Thin–tall Boolean Algebras David Asper´ o University of East Anglia Workshop on Pure and Descriptive Set Theory (Poi) Torino, 25, 26 Sep. 2015
Finite–support iterations with symmetric systems as side conditions Proper forcing is nice: • Proper forcing notions preserve ! 1 . • Properness (due to Shelah) is preserved under countable support iterations. Hence, granted the existence of a supercompact cardinal, one can build a model of PFA, the forcing axiom for proper forcings relative to collection of @ 1 –many dense series (Baumgartner).
PFA has many consequences. One of them is 2 @ 0 = @ 2 . Problem: Force some consequence of PFA or, for that matter, something we can force by iterating proper forcing, together with 2 @ 0 > @ 2 .
Countable support iterations won’t do. In fact, at stages of uncountable cofinality we are adding generics, over all previous models, for Add ( 1 , ! 1 ) (= adding a Cohen subset of ! 1 ); in particular we are collapsing the continuum of all those previous models to @ 1 . Hence, in the final model necessarily 2 @ 0 @ 2 . Bigger support won’t work either: The preservation lemma for properness doesn’t work in this context. Finite–support iterations won’t work either; in fact, any finite–support iteration of non–c.c.c. forcings collapses ! 1 .
A solution : Use finite supports, together with countable elementary substructures of some H ( ✓ ) as side conditions affecting the whole iteration or initial segments of the iteration in order to ensure properness (the idea of using countable structures as side conditions in order to “force” a non–proper forcing to become proper is old (Todorˇ cevi´ c, 1980’s, implicit in work of Baumgartner (adding a club of ! 1 by finite con- ditions)), but this was not done in the context of actual iterations). Typically we will want our iteration to have the @ 2 –c.c. (after all we are interested in 2 @ 0 arbitrarily large).The natural approach of using finite 2 –chains of structures won’t work, though, since we have too many structures and would therefore lose the @ 2 –c.c. We will replace 2 –chains of structures by “matrices” of structures with suitable symmetry properties. If we start with CH and consider only iterands with the @ 2 –c.c., we might succeed.
Symmetric systems of elementary substructures Given a set N , � N will denote N \ ! 1 (the height of N ). Definition Let ✓ be a cardinal and T ✓ H ( ✓ ) (such that S T = H ( ✓ ) ). A finite set N ✓ [ H ( ✓ )] @ 0 is a T–symmetric system iff the following holds for all N , N 0 , N 1 2 N : (1) ( N ; 2 , Y ) 4 ( H ( ✓ ); 2 , T ) (2) If � N 0 = � N 1 , then there is a unique isomorphism Ψ N 0 , N 1 : ( N 0 ; 2 , T ) � ! ( N 1 ; 2 , T ) Furthermore, Ψ N 0 , N 1 is the identity on N 0 \ N 1 . (3) If � N 0 = � N 1 and N 2 N 0 \ N , then Ψ N 0 , N 1 ( N ) 2 N . (4) If � N 0 < � N 1 , then there is some N 0 1 2 N such that � N 0 1 = � N 1 and N 0 2 N 0 1 .
• Symmetric systems had previously been considered in (at least) work of Todorˇ cevi´ c, Abraham–Cummings and Koszmider. Again, not in the context of forcing iterations. • The def. of symmetric system guarantees that (4)’ if N 0 , N 1 2 N and � N 0 < � N 1 , then there is some N 0 0 2 N 1 \ N such that � N 0 0 = � N 0 and N 0 \ N 1 = N 0 \ N 0 0 . (In fact, N 0 1 , N 1 ( N 0 ) , where N 0 0 = Ψ N 0 1 2 N is such that 1 = � N 1 and N 0 2 N 0 � N 0 1 .) This property is important in many applications. Sometimes it is enough to keep (1)–(3) and weaken (4) to (4)’. The resulting object is called partial T–symmetric system .
Two amalgamation lemmas 1st amalgamation lemma : If N and N 0 are T –symmetric systems, ( S N ) \ ( S N 0 ) = X , and there are enumerations ( N i ) i < n and ( N 0 i ) i < n of N , N 0 , resp., for which there is an isomorphism [ [ N 0 ; 2 , N 0 Ψ : ( N ; 2 , N i , T , X ) � ! ( i , T , X ) then N [ N 0 is a T –symmetric system. 2nd amalgamation lemma : Let N be a T –symmetric system and M 2 N . Suppose M 2 M is a T –symmetric system such that N \ M ✓ M . Let N M ( M ) = N [ { Ψ M , M 0 ( N ) : N 2 M , M 0 2 N : � M 0 = � M } Then N M ( M ) is a T –symmetric system.
Corollaries Let Symm T = ( {N : N T –symmetric system } , ◆ ) Using 1st amalgamation lemma: Corollary 1 (CH) Symm T is @ 2 –Knaster. Corollary 2 (CH) Symm T adds new reals but preserves CH. Using 2nd amalgamation lemma: Corollary 3 Symm T is proper.
Iterating: General template of the constructions. Start with CH, let regular with 2 < κ = . Fix suitable T ✓ H ( ) . Let ( P α : ↵ ) be such that for all ↵ , a condition in P α is a pair q = ( F , ∆ ) such that: (1) F is a finite function such that dom ( F ) ✓ ↵ (dom ( F ) is the support of q ). (2) ∆ is a finite set of pairs ( N , � ) , where N 2 [ H ( )] @ 0 , � ↵ , � sup ( N \ ) , and where dom ( ∆ ) is a (partial) T –symmetric system ( � is the marker associated to N ). (3) For all � < ↵ , q | β := ( F � � , { ( N , min { � , � } ) : ( N , � ) 2 ∆ } ) is a P β –condition.
(4) For every ⇠ 2 dom ( F ) , q | ξ � P ξ F ( ⇠ ) 2 Φ ⇤ ( ⇠ ) where Φ ⇤ ( ⇠ ) is a P ξ –name for a suitable forcing, and Φ ⇤ ( ⇠ ) = Φ ( ⇠ ) if Φ ( ⇠ ) is a P ξ –name for a suitable forcing (and where Φ is a suitable bookkeeping function on ). (5) For every ⇠ 2 dom ( F ) and every ( N , � ) 2 ∆ , if ⇠ � and ⇠ 2 N , then q | ξ � P ξ F ( ⇠ ) is ( N [ ˙ G ξ ] , Φ ⇤ ( ⇠ )) –generic Given P α –conditions q 0 = ( F 0 , ∆ 0 ) , q 1 = ( F 1 , ∆ 1 ) , q 1 α q 0 iff (a) for every ( N , � ) 2 ∆ 0 there is some � 0 � � such that ( N , � 0 ) 2 ∆ 1 , (b) dom ( F 0 ) ✓ dom ( F 1 ) , and (c) for every ⇠ 2 dom ( F 0 ) , q 0 | ξ � P ξ F 1 ( ⇠ ) Φ ⇤ ( ξ ) F 0 ( ⇠ )
Typical properties (1) P β is always a complete suborder of P α whenever � < ↵ : Thanks to the markers � in ( N , � ) 2 ∆ . (2) Each P β is typically @ 2 –c.c.: This often uses CH and standard ∆ –system arguments as in the proof of Corollary 1. (3) Properness of P α : We define a sequence ( M α ) α κ of clubs of [ H ( )] @ 0 (of increasing “richness”); e.g., by picking increasing sequence ( ✓ α ) α κ of cardinals above and letting M α = { N ⇤ \ H ( ) : N ⇤ 4 H ( ✓ α ) , T , Φ , ( ✓ β ) β < α 2 N ⇤ }
Proof by induction on ↵ : Let p 2 P α \ N ⇤ , N ⇤ 4 H ( ✓ ) countable containing everything relevant. We build q ⇤ from q by essentially adding ( N , min { ↵ , sup ( N \ ) } ) to ∆ p . We argue that q ⇤ can be built and is ( N ⇤ , P α ) –generic. For this, let A 2 N ⇤ , A ✓ P α maximal antichain, and let q 2 P α extend both a condi- tion t 2 A and q ⇤ . We want to find r 2 N \ A , r compatible with q .
Case ↵ = 0: Corollary 3. Case ↵ = � + 1: Usually easy, since, by definition, q | β � P β F q ( � ) is N [ ˙ G β ] , Φ ⇤ ( � )) –generic (if � 2 dom ( F q ) ). Case ↵ 6 = 0 limit : The case when cf ( ↵ ) 6 = ! 1 is typically easy, since then there is � 2 � 2 N \ ↵ bounding the support of (some condition in A extended by) q (obvious when cf ( ↵ ) = ! , using that | A | @ 1 if cf ( ↵ ) � ! 2 ). Then we apply induction hypothesis to � : Working in N [ ˙ G σ ] , find r 2 A such that r | σ 2 ˙ G σ , r compatible with everything N [ ˙ G σ ] can see, dom ( F r ) \ [ � , ↵ ) = ; . By extending q | σ we can assume r 2 N (since, by induction hypothesis, q | σ is ( N [ ˙ G σ ] , P σ ) –generic). Now we can amalgamate q and r into a condition. In the case cf ( ↵ ) = ! 1 , go to the blackboard.
A typical application: Theorem (A.–Mota, A generalization of Martin’s Axiom, Israel J. Math., to appear) ( GCH) For every regular � ! 2 , there is a proper @ 2 –c.c. forcing notion forcing MA 1 . 5 < κ + 2 @ 0 = . Here MA 1 . 5 is the forcing axiom for the class of @ 1 . 5 –c.c. partial λ orders relative to collections of � –many dense sets, where P has the @ 1 . 5 –c.c. iff for every ✓ > | TC ( P ) | there is a club D ✓ [ H ( ✓ )] @ 0 such that for any finitely many N 0 , . . . , N m 2 D and every p 2 P , if p 2 N i for all i with � N i minimal among N 0 , . . . , N m , then there is q P p such that q is ( N k , P ) –generic for all k m . MA 1 . 5 of course extends MA λ but also has, for example, many λ consequences at the level of strong failures of Club Guessing at ! 1 .
Some nice spin–offs: For every , there is a homogeneous @ 2 –c.c. proper forcing Add B ( ) adding –many Baumgartner clubs to ! 1 (CH not needed!). Add B ( ) Add ( , ! ) Adding a Baumgartner club = Cohen forcing In particular, Add B ( ) has applications in the context of cardinal characteristics for ω 1 ! 1 and [ ! 1 ] @ 1 . Add B ( ) also figures prominently in the construction, in ZFC, of a forcing notion collapsing @ 3 but preserving all other cadinals (the existence of this forcing answers a 1983 question of Abraham, who built in ZFC a forcing collapsing @ 2 and preserving all other cardinals).
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