Cohen real or random real: effect on strong measure zero sets and strongly meager sets
aa r X i v : . [ m a t h . L O ] M a y Cohen real or random real: effect on strong measurezero sets and strongly meager sets
Miguel A. CardonaInstitute of Discrete Mathematics and Geometry.Faculty of Mathematics and Geoinformation.TU Wien.
Abstract
We show that the set of the ground-model reals has strong measure zero (isstrongly meager) after adding a single Cohen real (random real). As consequencewe prove that the set of the ground-model reals has strong measure zero after addinga single Hechler real.
Let N be the σ -ideal of measure zero subsets of 2 ω , and let M be the σ -ideal of meagersets in 2 ω . More concretely X ∈ M if there is some sequence h F n : n < ω i such that X = S n<ω F n and int(cl( F n )) = ∅ . Let C and B be the Cohen algebra and random algebrarespectively, let D be the Hechler forcing, let L be the Laver forcing, let M be Mathiasforcing, let V be Silver forcing and let S be Sacks forcing. Definition 1.1.
For each σ ∈ (2 <ω ) ω define ht ∈ ω ω by ht σ ( n ) := | σ ( n ) | .Say that X ⊆ ω has strong measure zero ( X ∈ SN ) if, for each function f ∈ ω ω thereis some σ ∈ (2 <ω ) ω with ht σ = f such that X ⊆ S n<ω [ σ ( n )].It is clear that SN ⊆ N .Galvin, Mycielski and Solovay [GMS73] gave a very important description of the strongmeasure zero sets.
Theorem 1.2 ([GMS73]) . The following are equivalent:(1) X ∈ SN ,(2) for every set F ∈ M , there is some x ∈ ω such that ( x + X ) ∩ F = ∅ . Using this characterization, we consider the following objects.
Definition 1.3.
We say that X ⊆ ω is strongly meager ( X ∈ SM ) if, for each N ∈ N ,there is x ∈ ω such that ( X + x ) ∩ N = ∅ .It is clear that SM ⊆ M . 1unen [Kun84] proved that after adding a single Cohen real (random real) the set ofthe ground-model reals becomes null (meager). More presicely,
Theorem 1.4 ([Kun84]) . If c and r are a Cohen real and a random real over V respec-tively, then(i) V [ c ] | = 2 ω ∩ V ∈ N and ω ∩ V / ∈ M . In particular, V [ c ] | = 2 ω ∩ V / ∈ SM .(ii) V [ r ] | = 2 ω ∩ V ∈ M and ω ∩ V / ∈ N . In particular, V [ r ] | = 2 ω ∩ V / ∈ SN . Motivated by Theorem 1.4. in this paper we prove that the set of the ground-modelreals has strong measure zero after adding a single Cohen real. This was mentioned byLaver [Lav76] (without proof), afterwards, Goldstern sketched this in [Gol11]. We alsoprove that the set of the ground-model reals is strongly meager after adding a singlerandom real. This was sketched in [Wei13]. The author present a complete proof of theseresults with some slight variations associated with his perpective.
This section is dedicated to prove the following main result.
Theorem A. If c and r are a Cohen real and a random real over V respectively, then(i) V [ c ] | = 2 ω ∩ V ∈ SN .(ii) V [ r ] | = 2 ω ∩ V ∈ SM .Proof. (i) Enumerate 2 <ω := { r n : n < ω } . For each f ∈ ω ω and F ∈ ω ω define B cf,F := [ n ∈ ω [ r c ( F ( n )) a h , . . . , i ]where for each n , the length of h , . . . , i is the greatest between f ( n ) − | r c F ( n ) | and0. Note that B cf,F is coded in V [ c ]. It is enough to prove that, for any C -name ˙ f in ω ω there is a function F ∈ ω ω such that (cid:13) C ω ∩ V ⊆ B ˙ c ˙ f,F .In V define a function F p ∈ ω ω for each p ∈ C by F p ( m ) := min n k ∈ ω : ∃ q ∈ C ( | q | = k ∧ q ≤ p ∧ ∃ l < ω ( q (cid:13) ˙ f ( m ) = l )) o , Choose F ∈ ω ω such that F p ≤ ∗ F for all p ∈ C . It remains to check that (cid:13) C ω ∩ V ⊆ B ˙ c ˙ f,F . To do this, let p be an arbitrary condition in C . Choose n < ω suchthat F p ( m ) ≤ F ( m ) for all m ≥ n . Now choose q ∈ C with | q | = F p ( n ) and l < ω such that q extends p and q (cid:13) ˙ f ( n ) = l . Let x ∈ ω ∩ V . Find i < ω such that r i := x ↾ l . Define a condition q ∗ ∈ C such that | q ∗ | = F ( n ) + 1, q ∗ (cid:13) ˙ c ( F ( n )) = i and q ∗ ≤ q .Then, q ∗ (cid:13) x ∈ [ r ˙ c ( F ( n )) ] ⊆ B ˙ c ˙ f,F (this contention holds because | r ˙ c ( F ( n )) | = l = ˙ f ( n )).2ii) For an increasing function f ∈ ω ω and a function x ∈ ω define x f ∈ ω as x f ( n ) := x ( f ( n )) for n ∈ ω . Let A be a Borel set in V [ r ] ∩ N . In V find a Borel null setsuch that B ⊆ ω × ω and A = B r . Since B has measure zero, choose sequences s n , t n ∈ <ω with | s n | = | t n | such that B ⊆ \ m<ω [ n ≥ m [ s n ] × [ t n ] and ∞ X n =1 − | s n | < ∞ . Find an increasing function f ∈ ω ω by induction on n such that(a) j ≤ f ( n ) → | s j | < f ( n + 1).(b) P j ≥ f ( n ) Lb ([ s j ] × [ t j ]) ≤ Lb ([ s n ] × [ t n ])2 n +2 From (a) and (b) it follows that( ⋆ ) X f ( n ) ≤ j Note that H zn = [ s n ] ∩ [( z ↾ | t n | + t n ) ◦ f − ]. Let t ′ := ( z ↾ | t n | + t n ) ◦ f − . Then H zn = [ s n ] ∩ [( z ↾ | t n | + t n ) ◦ f − ] = ∅ when s n and t ′ are incompatible. Otherwise, H zn = [ s n ∪ (( z ↾ | t n | + t n ) ◦ f − )]= [ s n ∪ t ′ ]3ence, Lb (cid:16) [ s n ∪ t ′ ] (cid:17) = 2 −| s n ∪ t ′ | = 2 −| s n |−|{ f ( n ): n< | t n |∧ f ( n ) ≥| s n |}| ≤ −| s n |−| t n | + | f − [ | s n | ] = 2 | f − ( | s n | ) | | s n | . This ends the proof of Claim 2.1.We continue the proof of (ii). It follows that T m<ω S n ≥ m H zn has measure zero bythe Claim 2.1 and ( ⋆ ). In V [ r ], since r is a random real over V , h r, r f + z i 6∈ B ,which means that r f + z A . Therefore (2 ω ∩ V ) + A = 2 ω in V [ r ].As a consequence of Theorem A, we get that the set of the ground-model reals hasstrong measure zero after adding a single Hechler real. Corollary 2.2. If d is a Hechler real, then V [ d ] | = 2 ω ∩ V ∈ SN . Palumbo [Pal13] proved that D ∗ C ≡ D , that is, V [ d ′ ][ c ] = V [ d ] for some D -genericreal d ′ over V and a Cohen real c over V [ d ′ ]. By Theorem A, V [ d ′ ][ c ] | = 2 ω ∩ V [ d ′ ] ∈ SN ,in particular V [ d ′ ][ c ] | = 2 ω ∩ V ∈ SN . Then V [ d ] | = 2 ω ∩ V ∈ SN .The next result appears implicit in [JMS92]. Theorem 2.3. If G and G ′ are a V -generic over V and a S -generic over V respectively,then(a) V [ G ] | = 2 ω ∩ V / ∈ N ∪ M , in particular, V [ G ] | = 2 ω ∩ V / ∈ SN ∪ SM .(b) V [ G ′ ] | = 2 ω ∩ V / ∈ N ∪ M , in particular, V [ G ′ ] | = 2 ω ∩ V / ∈ SN ∪ SM . Miller [Mil81] introduced the infinitely often equal real forcing I to prove that somecombinatorial properties of measure and category of the real line are consistent. He alsoproved that the set of ground-model reals does not become meager (strongly null) afteradding a single infinitely often equal real, in particular, the ground-model real does notbecome strongly meager. To summarize, Theorem 2.4. If G is I -generic over V , then(i) V [ G ] | = 2 ω ∩ V / ∈ SN , and(ii) V [ G ] | = 2 ω ∩ V / ∈ SM . We finish this section with results related to the Laver property. Theorem 2.5 ([BJ94],[BJ95, Theorem 8.5.20]) . Assume that P has the Laver property.Then (cid:13) P ω ∩ V / ∈ SM . As a corollary we get Corollary 2.6. If G and G ′ are M -generic over V and L -generic over V respectively, then i) V [ G ] | = 2 ω ∩ V / ∈ SM .(ii) V [ G ′ ] | = 2 ω ∩ V / ∈ SM . On the other hand, Laver [Lav76] proved that adding an M -generic over the ground-model V forces all uncountable sets of reals in V to not have strong measure zero in theextension, that is, V [ G ] | = 2 ω ∩ V / ∈ SN .It is known that the set of the ground-model reals does not have measure zero afteradding a L -generic over V , that is, V [ G ] | = 2 ω ∩ V / ∈ N , in particular V [ G ] | = 2 ω ∩ V / ∈ SN . Open problems Miller [Mil81] proved that, if c is a Cohen real over V and r is a random real over V [ c ],then V [ c ][ r ] | = 2 ω ∩ V [ r ] / ∈ M , in particular V [ c ][ r ] | = 2 ω ∩ V [ r ] / ∈ SM . Afterwards,Cicho´n and Palikowski [CP86] proved that, if r is a random real over V and c is a Cohenover V [ r ], then V [ r ][ c ] | = 2 ω ∩ V [ c ] ∈ N . Later Palikowski [Paw86] proved that(i) If r is a random real over V and c is a Cohen over V [ r ], then V [ r ][ c ] | = 2 ω ∩ V [ c ] / ∈ M . In particular, V [ r ][ c ] | = 2 ω ∩ V [ c ] / ∈ SM .(ii) If c is a Cohen real over V and r is a random over V [ c ], then V [ c ][ r ] | = 2 ω ∩ V [ r ] ∈ N . We ask the following problems. Question 2.7. If c is a Cohen real over V and r is a random over V [ c ] , does V [ c ][ r ] | = 2 ω ∩ V [ r ] ∈ SN ? Question 2.8. If r is a random real over V and c is a Cohen over V [ r ] , does V [ r ][ c ] | = 2 ω ∩ V [ c ] ∈ SN ?In Corollary 2.2 it was proved that the ground-model real become strongly null afteradding a single Hechler real, but it is still open the following question. Question 2.9. If d is a Hechler real over V , does V [ d ] | = 2 ω ∩ V ∈ SM ? It is known that(a) (cid:13) ω ∩ V ∈ N .(b) (cid:13) ω ∩ V ∈ M .for the following posets:(1) The eventually different real forcing E .52) The localization forcing LOC .(3) Amoeba forcing A .It is natural to ask: Question 2.10. For the posets in the list above do we have(i) (cid:13) ω ∩ V ∈ SN ?(ii) (cid:13) ω ∩ V ∈ SM ? Acknowledgments This work was supported by the Austrian Science Fund (FWF) P30666 and the author isa recipent of a DOC Fellowship of the Austrian Academy of Sciences at the Institute ofDiscrete Mathematics and Geometry, TU Wien.This paper was developed for the conference proceedings corresponding to the SetTheory Workshop that Professor Daisuke Ikegami organized in November 2019. Theauthor is very thankful to Professor Ikegami for letting him participate in such wonderfulworkshop.The author thanks Dr. Diego Mej´ıa for his valuable comments while working on thispaper. The author also thanks Dr. Martin Goldstern for reading this work and for hisremark on references and grammar corrections. References [BJ94] Tomek Bartoszyski and Haim Judah. Borel images of sets of reals. Real AnalysisExchange , 20(2):536–558, 1994.[BJ95] Tomek Bartoszy´nski and Haim Judah. Set Theory: On the Structure of the Real Line .A K Peters, Wellesley, Massachusetts, 1995.[CP86] Jacek Cicho´n and Janusz Pawlikowski. On ideals of subsets of the plane and on Cohenreals. J. Symbolic Logic , 51(3):560–569, 09 1986.[GMS73] Fred Galvin, Jan Mycielski, and Robert Solovay. Strong measure zero sets. NoticesAmer. Math. Soc , 26, 1973.[Gol11] Martin Goldstern. https://mathoverflow.net/questions/64111, 2011.[JMS92] Haim Judah, Arnold W. Miller, and Saharon Shelah. Sacks forcing, Laver forcing, andMartin’s axiom. Arch. Math. Logic , 31(3):145–161, 1992.[Kun84] Kenneth Kunen. Random and Cohen reals. In Handbook of set-theoretic topology ,pages 887–911. North-Holland, Amsterdam, 1984.[Lav76] Richard Laver. On the consistency of Borel’s conjecture. Acta Math. , 137(3-4):151–169, 1976.[Mil81] Arnold W. Miller. Some properties of measure and category. Trans. Amer. Math.Soc. , 266(1):93–114, 1981. Pal13] Justin Palumbo. Unbounded and dominating reals in Hechler extensions. J. SymbolicLogic , 78(1):275–289, 03 2013.[Paw86] Janusz Pawlikowski. Why Solovay real produces Cohen real. J. Symbolic Logic ,51(4):957–968, 1986.[Wei13] Tomas Weiss. https://mathoverflow.net/questions/123693, 2013. Institute of Discrete Mathematics and GeometryFaculty of Mathematics and GeoinformationTU WienWiedner Hauptstrasse 8–10/104 A–1040 Wien Austria E-mail address : [email protected] URL