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Let m be the least cardinal θ such that MAθ fails. The only known model for “m

arXiv:math/9207205v1 [math.LO] 31 Jul 1992REMARK ON THE FAILURE OFMARTIN’S AXIOMAvner LandverLet m be the least cardinal θ such that MAθ fails. The only known model for “mis singular” was constructed by Kunen [K1].

In Kunen’s model cof(m) = ω1. It isunknown whether “ω1 < cof(m) < m” is consistent.

The purpose of this paper isto present a proof of Kunen’s result and to identify the difficulties of generalizingthis result to an arbitrary uncountable cofinality. The following material is basedon [K1].

We would like to thank K. Kunen and S. Shelah [S] for their input.§0 Definitions and some facts.For undefined terminology consult [K]. Let θ be a fixed singular cardinal withuncountable cofinality.

Let κ = cof(θ), and fix {θα : α < κ} an increasing sequenceof cardinals converging to θ with θ0 > κ.Let A = {aξ : ξ ∈θ} ⊆℘(ω) and B = {bξ : ξ ∈θ} ⊆℘(ω). (A, B) is a (θ, θ) pairif for every ξ, η ∈θ, |aξ ∩bη| < ℵ0.

The pair (A, B) is disjoint if for every ξ ∈θ,aξ ∩bξ = ∅. The pair (A, B) is locally split if for every X ⊂A, and every Y ⊂B,such that |X ∪Y| ≤ℵ1, the pair (X , Y) splits (i.e.

(∃c ⊆ω)[(∀a ∈X )|a \ c| < ℵ0and (∀b ∈Y)|c ∩b| < ℵ0]). Notice that if (A, B) is locally split in V, then (A, B) islocally split in every c.c.c.

extension of V.For a disjoint pair (A, B) we define the partial orderS(A, B) = {f : θ →ω : |f| < ℵ0 and (∀ξ, η ∈dom(f)) (aξ−f(ξ))∩(bη −f(η)) = ∅}.S(A, B) is partially ordered by inclusion. It is well known that if (A, B) is locallysplit, then S(A, B) is c.c.c., and therefore, if in addition MAθ holds, then (A, B)can be split (to split (A, B) use the set c = S{aξ −f(ξ) : f ∈G, ξ ∈dom(f)}Where G is a sufficiently generic filter on S(A, B)).Tt bAMS T X

If x ∈[θ]<ℵ0, then we definea(x) =\ξ∈xaξ.a(∅) = ω, and b(x) is defined analogously.The pair (A, B) is nice if for every X = {xi : i ∈θ}, a family of disjoint finitesubsets of θ, there are i, j ∈θ such that a(xi) ∩b(xj) ̸= ∅. It is not hard to seethat a nice disjoint (θ, θ) pair is also a gap (i.e.

can not be split). Therefore, theexistence of a nice, locally split, disjoint (θ, θ) pair contradicts MAθ.

“Nice” is avariation of Kunen’s “strong gap” [K1].To add a nice, disjoint (θ, θ) gap that is locally split, one uses the partial orderP defined by:p ∈P ⇐⇒p = ⟨np, sp, ap, bp, Zp, ⟨czp : z ∈Zp⟩⟩where np ∈ω, sp ∈[θ]<ℵ0, ap & bp ⊆np × sp, ap ∩bp = ∅, Zp ∈[[θ]≤ℵ1]<ℵ0, and(∀z ∈Zp) czp ⊆np. The ordering on P is defined by putting q ≤p if and only if thefollowing five conditions hold:(1) sq ⊇sp ∧nq ≥np ∧Zq ⊇Zp.

(2) aq ∩(np × sp) = ap and bq ∩(np × sp) = bp. (3) (∀z ∈Zp) czq ∩np = czp.

(4) (∀ξ, η ∈sp)(∀l ∈[np, nq)) [⟨l, ξ⟩∈aq →⟨l, η⟩̸∈bq]. (5) (∀z ∈Zp)(∀ξ ∈z ∩sp)(∀l ∈[np, nq) [(⟨l, ξ⟩∈aq →l ∈czq) ∧(⟨l, ξ⟩∈bq →l ̸∈czq)].P is c.c.c., and forcing with P adds the disjoint pair A = {aξ : ξ ∈θ}, B = {bξ :ξ ∈θ}, where aξ = {l ∈ω : (∃p ∈G) ⟨l, ξ⟩∈ap} (bξ is defined similarly), andG is the generic filter.To see that (A, B) is locally split it is enough to showthat for every z ∈[θ]ℵ1 ∩V, the pair ({aξ : ξ ∈z}, {bξ : ξ ∈z}) splits (this isenough because P is c.c.c., and therefore every subset of θ with cardinality ℵ1 inthe extension is contained in such a subset from V).

But this pair get split byS{czp : p ∈G and z ∈Zp}. Finally let us show thatFact 0.

The pair (A, B) is nice.2

Proof. Assume that p ∈P is such thatp ⊩“X = {xi : i ∈θ} are finite disjoint subsets of θ”.For every i ∈θ, let pi ≤p and yi ∈[θ]<ℵ0 be such that pi ⊩“xi = yi”.

Letκ < λ < θ be an arbitrary regular cardinal. Let A ∈[θ]λ and n ∈ω be such that:(a) (∀i ∈A)npi = n.(b) {spi : i ∈A} form a delta system with root s and (∀i, j ∈A)[api ∩(n × s) =apj ∩(n × s) and bpi ∩(n × s) = bpj ∩(n × s)].

(c) {Zpi : i ∈A} form a delta system with root Z and (∀z ∈Z)czpi = czpj. (d) (∀i ∈A)spi ⊃yi.Notice that (a)-(c) imply that {pi : i ∈A} are linked (i.e.

pairwise compatible)and therefore {yi : i ∈A} are disjoint. Let ¯Z = S Z, then | ¯Z| ≤ℵ1.

Therefore wecan find i ̸= j ∈A such thatyi ∩s = yi ∩¯Z = yj ∩s = yj ∩¯Z = ∅.Finally define q ≤pi, pj as follows: let nq = n + 1, let sq = spi ∪spj, and Zq =Zpi ∪Zpj. We put⟨n, ξ⟩∈aq ⇐⇒ξ ∈yi⟨n, ξ⟩∈bq ⇐⇒ξ ∈yj.We also make sure that for every z ∈(Zpi \ Z), if ξ ∈z ∩yi, then n ∈czq, and thatfor every z ∈(Zpj \ Z), if ξ ∈z ∩yj, then n /∈czq.

Notice that this does not causea contradiction since only z /∈Z are involved. We conclude that q ≤p and thatq ⊩“n ∈a(xi) ∩b(xj)”.□Given a disjoint (θ, θ) pair (A, B) and X = {xi : i ∈θ} a family of disjointfinite subsets of θ, we make the following definition.

We call S = {Sα : α < κ}an X-sequence if the Sα’s are disjoint subsets of θ and (∀α < κ) [ |Sα| > θα and(∀i, j ∈Sα) a(xi) ∩b(xj) = ∅].Next, for every X = {xi : i ∈θ} disjoint finite subsets of θ, and every S = {Sα :α < κ} an X-sequence, we define:Q(X, S) = {F ∈[κ]<ℵ0 : (∀α ̸= β ∈F)(∃i ∈Sα)(∃j ∈Sβ)[a(xi) ∩b(xj) ̸= ∅∨b(xi) ∩a(xj) ̸= ∅]}3

andP(X, S) = {F ∈[κ]<ℵ0 : (∀α ̸= β ∈F)(∀i ∈Sα)(∀j ∈Sβ)[a(xi) ∩b(xj) = ∅∧b(xi) ∩a(xj) = ∅]}.Q(X, S) and P(X, S) are both partially ordered by inclusion. Notice that the lastthree definitions depend on A = {aξ : ξ ∈θ} and B = {bξ : ξ ∈θ}.

P(X, S) is atypical “dangerous” partial order (see definition 1), and Q(X, S) will be used in theproof of Kunen’s result to “kill” dangerous partial orders. Notice that P(X, S) ×Q(X, S) is not κ.c.c.

(the set {⟨{α}, {α}⟩: α < κ} is an antichain of size κ).§1 A proof of Kunen’s result.The first step is to show that the niceness of the pair is in fact a statementconcerning the κ.c.c. of the various Q(X, S)’s.Lemma 1.

Assume that (A, B) is a (θ, θ) pair. Then (A, B) is nice if and only iffor every X = {xu : u ∈θ}, Y = {yv : v ∈θ} disjoint finite subsets of θ and everyS = {Sα : α < κ} an X-sequence, and T = {Tα : α < κ} a Y -sequence, the partialorder Q(X, S) × Q(Y, T) is κ.c.c.Proof.

The reverse implication is easy to check. Let us prove the direct implication.Assume that {⟨Kγ, Fγ⟩: γ < κ} is an antichain in Q(X, S) × Q(Y, T).

We mayassume that {Kγ : γ < κ} are disjoint and all have size n, and that the Fγ’s aredisjoint and all have size p (use delta systems).For t ∈[θ]<ℵ0, we call t a γ-transversal if |t| = n+p, and (∀α ∈Kγ) |t∩Sα| = 1,and (∀β ∈Fγ) |t∩Tβ| = 1. Choose {ti : i ∈θ} pairwise disjoint, with each ti being aγ-transversal for some γ < κ.

Next, let Z = {zi : i ∈θ} be defined in the followingway. If ti is a γ-transversal, thenzi = (∪{xu : u ∈(ti ∩Sα) ∧α ∈Kγ}) ∪(∪{yv : v ∈(ti ∩Tβ) ∧β ∈Fγ}).Since both {xu : u ∈θ} and {yv : v ∈θ} are pairwise disjoint, and the transversals{ti : i ∈θ} are pairwise disjoint, we may assume that Z = {zi : i ∈θ} are pairwisedisjoint.4

Finally, we show that the existence of Z contradicts the niceness of (A, B). Leti ̸= j ∈θ.Case 1: ti, tj are both γ-transversals.

Let α ∈Kγ (if n = 0, then work with theFγ’s). Let u ∈(ti ∩Sα) and w ∈(tj ∩Sα) with u ̸= w. Clearly, a(xu) ∩b(xw) =b(xu)∩a(xw) = ∅(because S is an X-sequence).

But xu ⊂zi and xw ⊂zj thereforea(zi) ∩b(zj) = b(zi) ∩a(zj) = ∅.Case 2: ti is a γ-transversal, and tj is a δ-transversal, and γ ̸= δ. In this case⟨Kγ, Fγ⟩⊥⟨Kδ, Fδ⟩.

Assume w.l.o.g. that Kγ ⊥Kδ.

This means that(∃α ∈Kγ)(∃β ∈Kδ)(∀u ∈Sα)(∀w ∈Sβ) a(xu) ∩b(xw) = b(xu) ∩a(xw) = ∅.Now let u ∈(ti ∩Sα) and w ∈(tj ∩Sβ). By the above, a(xu) ∩b(xw) = b(xu) ∩a(xw) = ∅.

But xu ⊂zi and xw ⊂zj, therefore a(zi)∩b(zj) = b(zi)∩a(zj) = ∅.□Similarly, it can be shown that (A, B) is nice if and only if every Q(X, S) is κ.c.c., and also if and only if the product of any finitely many partial orders of the formQ(X, S) is κ.c.c.It is also true that if ˜S = {Sα : α < κ} is defined by Sα = [θα, θ+α ), then: (A, B)is nice if and only if for every X = {xu : u ∈θ} disjoint finite subsets of θ, if ˜S isan X-sequence, then Q(X, ˜S) is κ.c.c.Definition 1. Assume that (A, B) is a nice pair.

The partial order R is calleddangerous for (A, B) if there exsits r ∈R such thatr ∥−R “(A, B) is not nice”.The following style of proof was motivated by [S].Lemma 2. Assume that (A, B) is a nice (θ, θ) pair, and R is a κ.c.c.

partialorder with |R| < θ.Then R is dangerous for (A, B) if and only if there existY = {yi : i ∈θ} disjoint finite subsets of θ, and T = {Tα : α < κ} a Y -sequence,such that R × Q(Y, T) is not κ.c.c.Proof. (⇐): If R × Q(Y, T) is not κ.c.c., then there exists r ∈R such thatr ∥−R “Q(Y, T) is not κ.c.c.

”.5

Therefore, by Lemma 1, R is dangerous. (⇒): Let r ∈R be such thatr ∥−“X = {xi : i ∈θ} are disjoint finite subsets of θ and (∀i, j ∈θ) a(xi)∩b(xj) = ∅”.

(∀i ∈θ) let ri ≤r and yi ∈[θ]<ℵ0 be such that ri ∥−“xi = yi”.|R| < θ, therefore (∃β0 < κ)(∀β ≥β0)(∃pβ ≤r) such that |{i ∈[θβ, θ+β ) : pβ =ri}| = θ+β .Now, for every α < κ, let rα = pβ0+α andTα = {i ∈[θβ0+α, θ+β0+α) : rα = ri}.Notice that for every α < κ, {yi : i ∈Tα} are disjoint. Furthermore we may assumethat {yi : i ∈Sα<κ Tα} are disjoint (otherwise, by induction on α < κ, pass to asubset of Tα of cardinality θ+α ).

If i /∈Sα<κ Tα, then redefine yi = ∅. We now haveY = {yi : i ∈θ} disjoint finite subsets of θ, and {Tα : α < κ} a Y -sequence.Let B be an R-name for the set {α < κ : rα ∈G}, where G is a name for thegeneric filter.

R is κ.c.c., therefore there exists p ≤r such that p ∥−“|B| = κ”.Finally it is not hard to check thatp ∥−“{{α} : α ∈B} is an antichain in Q(Y, T)”.□Corollary. If (A, B) is a nice (θ, θ) pair, then for every X = {xi : i ∈θ} disjointfinite subsets of θ, and every S = {Sα : α < κ} an X-sequence, Q(X, S) is notdangerous for (A, B).Lemma 3.

(See Lemma 8 [K1].) Let (A, B) be a nice (θ, θ) pair.

Let γ be a limitordinal and Pγ a finite support iteration of c.c.c. partial orders.

If Pγ is dangerousfor (A, B), then there exists α < γ such that Pα is dangerous for (A, B).Proof. Assume that Pγ is dangerous for (A, B).

There are two cases.cof(γ) ̸= κ: Let X = {xi : i ∈θ} ∈V[Gγ], disjoint finite subsets of θ, be awitness for the failure of niceness, where Gγ is a Pγ-generic filter. Now, there arethree subcases: cof(γ) > θ, κ < cof(γ) < θ, and cof(γ) < κ.

It is not hard to see6

that in each of these subcases there exists α < γ and there is A ∈[θ]θ such that{xi : i ∈A} ∈V[Gα], contradicting niceness in V[Gα].cof(γ) = κ: Let X be as in the previous case. Let c = ∪{a(xi) : i ∈θ} ⊂ω.Let α < γ be such that c ∈V[Gα].

(Such an α exists since κ > ω.) In V[Gα], letY be a disjoint family of finite subsets of θ which is maximal with respect to theproperty(∀y ∈Y ) [a(y) ⊂c ∧b(y) ∩c = ∅].Y remains maximal in V[Gγ] and therefore must have cardinality θ, which contra-dicts niceness in V[Gα].□Theorem (Kunen).

It is consisitent to have m singular with cof(m) = ω1.Proof. Let θ be singular with cof(θ) = κ = ω1 and force with P (see §0) to startwith (A, B), a nice, disjoint, locally split (θ, θ) pair.

Let us now iterate c.c.c. partialorders of size < θ in the following way.

Assume that the part of the iteration that hasbeen defined thus far is non-dangerous. Assume that the next partial order on thelist (of all c.c.c.

partial orders of size < θ) is R, but R is dangerous (otherwise justforce with R). Then by Lemma 2, there are Y and T such that R × Q(Y, T) is notc.c.c.

In addition, by Lemma 1, Q(Y, T) is c.c.c. and by the corollary, Q(Y, T) is non-dangerous for (A, B).

So instead of forcing with R, let us force with Q(Y, T) to addan uncountable antichain to R. By Lemma 3,the iteration of non-dangerous c.c.c.partial orders is a non-dangerous c.c.c. partial order.

Therefore, in the extension,(A, B) remains nice and m = θ.□In the general case (κ is any regular uncountable cardinal), all we know is thatQ(Y, T) is κ.c.c. and not necessarily c.c.c.

So if κ > ω1, and we perform the iterationas in the proof of the theorem, then cardinals below κ may be collapsed, and wemay end up with a model for Kunen’s result, in which κ = ω1.§2 Beyond niceness.Let us define a condition which implies niceness, and which is, in the presenceof MAκ, equivalent to niceness.Definition 2. Let (A, B) be a nice (θ, θ) pair.

We say that (∗) holds for (A, B) if7

there is no c.c.c. partial order of cardinality < θ which is dangerous for (A, B).The following Lemma shows that (∗) is in fact a statement concerning the exis-tence of certain dangerous c.c.c.

suborders of the various P(Y, T)’s.Lemma 4. Assume that (A, B) is a nice (θ, θ) pair.

Then (∗) fails for (A, B) if andonly if there are Y = {yi : i ∈θ} disjoint finite subsets of θ, and T = {Tα : α < κ}a Y -sequence, and there is P′ ∈[P(Y, T)]κ such that P′, equipt with the orderingof P(Y, T), is c.c.c. and closed under subsets.Proof.

(⇐): P′ × Q(Y, T) is not κ.c.c. and hence, by Lemma 1, P′ is dangerous for(A, B).

(⇒): Let R be a c.c.c. dangerous partial order of cardinality < θ.

By the proof ofLemma 2, there are Y = {yi : i ∈θ} disjoint finite subsets of θ, and T = {Tα : α <κ} a Y -sequence, and there are B an R-name for a subset of κ, and p ∈R suchthatp ∥−“|B| = κ and {{α} : α ∈B} is an antichain in Q(Y, T)”.Let B(R) be the boolean completion of R. For every F ∈[κ]<ℵ0 let r(F) = [[F ⊂B]] · p, where [[F ⊂B]] is the boolean value of “F ⊂B” in B(R). Now defineP′ = {F ∈[κ]<ℵ0 : r(F) > 0}.P′ ⊂P(Y, T), and |P′| = κ (because p ∥−“|B| = κ”.

)Finally, assume that F ⊥P′ K. Then r(F ∪K) = 0, and therefore r(F)·r(K) = 0.This proves that P′ is c.c.c. since B(R) is c.c.c.□Similar to the proof of Fact 0, and using Lemma 4, one can now show that thenice disjoint locally split pair (A, B), that was added using P in §0, also satisfiesthe property (∗).

Let us check how well property (∗) is preserved through a c.c.c.iteration of partial orders of size < θ.We first show that (∗) is preserved through successor steps of the iteration. Moreprecisely, if (∗) holds for (A, B) and R is a c.c.c.

partial order with |R| < θ, then∥−R “(∗) holds for (A, B)”. To show this assume otherwise.

By Lemma 4, there is8

r ∈R and π an R-name such thatr ∥−R “there are Y = {yi : i ∈θ} disjoint finite subsets of θ, and T, a Y −sequence withπ ∈[P(Y, T)]κ, and π is c.c.c. and closed under subsets”.In particular R∗π is a dangerous c.c.c.

partial order. But R∗π has a dense subset ofcardinality < θ, namely {(r, F) : r ∈R, F ∈[κ]<ℵ0 and r ∥−′′ F ∈π”}.

Therefore(∗) fails for (A, B), which is a contradiction.As for the preservation of (∗) at limits, we can only show the cases where cof(γ) ̸=κ (see Lemma 3). We first remark that in these cases the following holds: if B ∈[κ]κ,Y = {yi : i ∈θ} disjoint finite subsets of θ, and T = {Tα : α < κ} a Y -sequence,are all in V[Gγ], then there exists β < γ such that in V[Gβ], there exists A ∈[B]κ,and for every α ∈A, T ′α ∈[Tα]θ+α are such that {yi : i ∈Sα∈A T ′α} ∈V[Gβ].The proof that this remark implies that (∗) is preserved proceeds as follows.Assume that (∗) fails in V[Gγ].

By Lemma 4, let Y, T be given and P′ ∈[P(Y, T)]κsuch that P′ is c.c.c. and closed under subsets.

Let B = {α ∈κ : {α} ∈P′}. Nowlet β < γ and A ∈[B]κ as discussed above.

In V[Gβ] define the partial orderP′′ = {F ∈[A]<ℵ0 : (∀α ̸= β ∈F)(∀i ∈T ′α)(∀j ∈T ′β)[a(yi) ∩b(yj) = ∅∧b(yi) ∩a(yj) = ∅]}.In V[Gγ], consider R′ = {F ∈[A]<ℵ0 : F ∈P′}. R′ ⊂P′′ and R′ is a c.c.c.

partialorder of cardinality κ. Finally, V[Gγ] is a forcing extension of V[Gβ], therefore inV[Gβ] we can define the partial order R′′ = {F ∈P′′ : [[F ∈P′]]·[[P′ is c.c.c. ]] > 0}.|R′′| = κ because R′ ⊂R′′.

R′′ is c.c.c. because V[Gγ] is a c.c.c.

extension of V[Gβ].Therefore, V[Gβ] |= “R′′ is c.c.c. and dangerous for (A, B)”, and hence (∗) fails inV[Gβ].Finally, let us look again at the case where cof(θ) = κ = ω1.

Let (A, B) be anice (θ, θ) pair. We claim that there exists a c.c.c.

partial order Q such that∥−Q “(∗) holds for (A, B)”.Q is simply the finite support iteration of all partial orders of the form (Q(X, S))ω(product with finite support). Q is c.c.c.

and, by the Corollary and Lemma 3, it9

preserves the niceness of (A, B). In the extension, all partial orders of the formQ(X, S) are σ-centered and therefore, by Lemma 2, (∗) holds for (A, B).This discussion suggests an alternative way of viewing Kunen’s result.

Start witha disjoint, locally split (θ, θ) pair (A, B) for which (∗) holds (add such a pair usingP which was defined in §0). Then iterate all c.c.c.

partial orders of cardinality < θ.By the remarks above, (∗) may first fail only at limits of cofinality ω1. In this case,by Lemma 3, (A, B) is still nice so first force with Q, as defined above, to get anextension in which (∗) holds for (A, B), and then force with the next c.c.c.

partialorder of cardinality < θ on the list.Concluding remarks. Given a disjoint (θ, θ) pair (A, B) we discussed two properties:(1) (∗) holds for (A, B).

(2) (A, B) is nice. (1) implies (2).

(1) is preserved while forcing with c.c.c. partial orders of cardi-nality less than θ (we do not have a similar result for (2)).

Both (1) and (2) arepreserved at limit stages of c.c.c. iterations, of cofinality ̸= κ.

(2) is also preservedat limit stages of cofinality = κ (we do not have a similar result for (1)).Roughly speaking, it seems desirable to have uncountable c.c.c. suborders of thevarious Q(X, S)’s.

This would enable us to kill the dangerous partial orders as theycome along, or alternatively, force (∗) to hold and cosequently kill all the dangerouspartial orders at once.It should be mentioned that if one starts with a nice (θ, θ) pair (A, B), and thentries to preserve the niceness along a c.c.c. iteration of all partial orders of size lessthan θ, then the only difficulty lies in getting MAκ to hold.

This is true because ifMAκ holds, then (A, B) is nice if and only if (∗) holds for (A, B), and therefore oneis free to force with the next partial order on the list.On the otherhand if λ < κ, then MAλ could be forced without destroying nicenessbecause c.c.c. partail orders of size λ are not dangerous.

So if κ > ω1, then onecan force MAℵ1 and preserve niceness. In this stage all c.c.c.

partail orders arec.c.c.-productive and if there are still dangerous ones, they can not be killed andthe iteration is stuck.10

References[K] K. Kunen, Set Theory, North-Holland, 1980.[K1]K. Kunen, Where MA First Fails, J.

Symbolic Logic 53 (1988). [S] S. Shelah, personal conversation.Department of Mathematics, The University of Kansas, Lawrence, KS 66045E-mail address: landver@kuhub.cc.ukans.edu11

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