Cicho\'n's maximum without large cardinals

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 $\sigma$-compact subsets of the irrationals. It is consistent that all entries of Cicho\'n's diagram are pairwise different (apart from $\textrm{add}(\mathcal{M})$ and $\textrm{cof}(\mathcal{M})$, 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.


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 r∈R {r} = R.
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 "cov(N )".
As we have argued, ℵ 0 < cov(N ) ≤ 2 ℵ0 . So if the Continuum Hypothesis (CH) holds, then cov(N ) = 2 ℵ0 . 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 ZFC+CH as well as ZFC+¬CH.
Some more characteristics associated with the σ-ideal N of null sets are defined: • add(N ) is the smallest number of null sets whose union is not null.
• non(N ) is the smallest cardinality of a non-null set.
• cof(N ) is the smallest size of a cofinal family of null sets, i.e., a family that contains for each null set N a superset of N . Replacing N with another σ-ideal I gives us the analogously defined characteristics for I. In particular, for the meager ideal M we get add(M), non(M), cov(M), cof(M).
For the σ-ideal ctbl of countable sets, it is easy to see that add(ctbl) = non(ctbl) = ℵ 1 and cov(ctbl) = cof(ctbl) = 2 ℵ0 , which is also called c (for "continuum").  In the version on the right, the two "dependent" values are removed; the "independent" ones remain (nine entries excluding ℵ 1 , or ten including it). It is consistent that these ten entries are pairwise different.
For K, the σ-ideal generated by the compact subsets of the irrationals, it turns out that add(K) = non(K). This characteristic is more commonly called b. We also have cov(K) = cof(K), called d.
These  [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 ℵ 1 and ℵ 2 to the nine "independent" characteristics in Cichoń's diagram (excluding ℵ 1 and including c) 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 ℵ 1 and c), are pairwise different. However, the proof required four Boolean ultrapower embeddings, constructed from four strongly compact cardinals. 1 A strongly compact cardinal is an example of a socalled "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 ϕ where one cannot prove The consistency of ZFC implies the consistency of (ZFC plus ϕ), but only: The consistency of (ZFC plus LC) implies the consistency of (ZFC plus ϕ) 1 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.
for some specific large cardinal axiom LC. In many cases, one can even show that ϕ is equiconsistent to LC (i.e., one can also prove that the consistency of (ZFC plus ϕ) implies the consistency of (ZFC plus LC)). For example, "there is an extension of Lebesgue measure to a σ-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 ϕ 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 σ-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 P pre for the left hand side exists, without proofs or much explanation.
Annotated contents: S. 1 We define the properties LCU and COB for a forcing P , 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 P with a σ-complete elementary submodel. S. 2 We just quote (without proof) the result from [GKS19] (or [BCM18]) that a suitable forcing P pre for the left hand side with suitable LCU and COB properties exists. S. 3 We prove the main result: There is a complete subforcing P fin of P pre 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 LCU and COB properties and σ-closed elementary submodels Let R be a binary relation on some basic set Y . The cardinal b R , the bounding number of R, is the minimal size of an unbounded family. I.e., Dually, d R , the dominating number of R, is the minimal size of a dominating family. I.e., d R := min |D| : D ⊆ Y, (∀f ∈ Y ) (∃g ∈ D) f Rg .
We will use these notions in two situations: On the one hand, R may be a directed partial order (or a linear order) without largest element, such as ([X] <κ , ⊆) or (κ, ∈). Then we will call b R the completeness of R and denote it by comp(R); and we call d R the cofinality of R and denote it by cof(R). Note that R is <λ-directed iff λ ≤ comp(R) (as we assume that R is directed). If in addition R is linear without a maximal element, then cof(R) = comp(R) is an infinite regular cardinal.
On the other hand, R 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 b R and d R . Note We investigate relations on the reals, and fix ω ω 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 (∀x ∈ ω ω ) in formulas.) Definition 1.1. Assume R is a binary relation on ω ω which is Borel, or just sufficiently absolutely defined. 3 • For a directed partial order (S, ≤ S ) without maximal elements, the "cone of bounds" property COB(P, S) says: There is a sequence 4 (g s ) s∈S of Pnames of reals such that for any P -name f of a real there is an s ∈ S such that P (∀t ≥ S s) f Rg t .
• For a linear order L without largest element, the "linear cofinal unbounded" property LCU R (P, L) is defined as: There is a sequence (c α ) α∈L of P -names of reals such that for each Pname g of a real there is an α 0 ∈ L such that (When writing P f Rg, we of course mean that we evaluate the definition of R in the extension.) Actually, LCU is a special case of COB: (again, R ⊥ denotes the dual of R). However, LCU and COB will play different roles in our arguments, so we prefer to have different notations for these two concepts. (1) Let S be a <λ-directed partial order without a largest element, and let A ⊆ S be cofinal. Then COB R (P, S) is equivalent to COB R (P, A), and implies (2) Let L be linear without a largest element and set λ := cof(L). (So λ is an infinite regular cardinal.) Then LCU R (P, L) is equivalent to LCU R (P, λ), Proof. Regarding the equivalence: Let (g s ) s∈S witness COB R (P, S). Then (g s ) s∈A witnesses COB R (P, A). On the other hand, if (g s ) s∈A witnesses COB R (P, A), then we assign to every s ∈ S some a(s) ∈ A above s, and set g s := g a(s) . Then (g s ) s∈S witnesses COB R (P, S). From now on assume that (g s ) s∈A witnesses COB R (P, A). Regarding d R , note that {g s : s ∈ A} is forced to be dominating.
Regarding b R , assume that p 0 forces that X ⊆ ω ω is of size less than (the ordinal) λ. Fix p 1 ≤ p 0 , κ < λ and P -names (f α ) α∈κ of reals such that p 1 X = {f α : α ∈ κ}. For each α let s α be an element of S satisfying the COB requirement for f α . As S is <λ-directed, there is some t ∈ S above all s α , i.e., P f α Rg t for all α ∈ κ. Accordingly, p 0 cannot force X to be unbounded.
The claims on LCU follow from the ones on COB by (1.2) (together with the fact that for linear orders L, comp(L) = cof(L) and that In the following results we show that when we restrict a poset P to a σ-closed elementary submodel N of some H(χ), then the LCU and COB properties still hold (when we intersect the parameter with N as well). These are simple technical tools we will use to prove the main results.
Assume that κ is regular, P κ-cc, N H(χ) is <κ-closed and P ∈ N . Then P ∩ N is again κ-cc and thus a complete subforcing of P . So given a P ∩ N -generic G over V , there is a P -generic G + over V extending G. Note that G + is P -generic over N as well, and that N [ There is a correspondence of P ∩ N -names σ for reals and P -names τ ∈ N for reals, such that σ[G] = τ [G + ] and for all p ∈ P ∩ N and sufficiently absolute ϕ, In a bit more detail: A "nice Q-name for a ζ-subset" (for an ordinal ζ) is a sequencē h := (h n , A n ) n<ζ such that A n is a maximal antichain in Q and h n : A n → 2 (evaluated in the generic extension as {n ∈ ζ : (∃a ∈ G Q ∩ A n ) h n (a) = 1}). As P ∩ N P , every nice P ∩ N -nameh for a ζ-subset is also a nice P -name, and furthermoreh ∈ N whenever ζ < κ (as N is <κ-closed). On the other hand, if ζ < κ then every nice P -nameh for a ζ-subset which is in N is actually a nice P ∩N -name. Note that if ϕ is Borel, then we are done with showing (1.4). For a more general formula ϕ, note that we have just shown that N [G + ] ∩ 2 <κ = V [G] ∩ 2 <κ , and using an absolute bijection between 2 <κ and H(κ), we get that . So (1.4) holds whenever ϕ is, e.g., (provably) absolute between the universe and H(χ) (for χ = κ as well as for χ sufficiently large), where ϕ may use elements of H(κ) (or names for such elements) as parameters.
Lemma 1.5. Assume P is κ-cc for some uncountable regular κ and N H(χ) is <κ-closed. Then P ∩N is a κ-cc complete subforcing of P . Assume in the following that P , S, L, κ, R are in N . ( Proof. Let (f s ) s∈S witness COB R (P, S) in N . Then (f s ) s∈S∩N witnesses COB R (P ∩ N, S ∩ N ): Assume g ∈ V is a P ∩ N -name for a real. As above we interpret it as a P -name in N . So N thinks there is some s ∈ S such that for all t ≥ S s, P gRf t . So by absoluteness (1.4), for every t ≥ S s in N we get P ∩ N gRf t .
Lemma 1.6. Let κ ≤ λ ≤ θ be cardinals with κ and λ uncountable regular, S a directed set without maximal elements, ζ a regular cardinal, and let P be a κ-cc poset. Assume (N i ) i<λ is an increasing sequence of <κ-closed elementary submodels of H(χ), where χ is a fixed, sufficiently large 6 regular cardinal. Assume that |N i | = θ, that θ∪{θ, P, R, S, ζ} ⊆ N i , and that N i ∈ N i+1 for any i < λ. Set N := i<λ N i (which is also a <κ-closed elementary submodel).
Proof. For (2), the assumptions of Lemma 1.5 are sufficient: Assume that A ⊆ S∩N has size less than min(κ, comp(S)). As N is <κ-closed, A ∈ N . By absoluteness, N knows that the set A (which is smaller than comp(S) after all) has an upper bound, so there is an upper bound of A in S ∩ N .
(3) only requires that θ ∪ {θ} ⊆ N and |N | = θ: In N , let A ⊆ S be a cofinal subset of size cof(S). Since cof(S) ≤ θ ⊆ N , we have A ⊆ N , so A ⊆ S ∩ N is cofinal in S. And it is clear that any cofinal subset of a partial order has the same completeness and cofinality as the order itself.

The forcing for the left hand side
We set (b i , d i ) to be the following pairs of dual characteristics in Cichoń's diagram: We will use for each i two Borel relations 7 on ω ω , R LCU i and R COB i , in such a way that ZFC proves It is useful to have relations satisfying (2.2), because in this way we get: Theorem 2.4. Assume GCH and fix regular cardinals ℵ 1 < µ 1 < µ 2 < µ 3 < µ 4 < µ ∞ such that each µ n is the successor of a regular cardinal. We can choose R LCU i , R COB i satisfying (2.2) and construct a ccc poset P such that the following holds for i = 1, 2, 3, 4: (a) If i < 4 then, for all regular κ such that µ i ≤ κ ≤ µ ∞ , LCU i (P, κ) holds.
In the case i = 4, LCU i (P, µ 4 ) and LCU i (P, µ ∞ ) hold. (b) There is a directed order S i with comp(S i ) = µ i and cof(S i ) = µ ∞ such that COB i (P, S i ) holds.
Accordingly, P forces 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 R i for i = 1, . . . , 4. These R i are, apart from i = 2, the "canonical" relations for b i , d i . They play the role of R LCU Remark 2.5. In [BCM18, Thm. 5.3] a different construction is presented, which gives a stronger conclusion and requires the weaker assumption that ℵ 1 ≤ µ 1 < µ 2 < µ 3 < µ 4 < µ ∞ = µ <µ3 ∞ are just regular cardinals. If we use this paper, then 7 Actually, in most cases we will use the same R LCU i and R COB i , which is moreover the "canonical" choice for (b i , d i ). See the explanation that follows Theorem 2.4.

Cichoń's maximum without large cardinals
Theorem 3.1. Assume GCH and (µ n ) 1≤n≤9 is a weakly increasing sequence of cardinals with µ n regular for n ≤ 8 and µ ℵ0 9 = µ 9 . Then there is a ccc poset P fin forcing that Full GCH is not actually required, see Remark 3.5. Note that the µ n 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 2 9 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) such that the following holds: (1) All cardinals are regular, with the possible exception of λ ∞ , (2) λ ∞ = λ ℵ0 ∞ .
• For 0 ≤ m ≤ 8, we set P m := P pre ∩ N 0 ∩ · · · ∩ N m and P fin := P 8 = P pre ∩ N * . Note that N 0 ∩ · · · ∩ N m is again an elementary submodel of H(χ), 9 and accordingly each P m is a complete subforcing of P pre .
Our setup. The cardinals λ n and θ n are increasing along the arrows (strictly increasing above λ ∞ ). The preparatory forcing P pre forces x = x pre for each left hand side characteristic x (and forces the whole right side to be θ ∞ ); while the final forcing P fin forces x = x fin for every characteristic x (on either side). I.e., the upper Cichoń's diagram shows the situation forced by P pre , and the lower diagram shows the one forced by P fin .
So we get half of the desired inequalities.
This may be more transparent if we consider an explicit example, say i = 3. In each line of the following table, each cardinal in the right column is guaranteed to be an element of the LCU 3 spectrum of the forcing notion in the left column: Since λ 3 is the smallest of these 4 cardinals, and λ 2 the largest, we get that P fin forces b i ≤ λ 3 and d i ≥ λ 2 .
10 Element is clear, as all N 's contain the sequence of λ's. Subset follows from the fact that each N contains λ∞ and thus λm as a subset, and that |Λ| = λm. Now we show, by induction on n ≥ m+1, that S i ∩N 0 ∩· · ·∩N n has completeness ≥λ m+1 and cofinality ≤λ m . The step n = m + 1 was done above; for the steps n > m + 1, by induction we know that S := S i ∩ N 0 ∩ · · · ∩ N n−1 has cofinality at most λ m and completeness at least λ m+1 . So by Lemma 1.6(3), the same holds for S ∩ N n .
Regarding the continuum: There is a sequence (x ξ ) ξ<θ∞ of (nice) P pre -names of reals that are forced to be pairwise different due to absoluteness (1.4). Note that this sequence belongs to N * , so (x ξ ) ξ∈θ∞∩N * is a sequence of P fin -names of reals that are forced (by P fin ) to be pairwise different. Hence, P fin forces c ≥ |θ ∞ ∩N * | = λ ∞ . 11 The converse inequality also holds because |P fin | = λ ∞ = λ ℵ0 ∞ .
Remark 3.5. If we base the left-hand forcing P pre on [BCM18] (see Remark 2.5), then our proof (when we change item (3) on p. 8 to the assumptions listed in Remark 2.5) shows that GCH can be weakened to the following: There are at least 9 cardinals θ > µ 9 satisfying θ <θ = θ. Or, to be even be more pedantic: There are regular cardinals θ 7 < . . . < θ 0 < θ ∞ larger than µ 9 such that θ <µ1

Variants
4.1. Another order. The paper [KST19] constructs (assuming GCH) a ccc forcing notion P which forces another ordering of the left hand side. More concretely, P is ccc and it has LCU and COB witnesses for the following: 12 add(N ) < b < cov(N ) < non(M) < cov(M) = c.
If we use this forcing P instead of P pre , then the same argument shows that we can find a complete subforcing P fin that extends the order to the right hand side: Theorem 4.1. Assume GCH and let (µ n ) 1≤n≤9 be a weakly increasing sequence of cardinals with µ n regular for n ≤ 8 and µ ℵ0 9 = µ 9 . Then there is a ccc poset P fin forcing that 11 This argument can be written in terms of the LCU property for the identity relation on ω ω : As LCU Id (P pre , κ) holds for all regular κ ≤ λ∞ (even up to θ∞), we get LCU Id (P fin , κ) for all these cardinals, which implies λ∞ ≤ c.
12 In (2.1), the order/numbering of (b, d) and (cov(N ), non(N )) is swapped; for this new ordering we again get Theorem 2.4. We use the same R LCU -and R COB -relations as in [GKS19], except for the R LCU -relation for the pair (cov(N ), non(N )): Now we have to use a relation which is an ω 1 -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 R; but this is no problem as R is sufficiently absolute in the sense described after (1.4).
Remark 4.2. As in Remark 3.5, full GCH is not needed, but it is enough that there are 9 regular cardinals larger than µ 9 satisfying some arithmetical properties. However, it is not enough that θ <θ = θ for these 9 cardinals, but it is required in addition that one of them is ℵ 1 -inaccessible. 13   (1) A set A ⊆ ω ω is κ-R-dominating if, whenever F ⊆ ω ω has size <κ, there is some real a ∈ A dominating over F , that is, (∀x ∈ F )xRa. Dually, we say that A is κ-R-unbounded if it is κ-R ⊥ -dominating.
(2) Assume that R is sufficiently absolutely defined and let P be a forcing notion. We define DOM R (P, κ, S) to mean the following: There is a sequence (f α ) α∈S of P -names of reals such that, whenever γ < κ and (x ξ ) ξ<γ is a sequence of P -names of reals, there is some α ∈ S such that P (∀ξ < γ)x ξ Rf α .
• DOM R (P, κ, S) implies P κ ≤ b R & d R ≤ |S| . (This generalizes Lemma 1.3.) For this weaker notion we have the following result similar to Lemma 1.6.
In particular, if S is directed and comp(S) > θ then COB R (P, S) implies COB R (P ∩ N, λ).
(1) If (x ξ ) ξ<γ is a sequence of P -names of reals and γ < min{κ, ν} then the sequence is in N , so there is some α ∈ S ∩ N such that P x ξ Rf α for all ξ < γ. By absoluteness, P ∩ N forces the same.
(3) Fix i < λ. Since |N i | ≤ θ < ν, there is some α i ∈ S such that P xRf αi for all x ∈ N i that are P -names for reals. In fact, we can find such α i in S ∩ N i+1 .
As in [Mej19b], a simpler version of P pre can be constructed in such a way that (a) of Theorem 2.4 holds, and (b') For i = 1, 2, 3, 4 there is some set S i of size µ ∞ such that DOM R COB i (P, µ i , S i ) 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. A first result for (d), namely adding ℵ 1 < m < p < h < add(N ), is done in [GKMS19] (which also gives a more complicated construction to achieve (b)).

Open questions
Of course, it would be interesting to add more characteristics. For example, we can ask: Question 1. Can we add the splitting number s and the reaping number r?
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 cov(N ) < b, then on the right we can only expect to force the dual inequality d < non(N )).
The case when cov(M) < non(M) seems to be more complex. 14 We do not even know how to force the consistency of ℵ 1 < cov(M) < non(M). J. Brendle however does, see [Bre19] for slides of a presentation of his method of "shattered iterations". Brute force counting shows that there are 57 configurations of ten different values in Cichoń's diagram (satisfying the obvious inequalities) where cov(M) < non(M), but none of them have been proved to be consistent so far.