Cicho\'n's maximum without large cardinals
Martin Goldstern, Jakob Kellner, Diego A. Mej\'ia, Saharon, Shelah

TL;DR
This paper proves the consistency of all twelve cardinal characteristics in Cichoń's diagram being pairwise different without relying on large cardinal assumptions, which was previously unachievable.
Contribution
It demonstrates the consistency of Cichoń's diagram with all entries pairwise distinct without large cardinal hypotheses, advancing understanding in set theory.
Findings
All entries of Cichoń's diagram can be pairwise different without large cardinals.
Established new consistency results in set theory.
Removed the need for large cardinal assumptions in this context.
Abstract
Cicho\'n's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and -compact subsets of the irrationals. It is consistent that all entries of Cicho\'n's diagram are pairwise different (apart from and , which are provably equal to other entries). However, the consistency proofs so far required large cardinal assumptions. In this work, we show the consistency without such assumptions.
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Cichoń’s maximum without large cardinals
Martin Goldstern
Institut für Diskrete Mathematik und Geometrie, TU Wien, 1040 Vienna, Austria.
[email protected] http://www.tuwien.ac.at/goldstern/ ,
Jakob Kellner
Institut für Diskrete Mathematik und Geometrie, TU Wien, 1040 Vienna, Austria.
[email protected] http://dmg.tuwien.ac.at/kellner/ ,
Diego A. Mejía
Creative Science Course (Mathematics), Faculty of Science, Shizuoka University, Ohya 836, Suruga-ku, Shizuoka-shi, Japan 422-8529.
[email protected] http://www.researchgate.com/profile/Diego_Mejia2 and
Saharon Shelah
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel, and Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA.
[email protected] http://shelah.logic.at
(Date: April 16, 2020)
Abstract.
Cichoń’s diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and -compact subsets of the irrationals.
It is consistent that all entries of Cichoń’s diagram are pairwise different (apart from and , which are provably equal to other entries). However, the consistency proofs so far required large cardinal assumptions.
In this work, we show the consistency without such assumptions.
2010 Mathematics Subject Classification:
03E17, 03E35, 03E40
This work was supported by the following grants: Austrian Science Fund (FWF): project number I3081, P29575 (first author) and P26737, P30666 (second author); Grant-in-Aid for Early Career Scientists 18K13448, Japan Society for the Promotion of Science (third author); European Research Council grant 338821 (fourth author). This is publication number 1177 of the fourth author.
Introduction
How many Lebesgue null sets do we need to cover the real line? Countably many are not enough, as the countable union of null sets is null; and continuum many are enough, as .
The answer to this question (and similar ones) is called a cardinal characteristic (sometimes also called cardinal invariant); in our case the characteristic is called “”.
As we have argued, . So if the Continuum Hypothesis (CH) holds, then . It has been shown by Gödel [Göd40] and Cohen [Coh63] that CH is independent of ZFC. I.e., one can prove: If ZFC is consistent, then so is ZFCCH as well as ZFCCH.
Under CH, could be some cardinal less than , and one can indeed show that , and are all consistent.
Some more characteristics associated with the -ideal of null sets are defined:
- •
is the smallest number of null sets whose union is not null.
- •
is the smallest cardinality of a non-null set.
- •
is the smallest size of a cofinal family of null sets, i.e., a family that contains for each null set a superset of .
Replacing with another -ideal gives us the analogously defined characteristics for . In particular, for the meager ideal we get , , , .
For the -ideal ctbl of countable sets, it is easy to see that and , which is also called (for “continuum”).
For , the -ideal generated by the compact subsets of the irrationals, it turns out that . This characteristic is more commonly called . We also have , called .
These characteristics are customarily displayed in Cichoń’s diagram, see Figure 1.
An arrow from to indicates that ZFC proves . Moreover, one can show that and . A series of results [Bar84, CKP85, BJS93, JS90, Kam89, Mil81, Mil84, RS83, RS85], summarized in [BJ95, Ch. 7], proves these (in)equalities in ZFC and shows that they are the only ones provable. More precisely, all assignments of the values and to the nine “independent” characteristics in Cichoń’s diagram (excluding and including ) are consistent with ZFC, provided they honor the inequalities given by the arrows.
This leaves the question on how to separate more than two entries simultaneously. There was a lot of progress in recent years, giving four and up to seven values [Mej13, FGKS17, GMS16, FFMM18, Mej19a]. Finally, it was shown [GKS19] that the following statement, which we call “Cichoń’s maximum”, is consistent:
The maximal possible number of entries of Cichoń’s diagram, i.e., all ten “independent” entries (including and ), are pairwise different.
However, the proof required four Boolean ultrapower embeddings, constructed from four strongly compact cardinals.111 A simpler example of this Boolean ultrapower construction, giving only eight different values and using three compacts, can be found in [KTT18]; and later a construction for Cichoń’s maximum requiring only three compacts was given in [BCM18]. [Git19] notes that superstrongs are sufficient for the constructions. However until now all proofs showing the consistency of eight or more different values needed some large cardinals assumptions. A strongly compact cardinal is an example of a so-called “large cardinal” (LC). Such cardinals turned out to be an important scale for measuring consistency strengths of mathematical (and in particular set theoretic) statements: There are many examples of statements where one cannot prove
The consistency of ZFC implies the consistency of (ZFC plus ),
but only:
The consistency of (ZFC plus LC) implies the consistency of (ZFC plus )
for some specific large cardinal axiom LC. In many cases, one can even show that is equiconsistent to LC (i.e., one can also prove that the consistency of (ZFC plus ) implies the consistency of (ZFC plus LC)). For example, “there is an extension of Lebesgue measure to a -complete measure which measures all sets of reals” is equiconsistent with a so-called measurable cardinal (a notion much weaker than a strongly compact).
In case of being Cichoń’s maximum, we previously could only prove an upper bound for the consistency strength, but conjectured that Cichoń’s maximum is actually equiconsistent with ZFC. This turns out to be correct.
In this work, we introduce a new method to control cardinal characteristics when modifying a finite support ccc iteration (by taking intersections with -closed elementary submodels). This method can replace the Boolean ultrapower embeddings in previous constructions, so in particular we can get Cichoń’s maximum without assuming large cardinals. Furthermore, we can get arbitrary regular cardinals as the values of the entries in Cichoń’s diagram. As the method is quite general, we expect that it can be applied to control the values of other characteristics, in other constructions, as well.
This paper should be reasonably self-contained (modulo an understanding of forcing, such as presented in [Kun11]). However, in Section 2 we just quote the result (from [GKS19] or alternatively from [BCM18]) that a suitable preparatory forcing for the left hand side exists, without proofs or much explanation.
Annotated contents:
- S. 1
We define the properties and for a forcing , which give us the “strong witnesses” that will guarantee the desired equalities (or rather: both sides of the required inequalities) for the respective cardinal characteristics. We show how these properties are preserved when intersecting with a -complete elementary submodel.
- S. 2
We just quote (without proof) the result from [GKS19] (or [BCM18]) that a suitable forcing for the left hand side with suitable and properties exists.
- S. 3
We prove the main result: There is a complete subforcing of which forces ten different values to Cichoń’s diagram (we can actually choose any desired regular values).
- S. 4
We remark that the same argument can be applied to alternative “initial forcings” for the left hand side. In particular, using a construction of [KST19], we get another ordering of the ten entries in Cichoń’s diagram.
- S. 5
We list some open questions regarding alternative orders of Cichoń’s diagram with ten values.
1. The and properties and -closed
elementary submodels
Let be a binary relation on some basic set . The cardinal , the bounding number of , is the minimal size of an unbounded family. I.e.,
[TABLE]
Dually, , the dominating number of , is the minimal size of a dominating family. I.e.,
[TABLE]
We will use these notions in two situations:
On the one hand, may be a directed partial order (or a linear order) without largest element, such as or . Then we will call the completeness of and denote it by ; and we call the cofinality of and denote it by . Note that is -directed iff (as we assume that is directed). If in addition is linear without a maximal element, then is an infinite regular cardinal.
On the other hand, may be a (possibly non-transitive) Borel relation on the reals (more generally: a sufficiently absolute definition of a binary relation on the reals), and we get the cardinal characteristics of the continuum and . Note that , where we define the dual relation by iff . All entries of Cichoń’s diagram are of this form, for quite natural relations . (For more details, see the references after Theorem 2.4.)
In the following we give definitions of and which are notational variants222There are other variants of these definitions that do not mention forcings ([GKMS19, Def. 2.11]) but are applied to the extension . These variants are basically equivalent. of the definitions given in [GKS19, Def. 1.8 & 1.15].
We investigate relations on the reals, and fix as representation of the reals. (This choice is irrelevant, and we could use any of the other usual representations as well. We just pick one so that we can later refer to the reals as a well defined object, and so that we can e.g. use in formulas.)
Definition 1.1**.**
Assume is a binary relation on which is Borel, or just sufficiently absolutely defined.333The discussion after (1.4) shows which amount of absoluteness is sufficient for us. We will need non-Borel relations only in Subsection 4.1.
- •
For a directed partial order without maximal elements, the “cone of bounds” property says: There is a sequence444If is a (partially) ordered set, we sometimes use “a sequence indexed by ” as synonym for “a function with domain ”. of -names of reals such that for any -name of a real there is an such that
[TABLE]
- •
For a linear order without largest element, the “linear cofinal unbounded” property is defined as:
There is a sequence of -names of reals such that for each -name of a real there is an such that
[TABLE]
(When writing , we of course mean that we evaluate the definition of in the extension.)
Actually, is a special case of :
[TABLE]
(again, denotes the dual of ). However, and will play different roles in our arguments, so we prefer to have different notations for these two concepts.
The following is basically the same as [GKS19, Lem. 1.9 & 1.16] (see also [GKMS19, Fact 2.14]):
Lemma 1.3**.**
- (1)
Let be a -directed partial order without a largest element, and let be cofinal. Then is equivalent to , and implies
[TABLE] 2. (2)
Let be linear without a largest element and set . (So is an infinite regular cardinal.) Then is equivalent to , and implies555We actually do mean and not just , i.e., if is not a cardinal in the extension anymore, then we have . But this is irrelevant in our application, as will preserve .**
[TABLE]
Proof.
Regarding the equivalence: Let witness . Then witnesses . On the other hand, if witnesses , then we assign to every some above , and set . Then witnesses .
From now on assume that witnesses . Regarding , note that is forced to be dominating.
Regarding , assume that forces that is of size less than (the ordinal) . Fix , and -names of reals such that . For each let be an element of satisfying the requirement for . As is -directed, there is some above all , i.e., for all . Accordingly, cannot force to be unbounded.
The claims on follow from the ones on by (1.2) (together with the fact that for linear orders , and that ). ∎
In the following results we show that when we restrict a poset to a -closed elementary submodel of some , then the and properties still hold (when we intersect the parameter with as well). These are simple technical tools we will use to prove the main results.
Assume that is regular, -cc, is -closed and . Then is again -cc and thus a complete subforcing of . So given a -generic over , there is a -generic over extending . Note that is -generic over as well, and that .
There is a correspondence of -names for reals and -names for reals, such that and for all and sufficiently absolute ,
[TABLE]
In a bit more detail: A “nice -name for a -subset” (for an ordinal ) is a sequence \bar{h}:=\big{(}(h_{n},A_{n})\big{)}_{n<\zeta} such that is a maximal antichain in and (evaluated in the generic extension as ). As , every nice -name for a -subset is also a nice -name, and furthermore whenever (as is -closed). On the other hand, if then every nice -name for a -subset which is in is actually a nice -name. Note that if is Borel, then we are done with showing (1.4). For a more general formula , note that we have just shown that , and using an absolute bijection between and , we get that . So (1.4) holds whenever is, e.g., (provably) absolute between the universe and (for as well as for sufficiently large), where may use elements of (or names for such elements) as parameters.
Lemma 1.5**.**
Assume is -cc for some uncountable regular and is -closed. Then is a -cc complete subforcing of . Assume in the following that , , , , are in .
- (1)
* implies .*
*So if we set and , then *
* implies .* 2. (2)
* implies .*
*So if we set , then *
* implies .*
Proof.
Let witness in . Then witnesses : Assume is a -name for a real. As above we interpret it as a -name in . So thinks there is some such that for all , . So by absoluteness (1.4), for every in we get .
Again, (2) is a special case of (1). ∎
Lemma 1.6**.**
Let be cardinals with and uncountable regular, a directed set without maximal elements, a regular cardinal, and let be a -cc poset. Assume is an increasing sequence of -closed elementary submodels of , where is a fixed, sufficiently large666It is enough to assume , , and are in . regular cardinal. Assume that , that , and that for any . Set (which is also a -closed elementary submodel).
- (1)
, where
*In particular implies . * 2. (2)
*. * 3. (3)
If , then is cofinal in , and in particular has the same cofinality and completeness as . 4. (4)
If , then .
In particular implies .
Proof.
For (2), the assumptions of Lemma 1.5 are sufficient: Assume that has size less than . As is -closed, . By absoluteness, knows that the set (which is smaller than after all) has an upper bound, so there is an upper bound of in .
(3) only requires that and : In , let be a cofinal subset of size . Since , we have , so is cofinal in . And it is clear that any cofinal subset of a partial order has the same completeness and cofinality as the order itself.
For (4), fix . Since , there is some bounding . In fact, we can find such in because . Hence, is a cofinal increasing sequence of , so . The claim on follows from Lemmas 1.5(1) and 1.3(1).
For (1), if then, by (4) applied to , ; if then , so . The claim on follows from Lemmas 1.5(2) and 1.3(2). ∎
2. The forcing for the left hand side
We set to be the following pairs of dual characteristics in Cichoń’s diagram:
[TABLE]
We will use for each two Borel relations777Actually, in most cases we will use the same and , which is moreover the “canonical” choice for . See the explanation that follows Theorem 2.4. on , and , in such a way that ZFC proves
[TABLE]
We write instead of and instead of .
It is useful to have relations satisfying (2.2), because in this way we get:
Corollary 2.3**.**
* for regular implies .*
* for and implies .*
Theorem 2.4**.**
Assume GCH and fix regular cardinals such that each is the successor of a regular cardinal.
We can choose satisfying (2.2) and construct a ccc poset such that the following holds for :
- (a)
If then, for all regular such that , holds. In the case , and hold.
- (b)
There is a directed order with and such that holds.
Accordingly, forces
[TABLE]
This theorem is proved in [GKS19]; we will not repeat the proof here but instead point out where to find the definitions and proofs in the cited papers (the italic labels in the following paragraph refer to the cited paper):
Def. 1.2 defines relations called for . These are, apart from , the “canonical” relations for . They play the role of and, apart from , also of . is implicitly defined in Def. 1.17 as the canonical relation: iff is not in the Borel null set coded by . Lem. 1.3 corresponds to (2.2) in this work, and Thm. 1.35 is our Theorem 2.4.
Remark 2.5**.**
In [BCM18, Thm. 5.3] a different construction is presented, which gives a stronger conclusion and requires the weaker assumption that are just regular cardinals. If we use this paper, then for all , see [BCM18, Exm. 2.16] (where corresponds to item ).
3. Cichoń’s maximum without large cardinals
Theorem 3.1**.**
Assume GCH and is a weakly increasing sequence of cardinals with regular for and . Then there is a ccc poset forcing that
[TABLE]
Full GCH is not actually required, see Remark 3.5.
Note that the are required to be only weakly increasing, i.e., we can replace each in the inequality of characteristics by either or at will. So we get the consistency of many different “sub-constellations” in Cichoń’s diagram. Of course several of these have been known to be consistent before (even without large cardinals). E.g., the sub-constellation where we always choose “” is just CH.
Proof.
We fix an increasing sequence of cardinals (see Figure 2)
[TABLE]
such that the following holds:
- (1)
All cardinals are regular, with the possible exception of , 2. (2)
. 3. (3)
GCH, plus is the successor of a regular cardinal for .
I.e., the assumptions for Theorem 2.4 are satisfied if we set
[TABLE]
So we can apply Theorem 2.4, resulting in the forcing . (Thus forces the situation shown in the upper Cichoń diagram of Figure 2.)
We will now construct a forcing (a complete subforcing of ) which forces for all , and (i.e., the situation shown in the lower Cichoń diagram of Figure 2).
We fix for , as well as , satisfying the following for any :
- •
Each as well as is an elementary submodel of and contains (as elements) the sequences of ’s and ’s, as well as and (the directed orders provided by Theorem 2.4) for .
- •
contains as well as .
contains .
- •
, and is -closed (thus ).888For , -closed is enough; for , -closed is sufficient.
- •
We set . Note that is -closed and has size .
- •
is -closed and has size .
- •
We set .
- •
For , we set and .
Note that is again an elementary submodel of ,999If and then and . and accordingly each is a complete subforcing of .
Regarding : We fix (the case , as an example, is described more explicitly below). Let us call the set of regular cardinals satisfying the “-spectrum of ”, and let be the -spectrum of . So
[TABLE]
- •
In the first step , let us consider the -spectrum of : As , we get , and as are in , they are in as well (both according to Lemma 1.6(1), using ).
- •
For the next step , we similarly get that the -spectrum of contains , and, if , also .
- •
In this way we get that the final -spectrum of contains .
- •
This implies (by Corollary 2.3) that forces
[TABLE]
So we get half of the desired inequalities.
This may be more transparent if we consider an explicit example, say . In each line of the following table, each cardinal in the right column is guaranteed to be an element of the spectrum of the forcing notion in the left column:
[TABLE]
Since is the smallest of these 4 cardinals, and the largest, we get that forces and .
Regarding : Again we fix . Let . In particular, , is even, so according to (3.2) we have and
[TABLE]
Recall that holds where and (cf. Theorem 2.4 and (3.3)).
We claim that
[TABLE]
satisfies
[TABLE]
Completeness is clear by applying Lemma 1.6(2) iteratively: , so . Then , and so on.
Regarding the cofinality:
- •
Let be the product . So by (3.4).
For , set . Note that , and that is an element, and thus a subset, of each elementary submodel.101010Element is clear, as all ’s contain the sequence of ’s. Subset follows from the fact that each contains and thus as a subset, and that .
- •
For , set . Since is -closed and , we get . Hence, by Lemma 1.6(4) applied to , , and , we conclude .
Choose cofinal in of size . Hence, is cofinal in because , so by (3.4).
Now we show, by induction on , that has completeness and cofinality . The step was done above; for the steps , by induction we know that has cofinality at most and completeness at least . So by Lemma 1.6(3), the same holds for .
To summarize: For any , the cofinality of is at most , and the completeness at least . By Lemmas 1.5(2) and 2.3(2) we get
[TABLE]
So we get the remaining inequalities we need.
Regarding the continuum: There is a sequence of (nice) -names of reals that are forced to be pairwise different due to absoluteness (1.4). Note that this sequence belongs to , so is a sequence of -names of reals that are forced (by ) to be pairwise different. Hence, forces .111111This argument can be written in terms of the property for the identity relation on : As holds for all regular (even up to ), we get for all these cardinals, which implies . The converse inequality also holds because .∎
Remark 3.5**.**
If we base the left-hand forcing on [BCM18] (see Remark 2.5), then our proof (when we change item (3) on p. 3 to the assumptions listed in Remark 2.5) shows that GCH can be weakened to the following: There are at least 9 cardinals satisfying . Or, to be even be more pedantic: There are regular cardinals larger than such that , and for .
4. Variants
4.1. Another order
The paper [KST19] constructs (assuming GCH) a ccc forcing notion which forces another ordering of the left hand side. More concretely, is ccc and it has and witnesses for the following:121212In (2.1), the order/numbering of and is swapped; for this new ordering we again get Theorem 2.4. We use the same - and -relations as in [GKS19], except for the -relation for the pair : Now we have to use a relation which is an -union of Borel relations (which was originally defined in [KO14] and fit into a formal preservation framework in [CM19]; see details in [KST19, Def. 2.3]). This is the only place in this paper where we have to use a non-Borel relation ; but this is no problem as is sufficiently absolute in the sense described after (1.4).
[TABLE]
If we use this forcing instead of , then the same argument shows that we can find a complete subforcing that extends the order to the right hand side:
Theorem 4.1**.**
Assume GCH and let be a weakly increasing sequence of cardinals with regular for and . Then there is a ccc poset forcing that
[TABLE]
Remark 4.2**.**
As in Remark 3.5, full GCH is not needed, but it is enough that there are regular cardinals larger than satisfying some arithmetical properties. However, it is not enough that for these cardinals, but it is required in addition that one of them is -inaccessible.131313Recall that a cardinal is -inaccessible if for every and . For details, refer to [Mej19b, GKMS19].
4.2. A weaker notion than sufficient for the proof
Several papers about constellations of Cichoń’s diagram preceding [GKS19, BCM18], such as [Bre91, Mej13, GMS16], have considered similar, but simpler, forcing constructions. While witnesses are added in the same way, these do not provide for . Instead, a weaker property, which we call below, is implicit in these constructions. We now show that this notion is sufficient to carry out the proof of the main result.
Definition 4.3**.**
Let be a relation on and let be a cardinal.
- (1)
A set is --dominating if, whenever has size , there is some real dominating over , that is, . Dually, we say that is --unbounded if it is --dominating. 2. (2)
Assume that is sufficiently absolutely defined and let be a forcing notion. We define to mean the following: There is a sequence of -names of reals such that, whenever and is a sequence of -names of reals, there is some such that .
(Note that is stronger than just saying “ adds a --dominating family”.)
The following is straightforward:
- •
implies .
- •
If is regular then implies .
- •
implies P\Vdash\bigl{(}\,\kappa\leq\mathfrak{b}_{R}\ \&\ \mathfrak{d}_{R}\leq|S|\,\bigr{)}.
(This generalizes Lemma 1.3.)
For this weaker notion we have the following result similar to Lemma 1.6.
Lemma 4.4**.**
With the same hypothesis as in Lemma 1.6, assuming also :
- (1)
* implies .* 2. (2)
If then implies . 3. (3)
If then implies .
In particular, if is directed and then implies .
Proof.
In the following, assume that witnesses .
(1) If is a sequence of -names of reals and then the sequence is in , so there is some such that for all . By absoluteness, forces the same.
(2) is clear because (as and ).
(3) Fix . Since , there is some such that for all that are -names for reals. In fact, we can find such in . Hence, witnesses . ∎
As in [Mej19b], a simpler version of can be constructed in such a way that
- (a)
of Theorem 2.4 holds, and
- (b’)
For there is some set of size such that holds.
Thanks to Lemma 4.4 (in particular item (3)), the same proof of Theorem 3.1 can be carried out in this simpler context.
5. Open questions
[GKS19, Sect. 3] asks the following questions: Can you show the consistency of Cichoń’s maximum …
- ✓(a)
…without using large cardinals? 2. ✓(b)
…for specific (regular) values, such as ? 3. (c)
…for other orderings of the ten entries? 4. (d)
…together with further distinct values of additional (”classical”) cardinal characteristics?
This work, more concretely Theorem 3.1, solves questions (a) and (b).
A first result for (d), namely adding , is done in [GKMS19] (which also gives a more complicated construction to achieve (b)).
Of course, it would be interesting to add more characteristics. For example, we can ask:
Question 1**.**
Can we add the splitting number and the reaping number ?
The pair might be most promising among the classical characteristics, as it is of the form for a Borel relation which is well understood.
Question (c) remains largely open. There are four possible configurations where , and at the moment only are known to be consistent (see Theorems 3.1 and 4.1).
Question 2**.**
Are the following two constellations consistent?
[TABLE]
[TABLE]
It is not clear whether our method in Section 3 can be applied to solve this question (the same applies to Boolean ultrapowers), since we start with a poset forcing an order of the left side of Cichoń’s diagram and our method only manages to dualize this order to the right side (e.g., if on the left we force , then on the right we can only expect to force the dual inequality ).
The case when seems to be more complex.141414Recall that finite support iterations add Cohen reals at limit steps, so they force (when the length has uncountable cofinality). We do not even know how to force the consistency of . J. Brendle however does, see [Bre19] for slides of a presentation of his method of “shattered iterations”. Brute force counting shows that there are configurations of ten different values in Cichoń’s diagram (satisfying the obvious inequalities) where , but none of them have been proved to be consistent so far.
Question 3**.**
Is any constellation of Cichoń’s maximum consistent where ?
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[Bar 84] Tomek Bartoszyński, Additivity of measure implies additivity of category , Trans. Amer. Math. Soc. 281 (1984), no. 1, 209–213.
- 2[BCM 18] Jörg Brendle, Miguel A. Cardona, and Diego A. Mejía, Filter-linkedness and its effect on preservation of cardinal characteristics , preprint, ar Xiv:1809.05004 , 2018.
- 3[BJ 95] Tomek Bartoszyński and Haim Judah, Set theory , A K Peters, Ltd., Wellesley, MA, 1995, On the structure of the real line.
- 4[BJS 93] Tomek Bartoszyński, Haim Judah, and Saharon Shelah, The Cichoń diagram , J. Symbolic Logic 58 (1993), no. 2, 401–423.
- 5[Bre 91] Jörg Brendle, Larger cardinals in Cichoń’s diagram , J. Symbolic Logic 56 (1991), no. 3, 795–810.
- 6[Bre 19] Brendle, Jörg, Forcing and cardinal invariants, parts 1 and 2 , tutorial at Advanced Class - Young Set Theory Workshop 2019, July 28 th and 29 th 2019, Slides: https://sites.google.com/view/estc 2019/advanced-class-yst/program .
- 7[CKP 85] J. Cichoń, A. Kamburelis, and J. Pawlikowski, On dense subsets of the measure algebra , Proc. Amer. Math. Soc. 94 (1985), no. 1, 142–146.
- 8[CM 19] Miguel A. Cardona and Diego A. Mejía, On cardinal characteristics of Yorioka ideals , Mathematical Logic Quarterly 65 (2019), no. 2, 170–199.
