordered sequence of elements of W is in W ; Geometries • if E and F are Borel equivalence relations such that E is Questions pinned and F is unpinned, then | F | �≤ | E | ; Models • uniformization fails for sets whose cross-sections are equivalence classes of a fixed unpinned Borel equivalence relation; • there are no infinite MAD families on ω ; • if P is σ -closed, there are no unbounded linearly ordered subsets of ( ω ω , ≤ ∗ ) or the Turing degrees.

P.B. Larson Products Balanced conditions Variations of balance Discontinuous homomor- phisms More forms of balance Geometries Balance is preserved under countable (support) products. Questions Models

P.B. Larson Weak balance Balanced conditions Variations of balance Discontinuous homomor- phisms More forms of We get a weaker notion of balance if we require compatibility balance Geometries for some pair of extensions V [ H 0 ], V [ H 1 ] existing in a common Questions extension instead of requiring compatibility in all mutually Models generic extensions.