Compact Cardinals and Eight Values in Cichoń's Diagram
aa r X i v : . [ m a t h . L O ] A p r COMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM
JAKOB KELLNER, ANDA RAMONA TĂNASIE, AND FABIO ELIO TONTIA
BSTRACT . Assuming three strongly compact cardinals, it is consistent that ℵ < add( ) < cov( ) < 𝔟 < 𝔡 < non( ) < cof( ) < ℵ . Under the same assumption, it is consistent that ℵ < add( ) < cov( ) < non( ) < cov( ) < non( ) < cof( ) < ℵ . I NTRODUCTION
We assume the reader is familiar with the definitions and some basic properties (whichcan all be found, e.g., in [BJ95]) of the cardinal characteristics in Cichoń’s diagram: cov( ) / / non( ) / / cof( ) / / cof( ) / / ℵ 𝔟 / / O O 𝔡 O O ℵ / / add( ) / / O O add( ) / / O O cov( ) / / O O non( ) O O An arrow between 𝔵 and 𝔶 indicates that ZFC proves 𝔵 ≤ 𝔶 . Moreover, max( 𝔡 , non( )) =cof( ) and min( 𝔟 , cov( )) = add( ) . These are the only “simple” restrictions in thefollowing sense: every assignment of ℵ and ℵ to the entries of Cichoń’s diagram thathonors these restrictions can be shown to be consistent. It is more challenging to getmore than two simultaneously different values, for recent progress in this direction see,e.g., [Mej13, GMS16, FGKS17].This paper consists of two parts: In the first one, we present a finite support ccc iteration 𝑃 forcing that ℵ < add( ) < cov( ) < 𝔟 < 𝔡 = 2 ℵ (and actually something stronger,cf. Lemmas 1.18 and 1.20). This is nothing new: The forcing and all required propertieswere presented in [Mej13]. We recall all the facts that are required for our result, in a formconvenient for our purposes.In the second part, we investigate the (iterated) Boolean ultrapower 𝑃 of 𝑃 . Assumingthree strongly compact cardinals, this ultrapower (again a finite support ccc iteration) forces ℵ < add( ) < cov( ) < 𝔟 < 𝔡 < non( ) < cof ( ) < ℵ , i.e., we get the following values in the diagram (for some increasing cardinals 𝜆 𝑖 ): 𝜆 / / / / / / 𝜆 / / 𝜆 𝜆 / / O O 𝜆 O O ℵ / / 𝜆 / / O O / / O O / / O O 𝜆 O O Date : April 18, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Set theory of the reals, Cichoń’s Diagram, Large continuum, Large cardinals.
It seems unlikely that the large cardinals assumption is actually needed, but we would expecta proof without it to be considerably more complicated.The kind of Boolean ultrapower that we use was investigated in [Man71], and recentlyapplied, e.g., in [MS16] and [RS] (where a Boolean ultrapower of a forcing notion is ap-plied to cardinal characteristics of the reals). Recently Shelah developed a method of usingBoolean ultrapowers to control characteristics in Cichoń’s diagram. The current paper is arelatively simple application of these methods. A more complicated one, in an upcomingpaper [GKSnt] by Goldstern, Shelah and the first author, shows that all entries in Cichońsdiagram can be pairwise different.
Acknowledgments.
We are grateful to Martin Goldstern and Saharon Shelah for valuablediscussions.We would like to thank an anonymous referee for their quick, insightful and kind review,and for pointing out a few embarrassing typos and mistakes.Supported by the Austrian Science Fund (FWF): P26737 and P30666. The second andthird authors are both recipients of a DOC Fellowship of the Austrian Academy of Sciencesat the Institute of Discrete Mathematics and Geometry, TU Wien.1. T
HE INITIAL FORCING
Good iterations.
The forcing 𝑃 we are about to define has many pleasant propertiesbecause it is “good”, a notion first explored in [JS90] and [Bre91]. We now recall the basicfacts of good iterations, and specify the instances of the relations we use. Assumption 1.1.
We will consider binary relations R on 𝑋 = 𝜔 𝜔 (or on 𝑋 = 2 𝜔 ) thatsatisfy the following: There are relations R 𝑛 such that R = ⋃ 𝑛 ∈ 𝜔 R 𝑛 , each R 𝑛 is a closedsubset (and in fact absolutely defined) of 𝑋 × 𝑋 , and for 𝑔 ∈ 𝑋 and 𝑛 ∈ 𝜔 , the set { 𝑓 ∈ 𝑋 ∶ 𝑓 R 𝑛 𝑔 } is nowhere dense. Also, for all 𝑔 ∈ 𝑋 there is some 𝑓 ∈ 𝑋 with 𝑓 R 𝑔 . We will actually use another space as well, the space of strictly positive rational se-quences ( 𝑞 𝑛 ) 𝑛 ∈ 𝜔 such that ∑ 𝑛 ∈ 𝜔 𝑞 𝑛 ≤ . It is easy to see that is homeomorphic to 𝜔 𝜔 ,when we equip the rationals with the discrete topology and use the product topology.We use the following instances of relations R on 𝑋 ; it is easy to see that they all satisfythe assumption (in case of 𝑋 = we use the homeomorphism mentioned above): Definition 1.2. 𝑋 = : 𝑓 R 𝑔 if (∀ ∗ 𝑛 ∈ 𝜔 ) 𝑓 ( 𝑛 ) ≤ 𝑔 ( 𝑛 ) .(We use “ ∀ ∗ 𝑛 ∈ 𝜔 ” for “ (∃ 𝑛 ∈ 𝜔 ) (∀ 𝑛 > 𝑛 ) ”.)2. 𝑋 = 2 𝜔 : 𝑓 R 𝑔 if (∀ ∗ 𝑛 ∈ 𝜔 ) 𝑓 ↾ 𝐼 𝑛 ≠ 𝑔 ↾ 𝐼 𝑛 ,where ( 𝐼 𝑛 ) 𝑛 ∈ 𝜔 is the increasing interval partition of 𝜔 with | 𝐼 𝑛 | = 2 𝑛 +1 .3. 𝑋 = 𝜔 𝜔 : 𝑓 R 𝑔 if (∀ ∗ 𝑛 ∈ 𝜔 ) 𝑓 ( 𝑛 ) ≤ 𝑔 ( 𝑛 ) .We say “ 𝑓 is bounded by 𝑔 ” if 𝑓 R 𝑔 ; and, for ⊆ 𝜔 𝜔 , “ 𝑓 is bounded by ” if (∃ 𝑦 ∈ ) 𝑓 R 𝑦 . We say “unbounded” for “not bounded”. (I.e., 𝑓 is unbounded by if (∀ 𝑦 ∈ ) ¬ 𝑓 R 𝑦 .) We call an R -unbounded family, if ¬(∃ 𝑔 ) (∀ 𝑥 ∈ ) 𝑥 R 𝑔 , and an R -dominating family if (∀ 𝑓 ) (∃ 𝑥 ∈ ) 𝑓 R 𝑥 . Let 𝔟 𝑖 be the minimal size of an R i -unboundedfamily, and 𝔡 𝑖 of an R i -dominating family.We only need the following connection between R i and the cardinal characteristics: Lemma 1.3. add( ) = 𝔟 and cof( ) = 𝔡 . cov( ) ≤ 𝔟 and non( ) ≥ 𝔡 . 𝔟 = 𝔟 and 𝔡 = 𝔡 . OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 3
Proof. (3) holds by definition. (1) can be found in [BJ95, 6.5.B]. To prove (2), note that forfixed 𝑔 ∈ 2 𝜔 the set { 𝑓 ∈ 2 𝜔 ∶ ¬ 𝑔 R 𝑓 } is a null set, call it 𝑁 𝑔 . Let be an R -unboundedfamily. Then { 𝑁 𝑔 ∶ 𝑔 ∈ } covers 𝜔 : Fix 𝑓 ∈ 2 𝜔 . As 𝑓 does not bound , there is some 𝑔 ∈ unbounded by 𝑓 , i.e., 𝑓 ∈ 𝑁 𝑔 . Let 𝑋 be a non-null set. Then 𝑋 is R -dominating:For any 𝑔 ∈ 2 𝜔 there is some 𝑥 ∈ 𝑋 ⧵ 𝑁 𝑔 , i.e., 𝑔 R 𝑥 . (cid:3) Definition 1.4. [JS90] Let 𝑃 be a ccc forcing, 𝜆 an uncountable regular cardinal, and R asabove. 𝑃 is (𝐑 , 𝝀 ) -good , if for each 𝑃 -name 𝑟 ∈ 𝜔 𝜔 there is (in 𝑉 ) a nonempty set ⊆ 𝜔 𝜔 of size <𝜆 such that every 𝑓 (in 𝑉 ) that is R -unbounded by is forced to be R -unboundedby 𝑟 as well.Note that 𝜆 -good trivially implies 𝜇 -good if 𝜇 ≥ 𝜆 are regular.How do we get good forcings? Let us just quote the following results: Lemma 1.5.
A FS iteration of Cohen forcing is good for any (R , 𝜆 ) , and the compositionof two (R , 𝜆 ) -good forcings is (R , 𝜆 ) -good.Assume that ( 𝑃 𝛼 , 𝑄 𝛼 ) 𝛼<𝛿 is a FS ccc iteration. Then 𝑃 𝛿 is (R , 𝜆 ) -good, if each 𝑄 𝛼 is forcedto satisfy the following: For
R = R : | 𝑄 𝛼 | < 𝜆 , or 𝑄 𝛼 is 𝜎 -centered, or 𝑄 𝛼 is a sub-Boolean-algebra ofthe random algebra. For
R = R : | 𝑄 𝛼 | < 𝜆 , or 𝑄 𝛼 is 𝜎 -centered. For
R = R : | 𝑄 𝛼 | < 𝜆 .Proof. (R , 𝜆 ) -goodness is preserved by FS ccc iterations (in particular compositions), asproved in [JS90], cf. [BJ95, 6.4.11–12]. Also, ccc forcings of size <𝜆 are (R , 𝜆 ) -good [BJ95,6.4.7], which takes care of the case of Cohens and of | 𝑄 𝛼 | < 𝜆 . So it remains to show that(for 𝑖 = 1 , ) the “large” iterands in the list are (R i , 𝜆 ) -good. For R this follows from [JS90]and [Kam89], cf. [BJ95, 6.5.17–18]. For R this is proven in [Bre91]. (cid:3) Lemma 1.6.
Let 𝜆 ≤ 𝜅 ≤ 𝜇 be uncountable regular cardinals. After forcing with 𝜇 many Cohen reals ( 𝑐 𝛼 ) 𝛼 ∈ 𝜇 , followed by an (R , 𝜆 ) -good forcing, we get: For every real 𝑟 in the final extension, the set { 𝛽 ∈ 𝜅 ∶ 𝑐 𝛽 is unbounded by 𝑟 } is cobounded in 𝜅 . I.e., (∃ 𝛼 ∈ 𝜅 ) (∀ 𝛽 ∈ 𝜅 ⧵ 𝛼 ) ¬ 𝑐 𝛽 R 𝑟 . (The Cohen real 𝑐 𝛽 can be interpreted both as Cohen generic element of 𝜔 and as Cohengeneric element of 𝜔 𝜔 ; we use the interpretation suitable for the relation R .) Proof.
Work in the intermediate extension after 𝜅 many Cohen reals, let us call it 𝑉 𝜅 . Theremaining forcing (i.e., 𝜇 ⧵ 𝜅 many Cohens composed with the good forcing) is good; soapplying Definition 1.4 we get (in 𝑉 𝜅 ) a set of size <𝜆 .As the initial Cohen extension is ccc, and 𝜅 ≥ 𝜆 is regular, we get some 𝛼 ∈ 𝜅 suchthat each element 𝑦 of already exists in the extension by the first 𝛼 many Cohens, callit 𝑉 𝛼 . The set of reals 𝑀 𝑦 bounded by 𝑦 is meager (and absolute). Any 𝑐 𝛽 for 𝛽 ∈ 𝜅 ⧵ 𝛼 is Cohen over 𝑉 𝛼 , and therefore not in 𝑀 𝑦 , i.e., not bounded by 𝑦 . As this holds for all 𝑦 , 𝑐 𝛽 is unbounded by , and thus, according to the definition of good, unbounded by 𝑟 aswell. (cid:3) In the light of this result, let us revisit Lemma 1.3 with some new notation:
Definition 1.7.
For 𝑖 = 1 , , , 𝜆 > ℵ regular, and 𝑃 a ccc forcing notion, let ⊚ i ( 𝑃 , 𝜆 ) stand for: “There is a sequence ( 𝑥 𝛼 ) 𝛼 ∈ 𝜆 of 𝑃 -names such that for every 𝑃 -name 𝑦 we have (∃ 𝛼 ∈ 𝜆 ) (∀ 𝛽 ∈ 𝜆 ⧵ 𝛼 ) 𝑃 ⊩ ¬ 𝑥 𝛽 R i 𝑦 .” Lemma 1.8. ⊚ i ( 𝑃 , 𝜆 ) implies 𝔟 𝑖 ≤ 𝜆 and 𝔡 𝑖 ≥ 𝜆 . In particular: OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 4 ⊚ ( 𝑃 , 𝜆 ) implies 𝑃 ⊩ ( add( ) ≤ 𝜆 & cof( ) ≥ 𝜆 ) . ⊚ ( 𝑃 , 𝜆 ) implies 𝑃 ⊩ ( cov( ) ≤ 𝜆 & non( ) ≥ 𝜆 ) . ⊚ ( 𝑃 , 𝜆 ) implies 𝑃 ⊩ ( 𝔟 ≤ 𝜆 & 𝔡 ≥ 𝜆 ) .Proof. The set { 𝑥 𝛼 ∶ 𝛼 ∈ 𝜆 } is certainly forced to be R i -unbounded; and given a set 𝑌 = { 𝑦 𝑗 ∶ 𝑗 < 𝜃 } of 𝜃 < 𝜆 many 𝑃 -names, each has a bound 𝛼 𝑗 , so for any 𝛽 ∈ 𝜆 aboveall 𝛼 𝑗 we get 𝑃 ⊩ ¬ 𝑥 𝛽 R i 𝑦 𝑗 for all 𝑗 ; i.e., 𝑌 cannot be dominating. (cid:3) Ground model Borel functions, partial random forcing.
The following lemmaseems to be well known (but we are not aware of a good reference or an established no-tation):
Definition 1.9.
Let 𝑄 be a forcing notion, and let 𝜂 be a 𝑄 -name for a real. We say that 𝑄 is “generically Borel determined (by 𝜂 , via 𝐵 )”, if ∙ 𝑄 consists of reals, ∙ the 𝑄 -generic filter is determined by the real 𝜂 , and moreover: ∙ 𝐵 ⊆ ℝ is a Borel relation such that for all 𝑞 ∈ 𝑄 , 𝑄 ⊩ ( 𝐵 ( 𝑞, 𝜂 ) ↔ 𝑞 ∈ 𝐺 ) .We investigate iterations of such forcings: Lemma 1.10.
Assume that ( 𝑃 𝛽 , 𝑄 𝛽 ) 𝛽<𝛼 is a FS ccc iteration such that each 𝑄 𝛽 is gener-ically Borel determined (in an absolute way already fixed in 𝑉 ). Then for each 𝑃 𝛼 -name 𝑟 of a real, there is (in the ground model) a Borel function 𝐹 ∶ ℝ 𝜔 → ℝ and a sequence ( 𝛼 𝑖 ) 𝑖 ∈ 𝜔 of ordinals in 𝛼 such that 𝑃 𝛼 forces 𝑟 = 𝐹 (( 𝜂 𝛼 𝑖 ) 𝑖 ∈ 𝜔 ) .Proof. We prove by induction on 𝛾 ≤ 𝛼 : ∙ For all 𝑝 ∈ 𝑃 𝛾 there is a Borel relation 𝐵 𝑝 ⊆ ℝ 𝜔 and a sequence ( 𝛼 𝑝𝑖 ) 𝑖 ∈ 𝜔 of elementsof 𝛾 such that 𝑃 𝛾 ⊩ 𝐵 𝑝 (( 𝜂 𝛼 𝑝𝑖 ) 𝑖 ∈ 𝜔 ) ↔ 𝑝 ∈ 𝐺 𝛾 . ∙ For each 𝑃 𝛾 -name 𝑟 of a real, there is a Borel function 𝐹 𝑟 and a sequence ( 𝛼 𝑟𝑖 ) 𝑖 ∈ 𝜔 of elements of 𝛾 such that 𝑃 𝛾 ⊩ 𝑟 = 𝐹 𝑟 (( 𝜂 𝛼 𝑝𝑖 ) 𝑖 ∈ 𝜔 ) .The second item follows from the first, as we can use the countable maximal antichains thatdecide 𝑟 ( 𝑛 ) = 𝑚 .If 𝛾 is a limit ordinal, then 𝑃 𝛾 has no new elements, so there is nothing to do.So assume 𝛾 = 𝜁 + 1 . By our assumption, 𝑄 𝜁 is generically Borel determined from 𝜂 𝜁 via a Borel relation 𝐵 𝜁 . Consider ( 𝑝, 𝑞 ) ∈ 𝑃 𝜁 ∗ 𝑄 𝜁 . This is in 𝐺 𝛾 iff 𝑝 ∈ 𝐺 𝜁 (which,by induction, is Borel) and 𝑞 ∈ 𝐺 ( 𝜁 ) . As 𝑞 is a real, it is forced that 𝑞 = 𝐵 𝑞 (( 𝛼 𝑞𝑖 ) 𝑖 ∈ 𝜔 ) .Moreover, 𝑃 𝜁 forces that 𝑄 𝜁 forces that 𝑞 ∈ 𝐺 ( 𝜁 ) iff 𝐵 𝜁 ( 𝜂 𝜁 , 𝑞 ) iff 𝐵 𝜁 ( 𝜂 𝜁 , 𝐵 𝑞 (( 𝛼 𝑞𝑖 ) 𝑖 ∈ 𝜔 )) . (cid:3) Definition 1.11.
Given ( 𝑃 𝛽 , 𝑄 𝛽 ) 𝛽<𝛼 as above, and some 𝑤 ⊆ 𝛼 , we define the 𝑃 𝛼 -name ℝ 𝑤 to consist of all reals 𝑟 such that in the ground model there are a Borel function 𝐹 anda sequence ( 𝛼 𝑖 ) 𝑖 ∈ 𝜔 of elements of 𝑤 such that 𝑟 = 𝐹 (( 𝜂 𝛼 𝑖 ) 𝑖 ∈ 𝜔 ) .The following is straightforward: Facts 1.12. ∙ Set (in 𝑉 ) 𝜇 = ( | 𝑤 | + 2) ℵ . Then it is forced that ℝ 𝑤 has cardinality ≤ 𝜇 . ∙ If 𝑤 ′ ⊇ 𝑤 , then (it is forced that) ℝ 𝑤 ′ ⊇ ℝ 𝑤 . ∙ If 𝑤 is the increasing union of ( 𝑤 𝛼 ) 𝛼 ∈ 𝛾 with cf ( 𝛾 ) ≥ 𝜔 , then (it is forced that) ℝ 𝑤 = ⋃ 𝛼 ∈ 𝛾 ℝ 𝑤 𝛼 . ∙ For every 𝑃 𝛼 -name 𝑟 of a real there is a countable 𝑤 such that (it is forced that) 𝑟 ∈ ℝ 𝑤 . OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 5
Definition 1.13.
Let 𝔹 be (the definition of) random forcing, i.e., positive pruned trees 𝑇 , ordered by inclusion. Given ( 𝑃 𝛽 , 𝑄 𝛽 ) 𝛽<𝛼 as above, 𝑤 ⊆ 𝛼 , we define the 𝑃 𝛼 -name 𝔹 𝑤 ∶= 𝔹 ∩ ℝ 𝑤 and call it “partial random forcing defined from 𝑤 ”.Clearly 𝔹 𝑤 is a subforcing (not necessarily a complete one) of 𝔹 , and if 𝑝, 𝑞 in 𝔹 𝑤 areincompatible in 𝔹 𝑤 then they are incompatible in random forcing. In particular 𝔹 𝑤 is ccc.Note that 𝔹 𝑤 is forced to be generically Borel determined, in way already fixed in 𝑉 :The generic real 𝜂 is defined by { 𝜂 } = ⋂ {[ 𝑠 ] ∈ 𝐺 ∶ 𝑠 ∈ 2 <𝜔 } , and the Borel relation by“ 𝜂 ∈ [ 𝑇 ] ”. Remark . In this section, we have provided a very explicit notion of “partial random”,using Borel functions. The use of Borel functions is not essential, we could use any othermethod of calculating reals from generic reals at certain restricted positions, provided thismethod satisfies Facts 1.12. One such alternative definition has been used in [GMS16]: Wecan define the sub-forcing 𝑃 𝛼 ↾ 𝑤 of 𝑃 𝛼 in a natural way, and require that it is a completesubforcing (which is a closure property of 𝑤 ). Then we can take 𝑄 𝛼 to be random forcingas evaluated in the 𝑃 𝛼 ↾ 𝑤 -extension.While this approach is basically equivalent (and may seem slightly more natural thanthe artificial use of Borel functions), it has the disadvantage that we have to take care of theclosure property of 𝑤 . Definition 1.15.
Analogously to “partial random”, we define the “partial Hechler” and“partial amoeba” forcings.These forcings are generically Borel determined as well.1.3.
The inital forcing 𝑃 . Assume that 𝜆 is regular uncountable and 𝜇 < 𝜆 implies 𝜇 ℵ <𝜆 . Then | 𝑤 | < 𝜆 implies that the size of a partial forcing defined by 𝑤 is <𝜆 . Definition 1.16.
Assume GCH and let 𝜆 < 𝜆 < 𝜆 < 𝜆 be regular cardinals. Set 𝛿 = 𝜆 + 𝜆 . Partition 𝛿 ⧵ 𝜆 into unbounded sets 𝑆 , 𝑆 , and 𝑆 . Fix for each 𝛼 ∈ 𝛿 ⧵ 𝜆 some 𝑤 𝛼 ⊆ 𝛼 such that each { 𝑤 𝛼 ∶ 𝛼 ∈ 𝑆 𝑖 } is cofinal in [ 𝛿 ] <𝜆 𝑖 . We now define 𝑃 = ( 𝑃 𝛼 , 𝑄 𝛼 ) 𝛼 ∈ 𝛿 to be the FS ccc iteration which first adds 𝜆 manyCohen reals, and such that for each 𝛼 ∈ 𝛿 ⧵ 𝜆 ,if 𝛼 is in ⎧⎪⎨⎪⎩ 𝑆 𝑆 𝑆 ⎫⎪⎬⎪⎭ , then 𝑄 𝛼 is the partial ⎧⎪⎨⎪⎩ amoebarandomHechler ⎫⎪⎬⎪⎭ forcing defined from 𝑤 𝛼 .The forcing results in ℵ = 𝜆 , which follows from the following easy and well-knownfact: Lemma 1.17.
Let ( 𝑃 𝛼 , 𝑄 𝛼 ) 𝛼<𝛿 be a FS ccc iteration of length 𝛿 such that each 𝑄 𝛼 is forcedto consist of real numbers, and set 𝜆 ( 𝛿 ) ≔ (2 + 𝛿 ) ℵ . Then 𝑃 𝛿 ⊩ ℵ ≤ 𝜆 ( 𝛿 ) .Proof. By induction on 𝛿 , we show that there is a dense subforcing of 𝐷 𝛿 ⊆ 𝑃 𝛿 of size ≤ 𝜆 ( 𝛿 ) . Then the continuum has size at most 𝜆 ( 𝛿 ) (as each name of a real corresponds toa countable sequence of antichains, labeled with , , in 𝑃 𝛿 , without loss of generality in 𝐷 𝛿 ).For 𝛿 + 1 , 𝐷 𝛿 ⊆ 𝑃 𝛿 is dense and has size ≤ 𝜆 ( 𝛿 ) , and 𝑄 𝛿 is forced to have size ≤ 𝜆 ( 𝛿 ) .Without loss of generality we can identify 𝑄 𝛿 with a subset of 𝜆 ( 𝛿 ) . Let 𝐷 𝛿 +1 consist of ( 𝑝, ̌𝛼 ) ∈ 𝑃 𝛿 +1 such that 𝑝 ∈ 𝐷 𝛿 forces 𝛼 ∈ 𝑄 𝛿 . I.e., if 𝛼 ∈ 𝑆 𝑖 then | 𝑤 𝛼 | < 𝜆 𝑖 , and for all 𝑢 ⊆ 𝛿 , | 𝑢 | < 𝜆 𝑖 there is some 𝛼 ∈ 𝑆 𝑖 with 𝑤 𝛼 ⊇ 𝑢 . OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 6
For 𝛿 limit, the union of 𝐷 𝛼 is dense in 𝑃 𝛿 = ⋃ 𝛼 ∈ 𝛿 𝑃 𝛼 . (cid:3) According to Lemma 1.5 𝑃 is (R i , 𝜆 𝑖 ) -good for 𝑖 = 1 , , , so Lemmas 1.6 and 1.8 givesus: Lemma 1.18. ⊚ i ( 𝑃 , 𝜅 ) holds for 𝑖 = 1 , , and each regular cardinal 𝜅 in [ 𝜆 𝑖 , 𝜆 ] .So in particular, 𝑃 forces add( ) ≤ 𝜆 , cov( ) ≤ 𝜆 , 𝔟 ≤ 𝜆 and cof( ) =non( ) = 𝔡 = 2 ℵ . Theorem 1.19. [Mej13, Thm. 2] 𝑃 forces add( ) = 𝜆 , cov( ) = 𝜆 , 𝔟 = 𝜆 , and 𝔡 = 𝜆 = 2 ℵ .Proof. It is easy to see that the partial amoebas take care of add( ) ≥ 𝜆 : Let ( 𝑁 𝑖 ) 𝑖 ∈ 𝜇 , ℵ ≤ 𝜇 < 𝜆 be a family of 𝑃 -names of null sets. Each 𝑁 𝑖 is a Borel code, i.e., a real, andtherefore Borel-computed from some countable set 𝑤 𝑖 ⊆ 𝛿 . The union of the 𝑤 𝑖 is a set 𝑤 ∗ of size ≤ 𝜇 that already Borel-decides all 𝑁 𝑖 . There is some 𝛽 ∈ 𝑆 such that 𝑤 𝛽 ⊇ 𝑤 ∗ ,so the partial amoeba forcing at 𝛽 sees all the null sets 𝑁 𝑖 and therefore covers their union.Analogously one proves cov( ) ≥ 𝜆 and 𝔟 ≥ 𝜆 . (cid:3) We will reformulate the proof for cov( ) in a cumbersome manner that can be conve-niently used later on: Lemma 1.20.
Let ⊞ ( 𝑃 , 𝜆, 𝜇 ) stand for: “ 𝑃 is a ccc forcing notion, and there is a <𝜆 -directed partial order ( 𝑆, ≺ ) of size 𝜇 and a sequence ( 𝑟 𝑠 ) 𝑠 ∈ 𝑆 of 𝑃 -names for reals suchthat for each 𝑃 -name 𝑁 of a null set (∃ 𝑠 ∈ 𝑆 ) (∀ 𝑡 ≻ 𝑠 ) 𝑃 ⊩ 𝑟 𝑡 ∉ 𝑁 .” ∙ ⊞ ( 𝑃 , 𝜆, 𝜇 ) implies 𝑃 ⊩ ( cov( ) ≥ 𝜆 & non( ) ≤ 𝜇 ) . ∙ ⊞ ( 𝑃 , 𝜆 , 𝜆 ) holds.Proof. cov( ) ≥ 𝜆 : Fix <𝜆 many 𝑃 -names 𝑁 𝛼 of null sets. Each real has a “lower bound” 𝑠 𝛼 ∈ 𝑆 , i.e., 𝑃 ⊩ 𝑟 𝑡 ∉ 𝑁 𝛼 whenever 𝑡 ≻ 𝑠 𝛼 . Let 𝑡 ≻ 𝑠 𝛼 for all 𝛼 (this is possible as 𝑆 isdirected). So 𝑃 ⊩ 𝑟 𝑡 ∉ 𝑁 𝛼 for every 𝛼 , i.e., the union doesn’t cover the reals. non( ) ≤ 𝜇 , as the set of all 𝑟 𝑠 is not null: For every name 𝑁 of a null set there is some 𝑠 ∈ 𝑆 such that 𝑃 ⊩ 𝑟 𝑠 ∉ 𝑁 .For 𝑃 , we set 𝑆 = 𝑆 , 𝑠 ≺ 𝑡 if 𝑤 𝑠 ⊆ 𝑤 𝑡 , and we let 𝑟 𝑠 be the partial random real addedat 𝑠 . A 𝑃 name for a null set 𝑁 depends (in a Borel way) on a countable index set 𝑤 ∗ ⊆ 𝛿 .Fix some 𝑠 ∈ 𝑆 such that 𝑤 𝑠 ⊇ 𝑤 ∗ , and pick any 𝑡 ≻ 𝑠 . Then 𝑤 𝑡 contains all informationto calculate the null set 𝑁 , and therefore the partial random 𝑟 𝑡 over 𝑤 𝑡 will avoid 𝑁 . (cid:3)
2. T HE B OOLEAN ULTRAPOWER OF THE FORCING
Boolean ultrapowers.
Boolean ultrapowers generalize regular ultrapowers by usingarbitrary Boolean algebras instead of the power set algebra.
Assumption 2.1. 𝜅 is strongly compact, 𝐵 is a 𝜅 -distributive, 𝜅 + -cc, atomless completeBoolean algebra. Lemma 2.2. [KT64]
Every 𝜅 -complete filter on 𝐵 can be extended to a 𝜅 -complete ultra-filter 𝑈 . Proof.
List the required properties of 𝑈 as a set of propositional sentences in 𝜅 (a propo-sitional language allowing conjunctions and disjunctions of any size <𝜅 ), using atomicformulas coding 𝑏 ∈ 𝑈 and 𝑏 ∉ 𝑈 for 𝑏 ∈ 𝐵 . (cid:3) For this, neither 𝜅 + -cc nor atomless is required, and it is sufficient that 𝐵 is 𝜅 -complete. OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 7
Assumption 2.3. 𝑈 is a 𝜅 -complete ultrafilter on 𝐵 . Lemma 2.4.
There is a maximal antichain 𝐴 in 𝐵 of size 𝜅 such that 𝐴 ∩ 𝑈 = ∅ . Inother words, 𝑈 is not 𝜅 + -complete.Proof. Let 𝐴 be a maximal antichain in the open dense set 𝐵 ⧵ 𝑈 . As 𝐵 is 𝜅 + -cc, 𝐴 hassize ≤ 𝜅 . It cannot have size <𝜅 , as 𝑈 is 𝜅 -complete and therefore meets every antichain ofsize <𝜅 . (cid:3) The Boolean algebra 𝐵 can be used as forcing notion. As usual, 𝑉 denotes the universewe start with, sometimes called the ground model. In the following, we will not actuallyforce with 𝐵 (or any other p.o.); we always remain in 𝑉 , but we still use forcing notation.In particular, we call the usual 𝐵 -names “forcing names”. Definition 2.5. A BUP-name (or: labeled antichain) 𝑥 is a function 𝐴 → 𝑉 whose domainis a maximal antichain. We may write 𝐴 ( 𝑥 ) to denote 𝐴 .Each BUP-name corresponds to a forcing-name for an element of 𝑉 . We will identifythe BUP-name and the corresponding forcing-name. In turn, every forcing name 𝜏 for anelement of 𝑉 has a forcing-equivalent BUP-name.In particular, we can calculate, for two BUP-names 𝑥 and 𝑦 , the Boolean value ⟦ 𝑥 = 𝑦 ⟧ . Definition 2.6. ∙ Two BUP-names 𝑥 and 𝑦 are equivalent , if ⟦ 𝑥 = 𝑦 ⟧ ∈ 𝑈 . ∙ For 𝑣 ∈ 𝑉 , let ̌𝑣 be a BUP-name-version of the standard name for 𝑣 (unique up toequivalence). ∙ The
Boolean ultrapower 𝑀 − consists of the equivalence classes [ 𝑥 ] of BUP-names 𝑥 ; and we define [ 𝑥 ] ∈ − [ 𝑦 ] by ⟦ 𝑥 ∈ 𝑦 ⟧ ∈ 𝑈 . ∙ 𝑗 − ∶ 𝑉 → 𝑀 − maps 𝑣 to [ ̌𝑣 ] .We are interested in the ∈ -structure ( 𝑀 − , ∈ − ) .Given BUP-names 𝑥 , … , 𝑥 𝑛 and an ∈ -formula 𝜑 , the truth value ⟦ 𝜑 𝑉 ( 𝑥 , … , 𝑥 𝑛 ) ⟧ iswell defined (it is the weakest element of 𝐵 forcing that in the ground model 𝜑 ( 𝑥 , … , 𝑥 𝑛 ) holds, which makes sense as 𝑥 , … , 𝑥 𝑛 are guaranteed to be in the ground model). Lemma 2.7. ∙ Łoś’s theorem: ( 𝑀 − , ∈ − ) ⊨ 𝜑 ([ 𝑥 ] , … , [ 𝑥 𝑛 ]) iff ⟦ 𝜑 𝑉 ( 𝑥 , … , 𝑥 𝑛 ) ⟧ ∈ 𝑈 . ∙ 𝑗 − ∶ ( 𝑉 , ∈) → ( 𝑀 − , ∈ − ) is an elementary embedding. ∙ In particular, ( 𝑀 − , ∈ − ) is a ZFC model.Proof. Straightforward by the definition of equivalence and of [ 𝑥 ] ∈ − [ 𝑦 ] , and by induction(using that 𝑈 is a filter for 𝜑 ∧ 𝜓 and for ∃ 𝑣 𝜑 ( 𝑣 ) , and that it is an ultrafilter for ¬ 𝜑 ).For elementarity, note that 𝑀 − ⊨ 𝜑 ([ ̌𝑥 ] , … , [ ̌𝑥 𝑛 ]) iff ⟦ 𝜑 𝑉 ( ̌𝑥 , … , ̌𝑥 𝑛 ) ⟧ ∈ 𝑈 iff 𝑉 ⊨𝜑 ( 𝑥 , … , 𝑥 𝑛 ) . (cid:3) Lemma 2.8. ( 𝑀 − , ∈ − ) is wellfounded. More specifically, to the forcing-name {( ̌𝑥 ( 𝑎 ) , 𝑎 ) ∶ 𝑎 ∈ 𝐴 ( 𝑥 )} . We can calculate ⟦ 𝑥 = 𝑦 ⟧ more explicitly as follows: Pick some common refinement 𝐴 ′ of 𝐴 ( 𝑥 ) and 𝐴 ( 𝑦 ) .This defines in an obvious way BUP-names 𝑥 ′ and 𝑦 ′ both with domain 𝐴 ′ : For 𝑎 ∈ 𝐴 ′ we set 𝑥 ′ ( 𝑎 ) = 𝑥 ( ̃𝑎 ) for ̃𝑎 the unique element of 𝐴 ( 𝑥 ) above 𝑎 . Then ⟦ 𝑥 = 𝑦 ⟧ is ⋁ { 𝑎 ∈ 𝐴 ′ ∶ 𝑥 ′ ( 𝑎 ) = 𝑦 ′ ( 𝑎 )} (which is independent of therefinement 𝐴 ′ ). Equivalently, we can explicitly calculate ⟦ 𝜑 𝑉 ( 𝑥 , … , 𝑥 𝑛 ) ⟧ as follows: Chose a common refinement 𝐴 ′ of 𝐴 ( 𝑥 ) , … , 𝐴 ( 𝑥 𝑛 ) , and set ⟦ 𝜑 𝑉 ( 𝑥 , … , 𝑥 𝑛 ) ⟧ to be ⋁ { 𝑎 ∈ 𝐴 ′ ∶ 𝜑 ( 𝑥 ′1 ( 𝑎 ) , … , 𝑥 ′ 𝑛 ( 𝑎 ))} ; where again the BUP-names 𝑥 ′ 𝑖 are the canonically defined BUP-names with domain 𝐴 ′ that are equivalent to 𝑥 𝑖 . OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 8
Proof.
This is the standard argument, using the fact that 𝑈 is 𝜎 -complete:Assume [ 𝑥 𝑛 +1 ] ∈ − [ 𝑥 𝑛 ] for 𝑛 ∈ 𝜔 . Choose a common refinement 𝐴 of the antichains 𝐴 ( 𝑥 𝑛 ) , Again, let 𝑥 ′ 𝑛 be the BUP-names with domain 𝐴 equivalent to 𝑥 𝑛 . So, by our as-sumption, 𝑢 𝑛 ≔ ⟦ 𝑥 𝑛 +1 ∈ 𝑥 𝑛 ⟧ = ⋁ { 𝑎 ∈ 𝐴 ∶ 𝑥 ′ 𝑛 +1 ( 𝑎 ) ∈ 𝑥 ′ 𝑛 ( 𝑎 )} is in 𝑈 for each 𝑛 . As 𝑈 is 𝜎 -complete, there is some 𝑢 ∈ 𝑈 stronger than all 𝑢 𝑛 . This implies: If 𝑎 ∈ 𝐴 is compatiblewith 𝑢 , then 𝑎 is compatible with 𝑢 𝑛 (for all 𝑛 ), and therefore 𝑥 ′ 𝑛 +1 ( 𝑎 ) ∈ 𝑥 ′ 𝑛 ( 𝑎 ) for all 𝑛 , acontradiction. (cid:3) Definition 2.9.
Let 𝑀 be the transitive collapse of ( 𝑀 − , ∈ − ) , and let 𝑗 ∶ 𝑉 → 𝑀 be thecomposition of 𝑗 − with the collapse. We denote the collapse of [ 𝑥 ] by 𝑥 𝑈 . So in particular ̌𝑣 𝑈 = 𝑗 ( 𝑣 ) . Lemma 2.10. ∙ 𝑀 ⊧ 𝜑 ( 𝑥 𝑈 , … , 𝑥 𝑈𝑛 ) iff ⟦ 𝜑 𝑉 ( 𝑥 , … , 𝑥 𝑛 ) ⟧ ∈ 𝑈 . In particular, 𝑗 ∶ 𝑉 → 𝑀 is an elementary embedding. ∙ If | 𝑌 | < 𝜅 , then 𝑗 ( 𝑌 ) = 𝑗 ′′ 𝑌 . In particular, 𝑗 restricted to 𝜅 is the identity. 𝑀 isclosed under <𝜅 -sequences. ∙ 𝑗 ( 𝜅 ) ≠ 𝜅 . I.e., 𝜅 = cr( 𝑗 ) .Proof. If [ 𝑥 ] ∈ 𝑗 − ( 𝑌 ) , then we can refine the antichain 𝐴 ( 𝑥 ) to some 𝐴 ′ such that each 𝑎 ∈ 𝐴 ′ either forces 𝑥 = 𝑦 for some 𝑦 ∈ 𝑌 , or 𝑥 ∉ 𝑌 . Without loss of generality (by takingsuprema), we can assume different elements 𝑎 of 𝐴 ′ giving different values 𝑦 ( 𝑎 ) ; i.e., 𝐴 ′ has size | 𝑌 | + 1 < 𝜅 . So 𝑈 selects an element 𝑎 of 𝐴 ′ , and as ⟦ 𝑥 ∈ 𝑌 ⟧ ∈ 𝑈 , this element 𝑎 proves that [ 𝑥 ] = 𝑗 − ( 𝑦 ( 𝑎 )) .We have already mentioned that there is a maximal antichain 𝐴 = { 𝑎 𝑖 ∶ 𝑖 ∈ 𝜅 } ofsize 𝜅 such that 𝐴 ∩ 𝑈 = ∅ . The BUP-name 𝑥 with 𝐴 ( 𝑥 ) = 𝐴 and 𝑥 ( 𝑎 𝑖 ) = 𝑖 satisfies [ 𝑥 ] ∈ − 𝑗 − ( 𝜅 ) , but is not equivalent to any ̌𝑣 ; so 𝜅 ≤ 𝑥 𝑈 < 𝑗 ( 𝜅 ) . (cid:3) As we have already mentioned, an arbitrary forcing-name for an element of 𝑉 has aforcing-equivalent BUP-name, i.e., a maximal antichain labeled with elements of 𝑉 . If 𝜏 isa forcing-name for an element of 𝑌 ( 𝑌 ∈ 𝑉 ), then without loss of generality 𝜏 correspondsto a maximal antichain labeled with elements of 𝑌 . We call such an object 𝑦 a “BUP-namefor an element of 𝑗 ( 𝑌 ) ” (and not “for an element of 𝑌 ”, for the obvious reason: unlike inthe case of a forcing extension, 𝑦 𝑈 is generally not in 𝑌 , but, by definition of ∈ − , it is in 𝑗 ( 𝑌 ) ).2.2. The algebra and the filter.
We will now define the concrete Boolean algebra we aregoing to use:
Definition 2.11.
Assume GCH, let 𝜅 be strongly compact, and 𝜃 > 𝜅 regular. 𝑃 𝜅,𝜃 is the forcing notion adding 𝜃 Cohen subsets of 𝜅 . More concretely: 𝑃 𝜅,𝜃 consists ofpartial functions from 𝜃 to 𝜅 with domain of size <𝜅 , ordered by extension. Let 𝑓 ∗ ∶ 𝜃 → 𝜅 be the name of the generic function. 𝜅,𝜃 is the complete Boolean algebra generated by 𝑃 𝜅,𝜃 .Clearly 𝜅,𝜃 is 𝜅 + -cc and 𝜅 -distributive, as 𝑃 𝜅,𝜃 is even 𝜅 -closed. Lemma 2.12.
There is a 𝜅 -complete ultrafilter 𝑈 on 𝐵 = 𝜅,𝜃 such that: (a) The Boolean ultrapower gives an elementary embedding 𝑗 ∶ 𝑉 → 𝑀 . 𝑀 is closedunder <𝜅 -sequences. (b) The elements 𝑥 𝑈 of 𝑀 are exactly (the collapses of equivalence classes of) 𝐵 -names 𝑥 for elements of 𝑉 ; more concretely, a function from an antichain (of size 𝜅 ) to 𝑉 . We sometimes say “ 𝑥 𝑈 is a mixture of 𝜅 many possibilities”. OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 9
Similarly, for 𝑌 ∈ 𝑉 , the elements 𝑥 𝑈 of 𝑗 ( 𝑌 ) correspond to the 𝐵 -names 𝑥 ofelements of 𝑌 , i.e., antichains labeled with elements of 𝑌 . (c) If | 𝐴 | < 𝜅 , then 𝑗 ′′ 𝐴 = 𝑗 ( 𝐴 ) . In particular, 𝑗 restricted to 𝜅 is the identity. (d) 𝑗 has critical point 𝜅 , cf ( 𝑗 ( 𝜅 )) = 𝜃 , and 𝜃 ≤ 𝑗 ( 𝜅 ) ≤ 𝜃 + . (e) If 𝜆 > 𝜅 is regular, then max( 𝜃, 𝜆 ) ≤ 𝑗 ( 𝜆 ) < max( 𝜃, 𝜆 ) + . (f) If 𝑆 is a <𝜆 -directed partial order, and 𝜅 < 𝜆 , then 𝑗 ′′ 𝑆 is cofinal in 𝑗 ( 𝑆 ) . (g) If cf( 𝛼 ) ≠ 𝜅 , then 𝑗 ′′ 𝛼 is cofinal in 𝑗 ( 𝛼 ) , so in particular cf ( 𝑗 ( 𝛼 )) = cf( 𝛼 ) .Proof. We have already seen (a)–(c).(d): For each 𝛿 ∈ 𝜃 , 𝑓 ∗ ( 𝛿 ) is a forcing-name for an element of 𝜅 , and thus a BUP-namefor an element of 𝑗 ( 𝜅 ) . Let 𝑥 be some other BUP-name for an element of 𝑗 ( 𝜅 ) , i.e., anantichain 𝐴 of size 𝜅 labeled with elements of 𝜅 . Let 𝛿 ∈ 𝜃 be bigger than the supremum of supp( 𝑎 ) for each 𝑎 ∈ 𝐴 . We call such a pair ( 𝑥, 𝛿 ) “suitable”, and set 𝑏 𝑥,𝛿 ≔ ⟦ 𝑓 ∗ ( 𝛿 ) > 𝑥 ⟧ .We claim that all these elements form a basis for a 𝜅 -complete filter. To see this, fix suitablepairs ( 𝑥 𝑖 , 𝛿 𝑖 ) for 𝑖 < 𝜇 where 𝜇 < 𝜅 ; we have to show that ⋀ 𝑖 ∈ 𝜇 𝑏 𝑥 𝑖 ,𝛿 𝑖 ≠ . Enumerate { 𝛿 𝑖 ∶ 𝑖 ∈ 𝜇 } increasing (and without repetitions) as 𝛿 𝑗 for 𝑗 ∈ 𝛾 ≤ 𝜇 . Set 𝐴 𝑗 = { 𝑖 ∶ 𝛿 𝑖 = 𝛿 𝑗 } .Given 𝑞 𝑗 , define 𝑞 𝑗 +1 ∈ 𝑃 𝜅,𝜃 as follows: 𝑞 𝑗 +1 ≤ 𝑞 𝑗 ; 𝛿 𝑗 ∈ supp( 𝑞 𝑗 +1 ) ⊆ 𝛿 𝑗 ∪{ 𝛿 𝑗 } ; and 𝑞 𝑗 +1 ↾ 𝛿 𝑗 decides for all 𝑖 ∈ 𝐴 𝑗 the values of 𝑥 𝑖 to be some 𝛼 𝑖 ; and 𝑞 𝑗 +1 ( 𝛿 𝑗 ) = sup 𝑖 ∈ 𝐴 𝑗 ( 𝛼 𝑖 ) + 1 .For 𝑗 ≤ 𝛾 limit, let 𝑞 𝑗 be the union of { 𝑞 𝑘 ∶ 𝑘 < 𝑗 } . Then 𝑞 𝛾 is stronger than each 𝑏 𝑥 𝑖 ,𝛿 𝑖 .As 𝜅 is strongly compact, we can extend the 𝜅 -complete filter generated by all 𝑏 𝑥 𝑖 ,𝛿 𝑖 to a 𝜅 -complete ultrafilter 𝑈 . Then the sequence ( 𝑓 ∗ ( 𝛿 ) 𝑈 ) 𝛿 ∈ 𝜃 is strictly increasing (as ( 𝑓 ∗ ( 𝛿 ) , 𝛿 ′ ) is suitable for all 𝛿 < 𝛿 ′ ) and cofinal in 𝑗 ( 𝜅 ) (as we have just seen); so cf ( 𝑗 ( 𝜅 )) = 𝜃 . (e): We count all BUP-names for elements of 𝑗 ( 𝜆 ) . As we can assume that the antichainsare subsets of 𝑃 𝜅,𝜃 , which has size 𝜃 , and as 𝜆 is regular and GCH holds, we get | 𝑗 ( 𝜆 ) | ≤ [ 𝜃 ] 𝜅 × 𝜆 𝜅 = max( 𝜃, 𝜆 ) .(f): An element 𝑥 𝑈 of 𝑗 ( 𝑆 ) is a mixture of 𝜅 many possibilities in 𝑆 . As 𝜅 < 𝜆 , there issome 𝑡 ∈ 𝑆 above all the possibilities. Then 𝑗 ( 𝑡 ) > 𝑥 𝑈 .(g): Set 𝜇 = cf ( 𝛼 ) , and pick an increasing cofinal sequence ̄𝛽 = ( 𝛽 𝑖 ) 𝑖 ∈ 𝜇 in 𝛼 . 𝑗 ( ̄𝛽 ) isincreasing cofinal in 𝑗 ( 𝛼 ) (as this is absolute between 𝑀 and 𝑉 ). If 𝜇 < 𝜅 , then 𝑗 ′′ ̄𝛽 = 𝑗 ( ̄𝛽 ) ,otherwise use (f). (cid:3) The ultrapower of a forcing notion.
We now investigate the relation of a forcingnotion 𝑃 ∈ 𝑉 and its image 𝑗 ( 𝑃 ) ∈ 𝑀 , which we use as a forcing notion over 𝑉 . (Thinkof 𝑃 as being one of the forcings of Section 1; it has no relation with the Boolean algebra 𝐵 .)Note that as 𝑗 ( 𝑃 ) ∈ 𝑀 and 𝑀 is transitive, every 𝑗 ( 𝑃 ) -generic filter 𝐺 over 𝑉 is triviallygeneric over 𝑀 as well, and we will use absoluteness between 𝑀 [ 𝐺 ] and 𝑉 [ 𝐺 ] to provevarious properties of 𝑗 ( 𝑃 ) . Lemma 2.13. If 𝑃 is 𝜅 -cc, then 𝑗 gives a complete embedding from 𝑃 into 𝑗 ( 𝑃 ) . I.e., 𝑗 ′′ 𝑃 is a complete subforcing of 𝑗 ( 𝑃 ) , and 𝑗 is an isomorphism from 𝑃 to 𝑗 ′′ 𝑃 .Proof. It is clear that 𝑗 is an isomorphism onto 𝑗 ′′ 𝑃 : By definition the order < 𝑗 ( 𝑃 ) on 𝑗 ( 𝑃 ) is 𝑗 ( < 𝑃 ) , and by elementarity 𝑝 ≤ 𝑃 𝑞 iff 𝑗 ( 𝑞 ) < 𝑗 ( 𝑃 ) 𝑗 ( 𝑝 ) . Also, 𝑝 ⟂ 𝑞 is preserved: 𝑀 ⊨ 𝑝 ⟂ 𝑗 ( 𝑃 ) 𝑞 by elementarity, so 𝑝 ⟂ 𝑗 ( 𝑃 ) 𝑞 holds in 𝑉 (as 𝑗 ( 𝑃 ) ∈ 𝑀 and 𝑀 is transitive).It remains to be shown that each maximal antichain 𝐴 of 𝑃 is preserved, i.e., 𝑗 ′′ 𝐴 ⊆ 𝑗 ( 𝑃 ) is predense.By our assumption, | 𝐴 | < 𝜅 , so 𝑗 ′′ 𝐴 = 𝑗 ( 𝐴 ) (by Lemma 2.12(c)), which is maximal in 𝑀 (by elementarity) and thus maximal in 𝑉 (by absoluteness). (cid:3) OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 10
Accordingly, we can canonically translate 𝑃 -names into 𝑗 ( 𝑃 ) -names, etc.For later reference, let us make this a bit more explicit: Let 𝑔 be a 𝑃 -name for a real (i.e.,an element of 𝜔 𝜔 ). Each 𝑔 ( 𝑛 ) is decided by a maximal antichains 𝐴 𝑛 , where 𝑎 ∈ 𝐴 𝑛 forces 𝑔 ( 𝑛 ) = 𝑔 𝑛,𝑎 ∈ 𝜔 . Then the 𝑗 ( 𝑃 ) -name 𝑗 ( 𝑔 ) corresponds to the antichains(2.14) 𝑗 ( 𝐴 𝑛 ) = 𝑗 ′′ 𝐴 𝑛 , and 𝑗 ( 𝑎 ) forces 𝑗 ( 𝑔 )( 𝑛 ) = 𝑔 𝑛,𝑎 for each 𝑎 ∈ 𝐴 𝑛 . Lemma 2.15. If 𝑃 = ( 𝑃 𝛼 , 𝑄 𝛼 ) 𝛼<𝛿 is a finite support (FS) ccc iteration of length 𝛿 , then 𝑗 ( 𝑃 ) is a FS ccc iteration of length 𝑗 ( 𝛿 ) (more formally: it is canonically equivalent to one).Proof. 𝑀 certainly thinks that 𝑗 ( 𝑃 ) = ( 𝑃 ∗ 𝛼 , 𝑄 ∗ 𝛼 ) 𝛼<𝑗 ( 𝛿 ) is a FS iteration of length 𝑗 ( 𝛿 ) .By induction on 𝛼 we define the FS ccc iteration ( ̃𝑃 𝛼 , ̃𝑄 𝛼 ) 𝛼<𝑗 ( 𝛿 ) and show that 𝑃 ∗ 𝛼 is adense subforcing of ̃𝑃 𝛼 : Assume this is already the case for 𝑃 ∗ 𝛼 . 𝑀 thinks that 𝑄 ∗ 𝛼 is a 𝑃 ∗ 𝛼 -name, so we can interpret it as a ̃𝑃 𝛼 -name and use it as ̃𝑄 𝛼 . Assume that ( 𝑝, 𝑞 ) is anelement (in 𝑉 ) of ̃𝑃 𝛼 ∗ ̃𝑄 𝛼 . So 𝑝 forces that 𝑞 is a name in 𝑀 ; we can increase 𝑝 to some 𝑝 ′ that decides 𝑞 to be the name 𝑞 ′ ∈ 𝑀 . By induction we can further increase 𝑝 ′ to 𝑝 ′′ ∈ 𝑃 ∗ 𝛼 ,then ( 𝑝 ′′ , 𝑞 ′ ) ∈ 𝑃 ∗ 𝛼 +1 is stronger than ( 𝑝, 𝑞 ) . (At limits there is nothing to do, as we use FSiterations.) 𝑗 ( 𝑃 ) is ccc, as any 𝐴 ⊆ 𝑗 ( 𝑃 ) of size ℵ is in 𝑀 (and 𝑀 thinks that 𝑗 ( 𝑃 ) is ccc). (cid:3) Similarly, we get: ∙ If 𝜏 = 𝑥 𝑈 is in 𝑀 a 𝑗 ( 𝑃 ) -name for an element of 𝑗 ( 𝑍 ) , then 𝜏 is a mixture of 𝜅 many 𝑃 -names for an element of 𝑍 (i.e., the BUP-name 𝑥 consists of an antichain 𝐴 ⊆ 𝐵 labeled, without loss of generality, with 𝑃 -names for elements of 𝑍 ).(This is just the instance of “each 𝑥 𝑈 ∈ 𝑗 ( 𝑌 ) is a mixture of elements of 𝑌 ”,where we set 𝑌 to be the set of 𝑃 -names for elements of 𝑍 .) ∙ A 𝑗 ( 𝑃 ) -name 𝜏 for an element of 𝑀 [ 𝐺 ] has an equivalent 𝑗 ( 𝑃 ) -name in 𝑀 .(There is a maximal antichain 𝐴 of 𝑗 ( 𝑃 ) labeled with 𝑗 ( 𝑃 ) -names in 𝑀 . As 𝑀 is countably closed, this labeled antichain is in 𝑀 , and gives a 𝑗 ( 𝑃 ) -name in 𝑀 equivalent to 𝜏 .) ∙ In 𝑉 [ 𝐺 ] , 𝑀 [ 𝐺 ] is closed under <𝜅 sequences.(We can assume the names to be in 𝑀 and use <𝜅 -closure.) ∙ In particular, every 𝑗 ( 𝑃 ) -name for a real, a Borel-code, a countable sequence ofreals, etc., is in 𝑀 (more formally: has an equivalent name in 𝑀 ). ∙ If each iterand is forced to consist of reals, then 𝑗 ( 𝑃 ) forces the continuum to havesize at most | 𝑗 ( 𝛿 ) | ℵ .(This follows from Lemma 1.17 as 𝑗 ( 𝑃 ) also satisfies that each iterand consistsof reals.)2.4. Preservation of values of characteristics.Lemma 2.16.
Let 𝜆 be a regular uncountable cardinal and 𝑃 a ccc forcing. (a) Let 𝔵 be either add( ) or 𝔟 . If 𝑃 ⊩ 𝔵 = 𝜆 and 𝜅 ≠ 𝜆 , then 𝑗 ( 𝑃 ) ⊩ 𝔵 = 𝜆 . (b) Let 𝔶 be either cof( ) or 𝔡 . If 𝑃 ⊩ 𝔶 ≥ 𝜆 and 𝜅 < 𝜆 , then 𝑗 ( 𝑃 ) ⊩ 𝔶 ≥ 𝜆 . (c) Let ( 𝔵 , 𝔶 ) be either ( 𝔟 , 𝔡 ) or (add( ) , cof( )) . Then we get:If 𝑃 ⊩ ( 𝜅 < 𝔵 & 𝔶 ≤ 𝜆 ) then 𝑗 ( 𝑃 ) ⊩ 𝔶 ≤ 𝜆 . Formally: We set 𝑌 to be some set that contains representatives of each equivalence class of 𝑃 -names ofelements of 𝑍 . OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 11
Proof. (a)
We formulate the proof for add( ) ; the proof for 𝔟 is the same.Let ̄𝑁 = ( 𝑁 𝑖 ) 𝑖<𝜆 be 𝑃 -names for an increasing sequence of null sets such that ⋃ 𝑖<𝜆 𝑁 𝑖 is not null. So in particular for every 𝑃 -name 𝑁 of a null set: (∃ 𝑖 ∈ 𝜆 ) (∀ 𝑖 ∈ 𝜆 ⧵ 𝑖 ) 𝑃 ⊩𝑁 𝑖 ⊈ 𝑁 . (We can choose the 𝑖 in 𝑉 due to ccc.)Therefore 𝑀 thinks that the same holds for the sequence 𝑗 ( ̄𝑁 ) of 𝑗 ( 𝑃 ) -names of length 𝑗 ( 𝜆 ) . So whenever 𝑁 is a 𝑗 ( 𝑃 ) -name of a null set, we can assume without loss of generalitythat 𝑁 ∈ 𝑀 , so 𝑀 thinks that from some 𝑖 on it is forced that 𝑁 𝑖 ⊈ 𝑁 , which is absolute.As 𝜅 ≠ 𝜆 , we know that 𝑗 ′′ 𝜆 is cofinal in 𝑗 ( 𝜆 ) . So (since the sequence 𝑗 ( ̄𝑁 ) is increasing)we can use ( 𝑗 ( 𝑁 𝑖 )) 𝑖 ∈ 𝜆 and get the same property.This shows that 𝑗 ( 𝑃 ) ⊩ add( ) ≤ 𝜆 For the other inequality, fix some 𝜒 < 𝜆 , and ( 𝑁 𝑖 ) 𝑖<𝜒 a family of 𝑗 ( 𝑃 ) -names for nullsets (without loss of generality each name is in 𝑀 ), and 𝑝 ∈ 𝑗 ( 𝑃 ) . ∙ Case 1: 𝜅 ≥ 𝜆 . Then the sequence ( 𝑁 𝑖 ) 𝑖<𝜒 (as well as 𝑝 ) is in 𝑀 , and 𝑀 ⊧ ( 𝑝 ⊩ ⋃ 𝑁 𝑖 null ) ; which is absolute. ∙ Case 2: 𝜅 < 𝜆 . Every 𝑁 𝑖 is a “mixture” of 𝜅 many 𝑃 -names for null sets, so thereis a single 𝑃 -name 𝑁 ′ 𝑖 such that 𝑃 forces 𝑁 ′ 𝑖 is superset of all the names involved.Therefore, 𝑗 ( 𝑃 ) forces that 𝑗 ( 𝑁 ′ 𝑖 ) ⊇ 𝑁 𝑖 . And 𝑃 forces that ⋃ 𝑖<𝜒 𝑁 ′ 𝑖 is null, i.e.,covered by some null set 𝑁 ∗ . Then 𝑗 ( 𝑃 ) forces that 𝑗 ( 𝑁 ∗ ) covers ⋃ 𝑖<𝜒 𝑁 𝑖 . (b) We show that a small set cannot be dominating: Fix a sequence ( 𝑓 𝑖 ) 𝑖<𝜒 of 𝑗 ( 𝑃 ) -names of reals, with 𝜒 < 𝜆 . Each 𝑓 𝑖 corresponds to 𝜅 < 𝜆 many possible 𝑃 -names. As 𝜒 < 𝜆 , there is a 𝑃 -name 𝑔 unbounded by all 𝜒 × 𝜅 < 𝜆 many possible 𝑃 -names. So if 𝑓 is any of the possibilities, then 𝑃 forces 𝑔 ≰ ∗ 𝑓 ; and thus 𝑗 ( 𝑃 ) forces 𝑗 ( 𝑔 ) ≰ ∗ 𝑓 𝑖 for all 𝑖 .So 𝑗 ( 𝑃 ) forces 𝔡 ≥ 𝜆 .The same proof works for cof ( ) (using “the null set 𝑔 is not a subset of any of thepossible null sets”). (c) For ( 𝔵 , 𝔶 ) = ( 𝔟 , 𝔡 ) : Fix a 𝑃 -name of a dominating family ̄𝑓 = ( 𝑓 𝑖 ) 𝑖 ∈ 𝜆 .We claim that 𝑗 ( 𝑃 ) forces that 𝑗 ′′ ̄𝑓 = ( 𝑗 ( 𝑓 𝑖 )) 𝑖<𝜆 is dominating. Let 𝑟 be a 𝑗 ( 𝑃 ) -name ofa real, i.e., a mixture of 𝜅 many possibilities (each possibility corresponding to a 𝑃 -namefor a real). As 𝑃 ⊩ 𝜅 < 𝔟 , 𝑃 forces that these reals cannot be unbounded, i.e., there isa 𝑃 -name 𝛼 ∈ 𝜆 such that 𝑓 𝛼 is forced to dominate all the possibilities. By absoluteness, 𝑗 ( 𝑃 ) ⊩ 𝑗 ( 𝑓 𝛼 ) > ∗ 𝑟 .It remains to be shown that 𝑗 ( 𝑃 ) ⊩ 𝑗 ( 𝑓 𝛼 ) ∈ 𝑗 ′′ ̄𝑓 . (Note that 𝛼 is just a 𝑃 -name.) Fix amaximal antichain 𝐴 in 𝑃 deciding 𝛼 , i.e., 𝑎 ∈ 𝐴 forces 𝛼 = 𝛼 ( 𝑎 ) . As 𝑗 maps 𝑃 completelyinto 𝑗 ( 𝑃 ) , 𝑗 ′′ 𝐴 is a maximal antichain in 𝑗 ( 𝑃 ) . So 𝑗 ( 𝑃 ) forces that exactly on 𝑗 ( 𝑎 ) for 𝑎 ∈ 𝐴 is in the generic filter, cf. (2.14). Accordingly 𝑗 ( 𝑓 𝛼 ) = 𝑗 ( 𝑓 𝛼 ( 𝑎 ) ) ∈ 𝑗 ′′ ̄𝑓 .The proof for cof( ) is the same. (cid:3) For the other direction of the invariants, and the pair (cov( ) , non( )) , we use thefollowing two lemmas, which are reformulations of results of Shelah. Recall Definition 1.7 (which is useful because of Lemma 1.8 and satisfied for the initalforcing according to Lemma 1.18).
Lemma 2.17.
Assume ⊚ i ( 𝑃 , 𝜆 ) . Then ⊚ i ( 𝑗 ( 𝑃 ) , cf( 𝑗 ( 𝜆 ))) .So if 𝜅 ≠ 𝜆 , then ⊚ i ( 𝑗 ( 𝑃 ) , 𝜆 ) , and if 𝜅 = 𝜆 , then ⊚ i ( 𝑗 ( 𝑃 ) , 𝜃 ) .Proof. Let ̄𝑦 = ( 𝑦 𝛼 ) 𝛼<𝜆 be the sequence of 𝑃 -names witnessing ⊚ i ( 𝑃 , 𝜆 ) . Note that 𝑗 ( ̄𝑦 ) is a sequence of length 𝑗 ( 𝜆 ) ; we denote the 𝛽 -th element by ( 𝑗 ( ̄𝑦 )) 𝛽 . So 𝑀 thinks: Forevery 𝑗 ( 𝑃 ) -name 𝑟 of a real (∃ 𝛼 ∈ 𝑗 ( 𝜆 )) (∀ 𝛽 ∈ 𝑗 ( 𝜆 ) ⧵ 𝛼 ) ¬( 𝑗 ( ̄𝑦 )) 𝛽 R i 𝑟 . This is absolute. In S. Shelah, personal communication.
OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 12 particular, pick in 𝑉 a cofinal subset 𝐴 of 𝑗 ( 𝜆 ) of order type cf ( 𝑗 ( 𝜆 )) =∶ 𝜇 . Then 𝑗 ( ̄𝑦 ) ↾ 𝐴 witnesses that ⊚ i ( 𝑗 ( 𝑃 ) , 𝜇 ) holds. (cid:3) We have seen in Lemma 1.20 that ⊞ ( 𝑃 , 𝜆 , 𝜆 ) holds and implies that 𝑃 forces cov( ) ≥ 𝜆 and non( ) ≤ 𝜆 (the latter being trivial in the case of 𝑃 ). Lemma 2.18.
Assume ⊞ ( 𝑃 , 𝜆, 𝜇 ) . If 𝜅 > 𝜆 , then ⊞ ( 𝑗 ( 𝑃 ) , 𝜆, | 𝑗 ( 𝜇 ) | ) ; if 𝜅 < 𝜆 , then ⊞ ( 𝑗 ( 𝑃 ) , 𝜆, 𝜇 ) .Proof. Let ( 𝑆, ≺ ) and ̄𝑟 witness ⊞ ( 𝑃 , 𝜆, 𝜇 ) . 𝑀 thinks that( ∗ ) for each 𝑗 ( 𝑃 ) -name 𝑁 of a null set (∃ 𝑠 ∈ 𝑗 ( 𝑆 )) (∀ 𝑡 ∈ 𝑗 ( 𝑆 )) 𝑡 ≻ 𝑠 → 𝑗 ( 𝑃 ) ⊩ ( 𝑗 ( ̄𝑟 )) 𝑡 ∉ 𝑁, which is absolute.If 𝜅 > 𝜆 , then 𝑗 ( 𝜆 ) = 𝜆 , and 𝑗 ( 𝑆 ) is 𝜆 -directed in 𝑀 and therefore in 𝑉 as well, and sowe get ⊞ ( 𝑗 ( 𝑃 ) , 𝜆, | 𝑗 ( 𝜇 ) | ) .So assume 𝜅 < 𝜆 . We claim that 𝑗 ′′ 𝑆 and 𝑗 ′′ ̄𝑟 witness ⊞ ( 𝑗 ( 𝑃 ) , 𝜆, 𝜇 ) . 𝑗 ′′ 𝑆 is isomorphicto 𝑆 , so directedness is trivial. Given a 𝑗 ( 𝑃 ) -name 𝑁 , without loss of generality in 𝑀 , thereis in 𝑀 a bound 𝑠 ∈ 𝑗 ( 𝑆 ) as in ( ∗ ). As 𝑗 ′′ 𝑆 is cofinal in 𝑗 ( 𝑆 ) (according to Lemma 2.12(f)),there is some 𝑠 ′ ∈ 𝑆 such that 𝑗 ( 𝑠 ′ ) ≻ 𝑠 . Then for all 𝑡 ′ ≻ 𝑠 ′ , i.e., 𝑗 ( 𝑡 ′ ) ≻ 𝑗 ( 𝑠 ′ ) , we get 𝑗 ( 𝑃 ) ⊩ 𝑗 ( 𝑟 𝑡 ) ∉ 𝑁 . (cid:3) The main theorem.
We now have everything required for the main result:
Theorem 2.19.
Assume GCH and that ℵ < 𝜅 < 𝜆 < 𝜅 < 𝜆 < 𝜅 < 𝜆 < 𝜆 <𝜆 < 𝜆 < 𝜆 are regular, 𝜅 𝑖 strongly compact for 𝑖 = 5 , , . Then there is a ccc order 𝑃 forcing add( ) = 𝜆 < cov( ) = 𝜆 < 𝔟 = 𝜆 << 𝔡 = 𝜆 < non( ) = 𝜆 < cof ( ) = 𝜆 < ℵ = 𝜆 . Proof.
Let 𝑗 𝑖 ∶ 𝑉 → 𝑀 𝑖 be the Boolean ultrapower embedding with cf ( 𝑗 ( 𝜅 𝑖 )) = 𝜆 𝑖 (for 𝑖 = 5 , , ). Recall that 𝑃 is an iteration of length 𝛿 . We set 𝑃 ≔ 𝑗 ( 𝑃 ) , 𝑃 ≔ 𝑗 ( 𝑃 ) ,and 𝑃 ≔ 𝑗 ( 𝑃 ) ; and 𝛿 ≔ 𝑗 ( 𝛿 ) , 𝛿 ≔ 𝑗 ( 𝛿 ) and 𝛿 ≔ 𝑗 ( 𝛿 ) .It is enough to show the following:(a) 𝑃 𝑖 is a FS ccc iteration of length 𝛿 𝑖 and forces ℵ = 𝜆 𝑖 for 𝑖 = 4 , , , .(b) 𝑃 𝑖 ⊩ ( add( ) = 𝜆 & 𝔟 = 𝜆 & 𝔡 = 𝜆 ) for 𝑖 = 4 , , , .(c) 𝑃 𝑖 ⊩ non( ) ≥ 𝜆 for 𝑖 = 5 , , . 𝑃 𝑖 ⊩ cof( ) ≥ 𝜆 for 𝑖 = 6 , . 𝑃 𝑖 ⊩ cov( ) ≤ 𝜆 for 𝑖 = 4 , , , .(d) 𝑃 𝑖 ⊩ cof( ) = 𝜆 for 𝑖 = 6 , .(e) 𝑃 𝑖 ⊨ ( cov( ) ≥ 𝜆 & non( ) ≤ 𝜆 ) for 𝑖 = 4 , , , .(a) was shown in Section 2.3.(b): For 𝑃 this is Theorem 1.19. For 𝑃 use Lemma 2.16 (using for 𝔡 that 𝜅 < 𝜆 ).Using the same lemma again we get the result for 𝑃 and 𝑃 (using that 𝜅 𝑖 < 𝜆 for 𝑖 = 6 , as well.)(c): As 𝜅 > 𝜆 , we have ⊚ ( 𝑃 , 𝜅 ) (by Lemma 1.18), and thus ⊚ ( 𝑃 , 𝜆 ) (byLemma 2.17, as cf( 𝑗 ( 𝜅 )) = 𝜆 ), so 𝑃 ⊩ non( ) ≥ 𝜆 (Lemma 1.8). Repeating thesame argument we get ⊚ ( 𝑃 𝑖 , 𝜆 ) for 𝑖 = 6 , (as 𝜅 𝑖 ≠ 𝜆 for 𝑖 = 6 , ). OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 13
Analogously, as 𝜅 > 𝜆 , we start with ⊚ ( 𝑃 , 𝜅 ) , get ⊚ ( 𝑃 , 𝜅 ) (as 𝜅 ≠ 𝜅 ) andthen ⊚ ( 𝑃 , 𝜆 ) (as cf ( 𝑗 ( 𝜅 )) = 𝜆 ) and ⊚ ( 𝑃 , 𝜆 ) (again as 𝜅 ≠ 𝜆 ). So we get thus 𝑃 𝑖 ⊩ cof ( ) ≥ 𝜆 for 𝑖 = 6 , .Similarly, ⊚ ( 𝑃 , 𝜆 ) holds, which is preserved by all embeddings, so we get cov( ) ≤ 𝜆 .(d): As 𝑃 forces the continuum to have size 𝜆 , the previous item implies 𝑃 ⊩ cof( ) = 𝜆 . And as in (b), this implies the same for 𝑃 (as 𝜅 < 𝜆 , the value of add( ) ).(e): ⊞ ( 𝑃 , 𝜆 , 𝜆 ) holds (cf. Lemma 1.20). So by Lemma 2.18 for the case 𝜅 > 𝜆 ,and as | 𝑗 ( 𝜆 ) | = 𝜆 , according to Lemma 2.12(e), ⊞ ( 𝑃 , 𝜆 , 𝜆 ) holds. I.e., 𝑃 forces cov( ) ≥ 𝜆 and non( ) ≤ 𝜆 (the latter being trivial as the continuum has size 𝜆 ).For 𝑖 = 6 , , the same lemma, now for the case 𝜅 < 𝜆 , gives ⊞ ( 𝑃 𝑖 , 𝜆 , 𝜆 ) , i.e., 𝑃 𝑖 forces cov( ) ≥ 𝜆 and non( ) ≤ 𝜆 . (cid:3) An alternative.
In the same way we can prove the consistency of ℵ < add( ) < cov( ) < non( ) < cov( ) < non( ) < cof ( ) < ℵ . (I.e., we can replace 𝔟 and 𝔡 by non( ) and cov( ) , respectively.)For this, we use the following relation as R : 𝑓 R 𝑔 , if 𝑓 , 𝑔 ∈ 𝜔 𝜔 and (∀ ∗ 𝑛 ∈ 𝜔 ) 𝑓 ( 𝑛 ) ≠ 𝑔 ( 𝑛 ) . By a result of [Mil82, Bar87] (cf. [BJ95, 2.4.1 and 2.4.7]) we have non( ) = 𝔟 and cov( ) = 𝔡 . As before, we use that an iteration where each iterand has size <𝜆 is (R , 𝜆 ) -good.To define 𝑃 , we use partial eventually different (instead of partial Hechler) forcings.Unlike for ( 𝔟 , 𝔡 ) , we do not know whether non( ) = 𝜆 is generally preserved if 𝜅 ≠ 𝜆 and cov( ) = 𝜆 is preserved if 𝜅 is small; but we can use the same argument for (non( ) , cov( )) that we have used for (cov( ) , non( )) . So we can get the analogof Lemma 1.20 that proves that non( ) is large and cov( ) small; and ⊚ implies that non( ) is small and cov( ) large. R EFERENCES[Bar87] Tomek Bartoszyński. Combinatorial aspects of measure and category.
Fund. Math. , 127(3):225–239,1987.[BJ95] Tomek Bartoszyński and Haim Judah.
Set theory . A K Peters, Ltd., Wellesley, MA, 1995. On thestructure of the real line.[Bre91] Jörg Brendle. Larger cardinals in Cichoń’s diagram.
J. Symbolic Logic , 56(3):795–810, 1991.[FGKS17] Arthur Fischer, Martin Goldstern, Jakob Kellner, and Saharon Shelah. Creature forcingand five cardinal characteristics of the continuum.
Arch. Math. Logic , 56:1045–1103, 2017.doi:10.1007/s00153-017-0553-8.[GKSnt] Martin Goldstern, Jakob Kellner, and Saharon Shelah. Cichoń’s maximum. https://arxiv.org/abs/1708.03691 , preprint.[GMS16] Martin Goldstern, Diego Alejandro Mejía, and Saharon Shelah. The left side of Cichoń’s diagram.
Proc. Amer. Math. Soc. , 144(9):4025–4042, 2016.[JS90] Haim Judah and Saharon Shelah. The Kunen-Miller chart (Lebesgue measure, the Baire property,Laver reals and preservation theorems for forcing).
J. Symbolic Logic , 55(3):909–927, 1990.[Kam89] Anastasis Kamburelis. Iterations of Boolean algebras with measure.
Arch. Math. Logic , 29(1):21–28,1989.[KT64] H. J. Keisler and A. Tarski. From accessible to inaccessible cardinals. Results holding for all accessiblecardinal numbers and the problem of their extension to inaccessible ones.
Fund. Math. , 53:225–308,1963/1964.[Man71] Richard Mansfield. The theory of Boolean ultrapowers.
Ann. Math. Logic , 2(3):297–323, 1970/71.
OMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM 14 [Mej13] Diego Alejandro Mejía. Matrix iterations and Cichon’s diagram.
Arch. Math. Logic , 52(3-4):261–278,2013.[Mil82] Arnold W. Miller. A characterization of the least cardinal for which the Baire category theorem fails.
Proc. Amer. Math. Soc. , 86(3):498–502, 1982.[MS16] M. Malliaris and S. Shelah. Existence of optimal ultrafilters and the fundamental complexity of simpletheories.
Adv. Math. , 290:614–681, 2016.[RS] D. Raghavan and S. Shelah. Boolean ultrapowers and iterated forcing. In preparation.
E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected] I NSTITUTE OF D ISCRETE M ATHEMATICS AND G EOMETRY , T
ECHNISCHE U NIVERSITÄT W IEN (TU W