Inaccessible Cardinals
J¨
org Brendle
1, Benedikt L¨
owe
2,3,4∗1Graduate School of Engineering, Kobe University, Rokko-dai 1-1, Nada, Kobe 657-8501, Japan
2 Institute for Logic, Language and Computation, Universiteit van Amsterdam, Plan-tage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
3 Department Mathematik, Universit¨at Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
4 Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Bering-straße 1, 53115 Bonn, Germany
[email protected], [email protected]
1
Introduction
Statements about regularity properties at the second level of the projective hierarchy such as “Every∆12set of reals has propertyP” or “EveryΣ12set of reals has propertyP” are complicated enough not to be ZFC theorems (typically, they fail to hold in L) and thus it is interesting to investigate their relative logical strength. The strongest such statement is “∀x(ω1L[x]< ω1)” (or “ω1 is inaccessible by reals”) which typically implies all of the
above mentioned properties and the weakest nontrivial such statement is “∀x(ωω\L[x]6=
∅)” (which by [BL99, Theorem 7.1] is equivalent to “every
Σ12 set of reals is Sacks-measurable”).
Most of the regularity properties investigated in this context are derived from forcing notions, and the computation of relative logical strength has been done for many such properties, e.g., in [Sol70, JS89, BL99, BHL05]. In this paper, we continue this work by looking at the Baire property in the eventually different topology (cf.§2) and the statementsΣ12(E) “every
Σ12 set of reals has the Baire property in the eventually different topology” and∆12(E) “every∆12 set of reals has the Baire property in the eventually different topology”. Based on preliminaries on definability of ideals and
∗This work was started in 1997 when the second author was supported by DAAD-Grant
forcing absoluteness (§3), we prove in§4 thatΣ12(E) is equivalent to “ω1is
inaccessible by reals” (Theorem 7). This result was not unexpected, as the present authors showed the same for the Baire property in the dominating topology in [BL99, Theorem 5.11], based on a combinatorial property of the Hechler ideal. In our proof here, we use the analogue of that property for eventually different forcing (Theorem 2). In§5, we then move on to∆12(E)
and show that it fails in the ω1-stage finite support iteration of Hechler
forcing (Theorem 18). With this ingredient, we are then (§6) able to place ∆12(E) in the diagram of regularity statements on the second level of the projective hierarchy and prove all implications and non-implications.
The technical result leading to Theorem 18 (cf. Corollary 13 and Theo-rem 17) says that eventually different reals in the (iterated) Hechler exten-sion are necessarily dominating reals. This may be of independent interest.
2
Eventually different forcing
The conditions of eventually different forcing, denoted by E, are pairs
hs, Fi, where s ∈ ω<ω is a finite sequence of natural numbers and F is
a finite set of reals. We say that hs, Fi ≤ ht, Gi if and only if t ⊆ s,
G⊆F and for alli ∈ dom(s\t) and all g ∈ G, we have that s(i) 6=g(i). In [ Lab96], Lab¸edzki discusses all basic properties of this forcing partial order, and we refer to this paper for details. Eventually different forcing is a c.c.c. and evenσ-centered forcing that generates a topologyE refining the standard topology on Baire space. Basic open sets ofE are of the form [s, F] = {x ∈ ωω;s ⊆ x and ∀f ∈ F ∀n ≥ |s| (x(n) 6= f(n))} where
hs, Fi ∈E. These sets are in fact closed in the standard topology and thus
E-clopen. HenceE-open dense sets are Fσ, andE-closed nowhere dense sets
Gδ, in the standard topology. Therefore the E-meager sets form an ideal
IE which has a basis ofΣ03 sets in the standard topology (cf. also [ Lab96, Theorem 3.1]).
We fix some coding of the Borel sets by real numbers, and write BM c for
the Borel set coded bycas interpreted in the modelM and Bc:= BVc . Note
that the statement “c is a code for a Borel set inIE” is absolute between models of set theory, allowing us to call a Borel codecE-meagerif BMc is meager in any modelM of set theory. For a given model of set theoryM, we writeEvD(M) for the set of realsE-generic over M. SinceE is a c.c.c.
forcing notion, we have the usual connection betweenE-meager Borel codes and the notion ofE-genericity:
Lemma 1 ([ Lab96, Theorem 3.3]). If M is a model of set theory and
x∈ ωω, thenx is
E-generic over M if and only if for all E-meager Borel
codesc∈M, we have thatx /∈Bc.
Let Eα := {x ∈ ωω;∃∞k ∈ ω(x(k) = fα(k))}. Note that these sets are E-nowhere dense.
Theorem 2 (Brendle). If Gis E-meager and hfα;α <2ωiis a family of
pairwise eventually different functions then the set{α;Eα⊆G} is count-able.
Proof. [ Lab96, Theorem 4.7] q.e.d.
This theorem is the main ingredient of the proof that the additivity of the meager ideal inE isℵ1. Its combinatorics is based on the construction
in [Bre95, Theorem 2.1]
3
Preliminaries
We shall work in the general framework introduced by Ikegami [Ike09] which we briefly review: We call a forcing notion P arboreal if its conditions
are (isomorphic to) a set of perfect trees ordered by inclusion and we call it strongly arboreal if for any T ∈ P and any t ∈ T, we have that
{s ∈ T;s ⊆ t∨t ⊆ s} ∈ P. For arboreal forcings, we say that a set
X ⊆ωω is
P-null if for any T ∈ P there is some S ∈ P such thatS ≤T
and [S]∩X =∅. We let IP be the σ-ideal generated by the P-null sets. UsingIP, we call a setX P-measurableif for anyT ∈Pthere is anS ∈P with S ≤ T such that either [S]∩X ∈ IP or [S]\X ∈ IP. We define an idealI∗
P ⊆ IP byI
∗
P :={X;∀T ∈P∃S ∈P(S ≤T∧[S]∩X ∈ IP}. If Γ is
a pointclass andP is an arboreal forcing, we writeΓ(P) for the statement “Every set inΓisP-measurable”.
For all classical forcing notions (Cohen, random, Hechler, Laver, Miller, Sacks, etc.),P-measurability coincides with the natural notion of
measura-bility. It is easy to see that for eventually different forcingE, the idealIEis
exactly the ideal of E-meager sets (thus allowing us to use the same nota-tion) and beingE-measurable coincides with having theE-Baire property.
In joint work with Halbeisen, the present authors introduced a notion of quasigenericity in [BHL05, §1.5]: given a model of set theory M, an ideal I and a real r, we say that r is I-quasigeneric over M if for all Borel codes c ∈ M such that Bc ∈ I, we have that r /∈ Bc.1 For classical c.c.c.
forcing notions P (Cohen, random, Hechler, eventually different, etc.), IP -quasigenericity agrees withP-genericity (cf. Lemma 1 for eventually different forcing). These are particular instances of a general fact [Ike09, Proposition 2.17].
Recall that if Γ is a projective pointclass, we say Γ-P-absoluteness holds if for every sentenceϕin Γwith parameters inV,V |=ϕiffVP|=ϕ.
1 Note that we are presupposing that “B
c∈ I” is absolute between models of set theory.
This is the case by Shoenfield absoluteness ifIis sufficiently definable, for instance if it isΣ1
Theorem 3 ([Ike09, Theorem 4.3]). IfPis a proper and strongly arboreal
forcing notion such that{c;c is a Borel code and Bc∈ IP∗}isΣ12, then the
following are equivalent:
(i) Σ13-P-absoluteness,
(ii) every∆12set is P-measurable, and
(iii) for every realxand everyT ∈P, there is aI∗
P-quasigeneric real in [T]
overL[x].
Theorem 4 ([Ike09, Theorem 4.4]). IfPis a proper and strongly arboreal forcing notion such that{c;c is a Borel code and Bc∈ IP∗} isΣ12 andIPis
Borel generated, then the following are equivalent:
(i) everyΣ12set is P-measurable, and
(ii) for every realx, the set{y;yis notI∗
P-quasigeneric overL[x]}belongs
toI∗
P.
Since the ideal I∗
P ⊇ IP is equal to IP for c.c.c. forcing notions [Ike09,
Lemma 2.13] and since we only deal with c.c.c. forcing, we can ignore the difference betweenI∗
P andIP.
Suppose thatΓis a projective pointclass. Aσ-idealI is calledΓonΣ11 if for every analytic set A ⊆ 2ω×ωω, the set {y ∈ 2ω; A
y ∈ I} is in Γ
(whereAy :={x;hy, xi ∈ A} is the vertical section at y). The notion of beingΠ11onΣ11is a crucial property of ideals, as discussed in [Zap08,§3.8]. Most ideals occurring in nature are∆12 onΣ11. The ideal ofE-meager sets isΠ11 onΣ11 [Zap08, Proposition 3.8.12].
Lemma 5. AssumeI is aΣ12onΣ11σ-ideal which has aΣ0αbasis for some
α. Then{c; cis a Borel code and Bc∈ I}isΣ12.
Proof. LetA⊆2ω×ωω be a universalΣ0
α set. Then Bc ∈ I iff there isx
such thatAx ∈ I and Bc⊆Ax. By assumption, the first statement isΣ12,
and the second is obviouslyΠ11. q.e.d. A similar argument, using a universal analytic set instead, shows that if I is∆12 onΣ11, then{c;c is a Borel code and Bc∈ I}isΠ12.
4
Σ
12sets
We start by proving a “Judah-Shelah-style characterization” connecting the E-Baire property of all sets in∆12 and the existence of generics.
(ii) Every∆12 set has theE-Baire property (i.e.,∆12(E)), and
(iii) for everyx, there is anE-generic overL[x].
Proof. This follows immediately from Theorem 3, keeping in mind that in
the case ofE, having theE-Baire property and beingE-measurable are the same, thatIE∗=IE, thatIE-quasigenericity and E-genericity are the same (Lemma 1), that IE has a basis consisting of Σ03 sets, that it is Π11 on Σ11[Zap08, Proposition 3.8.12], and that{c;cis a Borel code and Bc∈ IE}
therefore isΣ12 by Lemma 5. q.e.d. Here is the characterization of theE-Baire property of theΣ12 sets. Theorem 7. The following are equivalent:
(i) EveryΣ12set has theE-Baire property (i.e.,Σ 1 2(E)),
(ii) for everyx, the set ofE-generics overL[x] isE-comeager, and (iii) ω1 is inaccessible by reals.
Proof. The equivalence of (i) and (ii) follows from Theorem 4.
“(iii)⇒(ii)”: If ωL1[x] is countable, then there are at most countably many codes forE-meager sets in L[x]. By Lemma 1,ωω\EvD(L[x]) =S{B
c;c∈
L[x] is anE-meager Borel code}which now is a countable union ofE-meager sets, and thusE-meager. Consequently,EvD(L[x]) isE-comeager.
“(ii)⇒(iii)”: Towards a contradiction, let ωL1[x] = ω1 for some fixed x.
In L[x], there is a family hfα;α < ω1i of pairwise eventually different
functions. Recall that the sets Eα := {y ∈ ωω;∃∞k(y(k) = fα(k))} are
E-nowhere dense in L[x]. Therefore, ωω\EvD(L[x]) must contain all (i.e.,
uncountably many) of these setsEα. By Theorem 2,ωω\EvD(L[x]) cannot
beE-meager, and thusEvD(L[x]) cannot be E-comeager. q.e.d.
5
∆
12sets
In this section we compare theE-Baire property of ∆12 sets with measura-bility and the Baire property ofΣ12sets. We first show that the statement that allΣ12 are Lebesgue measurable,Σ12(B), implies∆12(E).
The partial order LOC of localization forcing consists of all pairs
hσ, Fi such thatσ∈([ω]<ω)<ω is a finite sequence with|σ(n)|=n for all
n <|σ| andF is a finite set of reals with |F| ≤ |σ|. The order is given by hτ, Gi ≤ hσ, Fi iff τ ⊇σ, G ⊇F, and f(n) ∈ τ(n) for allf ∈ F and all
n∈ |τ|\|σ|. The forcingLOCis c.c.c. and evenσ-linked (but notσ-centered)
and adds a genericslalomϕ∈([ω]<ω)ωgiven byϕ=S
Lemma 8. The product LOC×C adds an E-generic real. In particular
E<◦LOC×C.
Proof. Letϕbe theLOC-generic real. SinceLOC×C∼=LOC?C˙, we may
think of Cohen forcingCas adding a generic realcoverV[ϕ]. Furthermore,
we may think of C as being the order of finite partial functions s with
s(n)∈/ ϕ(n) for alln <|s|. That is, c(n)∈/ ϕ(n) for alln. We claim that thiscisE-generic overV.
Let D ⊆ E be open dense. Let (hσ, Fi, s) ∈ LOC×C. Without loss we may assume |σ| = |s|. By our stipulation in the preceding paragraph this means thats(n)∈/ σ(n) for alln < |s|. We need to find (hτ, Gi, t)≤ (hσ, Fi, s) with |τ|=|t|andht, Gi ∈D. To see that this suffices note that such (hτ, Gi, t) necessarily forces thatt⊆c˙ and ˙c(n)6=f(n) for all n≥ |t| andf ∈G.
Clearly, hs, Fi ∈ E. There is ht, Gi ≤ hs, Fi with ht, Gi ∈ D. By
extendingt, if necessary, we may assume that |t| ≥ |G|. Next extendσto
τ such that|τ|=|t| and for allnwith|s| ≤n <|t| and allf ∈F we have
f(n)∈ τ(n) andt(n)∈/ τ(n). This is possible because t(n)6=f(n) for all
f ∈F and all suchn. Then (hτ, Gi, t)≤(hσ, Fi, s) is as required. q.e.d.
In the following, we shall be interested in the statement “for all reals
x, there is aLOC-generic overL[x]”. Using Ikegami’s general methods, we can prove that this is equivalent to the statement∆12(LOC), i.e., every∆12 set isLOC-measurable, as in Theorem 6. We shall not go into details here, and just use the notation∆12(LOC) as a shorthand for the statement “for all realsx, there is aLOC-generic overL[x]”.
Corollary 9. ∆12(LOC) implies∆12(E).
Proof. It is easy to see thatLOCadds a Cohen real. ThereforeE<◦LOC?
˙
LOC by Lemma 8. In particular, if for allx there is aLOC-generic over
L[x], then for allxthere is an E-generic overL[x], and ∆12(E) follows by
Theorem 6. q.e.d.
Lemma 10. ∆12(LOC) is equivalent to Σ12(B), the statement “all Σ12 sets are Lebesgue measurable”.
Proof. By Bartoszy´nski’s characterization of additivity of measure [BJ95,
Theorems 2.3.11 and 2.3.12],∆12(LOC) impliesΣ12-Lebesgue measurability.
The same characterization yields thatΣ12(B) implies that for allx, there is
a slalom localizing all reals in L[x]. By [Bre06, Lemma 2.1], this in turn entails that there is is aLOC-generic overL[x] for allx, that is,∆12(LOC)
holds. q.e.d.
Proof. This follows from the two preceding results. q.e.d.
After Lebesgue measurability, we are now considering the (regular) Baire property. The statementΣ12(C), i.e., “allΣ12 sets have the Baire property
(in the ordinary topology)” is equivalent to∆12(D), i.e., “all∆ 1
2 sets have
the Baire property in the dominating topology by [BL99, Theorem 5.8]. The rest of this section will contain the proof that this statement does not imply∆12(E).
For technical purposes, we consider a slight variant of standard Hechler forcing: conditions of our D are trees T ⊆ ω<ω such that for any s ∈ T
beyond the stem, sˆn belongs to T for almost all n. Obviously D is a σ -centered forcing notion which adds a dominating real. So D×C adds a standard Hechler real [BJ95, Corollary 3.5.3]. On the other hand, it is easy to see that standard Hechler forcing adds aD-generic. (In fact, if d∈ωω
is a standard Hechler generic satisfying d(n)> n for all n, then d0 ∈ ωω
given recursively by d0(0) =d(0) andd0(n+ 1) =d(d0(n)) is aD-generic.)
Thus, the finite support iterations of the two partial orders have the same properties. In particular, the finite support iteration ofDforces ∆12(D).
The reason we use D is that this makes the rank analysis of Hechler
forcing (which is originally due to Baumgartner and Dordal [BD85]) a bit simpler. Recall that, ifs∈ω<ωandϕis a statement of the forcing language, we saysforcesϕif there isT ∈Dwith stemssuch thatT forcesϕ. Next,
define the rankρϕ by:
ρϕ(s) = 0 ⇐⇒ sforcesϕ
ρϕ(s)≤α ⇐⇒ ∃∞n ρϕ(sˆn)< α
Say thats favors ϕifρϕ(s)<∞. The following are well-known and easy: (i) A sequence can force at most one ofϕand¬ϕ.
(ii) Each sequence favors at least one ofϕand¬ϕ.
(iii) A sequencesforcesϕiffsdoes not favor¬ϕ.
(iv) A sequence s favors ϕiff for all T with stems there is U ≤T such that U ϕ.
We prove (and this is the main technical result of this section):
Theorem 12. LetW be a c.c.c. extension of V with the property that for all infinite partial functionsx:ω→ωin W which are not dominating over
V, there are infinite partial functions{xn;n∈ω}inV such that whenever
y∈ωω∩V is infinitely often equal to allxn, thenyis infinitely often equal to
inW[d] which are not dominating overV, there are infinite partial functions {xn;n∈ω} in V such that whenevery ∈ωω∩V is infinitely often equal
to allxn, theny is infinitely often equal tox.
Corollary 13 (Dichotomy for Hechler forcing). Let d be a Hechler real overV and letx∈V[d] be a real. Then
(i) eitherxis dominating overV
(ii) orxis not eventually different overV.
Proof. Apply the theorem withV =W. q.e.d.
The proof of Theorem 12 splits into three cases (see below); inCase 1
and Case 2, we find a ground model real that is infinitely often equal to
x, and inCase 3, we can prove thatxis dominating. In fact, we believe a different dichotomy (Conjecture 14) holds as well:
Conjecture 14. Letx∈V[d] be a new real. Then (i) either there is a dominating real (overV) inV[x]
(ii) orV[x] is a Cohen extension ofV.
However, the split would occur along different lines than in the proof of Theorem 12. It can be shown that inCase 2 of the proof of Theorem 12,
V[x] does contain a dominating real. We could strengthen Conjecture 14 to:
Conjecture 15. Letx∈V[d] be a new real. Then • eitherV[x] is a Hechler extension ofV
• orV[x] is a Cohen extension ofV.
Note that the original dichotomy Corollary 13 generalizes to iterated Hechler extensions (cf. Theorem 17). This could not be true for Conjecture 15, becauseD?D˙ 6∼=Dby [Paw86].
With respect to the eventually dominating order≤∗ on the Baire space
ωω, one may consider three different kinds of eventually different reals:
bounded reals, unbounded reals which are not dominating, and reals which are dominating. E.g., random forcingBadds a bounded eventually different
real and, since B is ωω-bounding, there are no other kinds of eventually
different reals. By Corollary 13,Dadds a dominating (and thus necessarily
eventually different) real, but no other eventually different reals. Finally,E
Proof of Theorem 12. Let ˙xbe a D-name for x. Calls∈ω<ω very good if
there is an infinite partial functionxs:ω→ωinW which is not dominating over V such that s favors ˙x(k) = xs(k) for all k ∈ dom(xs). By “not-dominating” we mean, of course, that there isz∈ωω∩V such thatxs(k)≤
z(k) for infinitely many k ∈ dom(xs). With respect to this notion we introduce a rank rkx˙ exactly as before:
rkx˙(s) = 0 ⇐⇒ sis very good
rkx˙(s)≤α ⇐⇒ ∃∞n rkx˙(sˆn)< α
We say thatsisgoodif rkx˙(s)<∞. Otherwise sisnot good.
Case 1. All sare good.
This is the easiest case. By assumption, there is {xn;n ∈ ω} ∈ V
such that whenevery∈ωω∩V is infinitely often equal to allx
n, then y is
infinitely often equal to allxsfor very goods. So, choosey∈ωω∩V which
is infinitely often equal to allxn. We claim that the trivial condition forces
thaty is infinitely often equal to ˙x.
For indeed, letk0andT ∈Dbe given. Letsbe its stem. By assumption,
rkx˙(s)<∞. Thus, by replacing T with a stronger condition if necessary,
we may assume without loss of generality that rkx˙(s) = 0, i.e., s is very
good. Choosek≥k0 such that y(k) =xs(k). Since sfavors ˙x(k) =xs(k),
there is aU≤T such thatU x˙(k) =y(k), as required. a
Hence we may assume somesis not good. Then we can easily construct a conditionT with stemssuch that allt∈T extendingsare not good. We now work below the condition T. Call such t not badif there are infinite partial functionsyt, ft:ω→ω with the same domain in W such thatyt is not dominating overV,ftis one-to-one, andsˆft(k) favors ˙x(k) =yt(k) for allk∈dom(yt). Define the rank Rkx˙ as before:
Rkx˙(t) = 0 ⇐⇒ t is not bad
Rkx˙(t)≤α ⇐⇒ ∃∞n Rkx˙(tˆn)< α
We say thattisnot very bad if Rkx˙(t)<∞. Otherwisetisvery bad.
Case 2. All t∈T are not very bad.
Again, there is {xn;n ∈ ω} ∈ V such that whenever y ∈ ωω∩V is
infinitely often equal to allxn, thenyis infinitely often equal to allytfort
which is not bad. Choosey ∈ωω∩V which is infinitely often equal to all xn. We claim thatT forces thaty is infinitely often equal to ˙x.
Assumek0andU ≤T are given. Lettbe the stem ofU. By assumption,
such that y(k) = yt(k) and sˆft(k) belongs to U. Since sˆft(k) favors ˙
x(k) =yt(k), there is aU0≤U such thatU0x˙(k) =y(k). a
Case 3. Somet∈T is very bad.
Construct a conditionU ≤T with stemtsuch that allu∈U extending
tare very bad. We claim thatU forces that ˙xis a dominating real overV. To see this letz∈ωω∩V andU0 ≤U. We need to findk
0 andU00≤U0
such thatU00x˙(k)≥z(k) for all k≥k
0.
Let t0 = stem(U0). For u0 ∈ U0, define the partial function xu0 by xu0(k) = min{`;u0favors ˙x(k) =`}if the latter set is non-empty; otherwise xu0(k) is undefined. Note that, since u0 is not (very) good, xu0 either has finite domain or dominatesV. Therefore there isk0such that for allk≥k0,
eitherxt0(k) is undefined orxt0(k)≥z(k).
Similarly, foru0∈U0, defineyu0 byyu0(k) = min{`; for somen, we have u0ˆn∈U0 andu0ˆnfavors ˙x(k) =`}if the latter set is non-empty; otherwise
yu0 is undefined. Again, since u0 is (very) bad, it is easy to see that yu0 either has finite domain or dominatesV.
Now we recursively construct U00 ≤U0 with stem(U00) =t0, as well as numbersku0 for allu0∈U00.
First put t0 into U00 and fix kt0 ≥k0 such that for all k ≥ kt0, either yt0(k) is undefined or yt0(k)≥z(k). Next, putt0ˆnintoU00 if for allkwith k0 ≤k < kt0, whenever t0ˆn favors ˙x(k) =`, then ` ≥z(k). This defines the successor level oft0 because for each such k and each ` < z(k), there are only finitely many nsuch that t0ˆn favors ˙x(k) = `. By replacing the trees Ut00ˆn by appropriate subtrees if necessary, we may assume without loss of generality thatUt00ˆn forces ˙x(k)≥z(k) for all kwith k0≤k < kt0. Thus, U00 will also force this. Notice that xt0ˆn ≥ yt0 everywhere so that xt0ˆn(k)≥z(k) for all k≥kt0.
In general, assumeu0 has been put intoU00. Fixku0 ≥ku0(|u0|−1) such that for allk≥ku0, eitheryu0(k) is undefined or yu0(k)≥z(k). Next, put u0ˆn into U00 if for all k with ku0(|u0|−1) ≤k < ku0, whenever u0ˆn favors
˙
x(k) = `, then `≥ z(k). Since xu0(k) ≥z(k) for all k ≥ku0(|u0|−1), this indeed defines the successor level ofu0. Again we may assume that Uu00ˆn forces ˙x(k)≥z(k) for allk with ku0(|u0|−1) ≤k < ku0. Notice again that xu0ˆn ≥yu0 everywhere so that xu0ˆn(k)≥z(k) for all k≥ku0.
This completes the construction of U00, and it is immediate from the construction thatU00 forces ˙x(k)≥z(k) for allk≥k
0. a
This completes the proof of the theorem. q.e.d.
Lemma 16. Letγ be a limit ordinal. Assume (Pα,Q˙α;α < γ) is a finite
support iteration of c.c.c. forcing such that for allα < γthe following holds:
For everyPα-name ˙x:ω→ωfor an infinite partial function
which is not dominating over V, there are infinite partial functions xn : ω → ω, n ∈ ω, in V such that whenever
y∈ωω∩V is infinitely often equal to allxn, thenyis forced to be infinitely often equal to ˙x.
(?α)
Then (?γ) holds as well.
Proof. If cf(γ)> ω, then no new real number occurs at stageγ, and so the
claim is trivially true. Therefore we can assume that cf(γ) =ω. Since an iteration of length γ with cf(γ) is isomorphic to one of length ω, we can without loss of generality assume thatγ=ω.
Let ˙xbe aPω-name for an infinite partial function. Assume the trivial
condition forces that ˙xis not dominating over V and fixn < ω. In the Pn
-generic extensionVn, define a partial functionxn byxn(k) = min{`; there is a pin the remainder forcing Pω/Pn such thatp x˙(k) =`} if this set
is non-empty; otherwise xn(k) is undefined. Notice that xn is an infinite partial function, and that it cannot be dominating overV.
In the ground modelV, we havePn-names ˙xnfor all thexn. By (?n), we
can find a countable family{ym;m∈ω}such that whenevery∈ωω∩V is infinitely often equal to allym, then y is forced to be infinitely often equal
to all ˙xn. We show that such a y is also forced to be infinitely often equal
to ˙x.
Fixk0 andp∈Pω. Letnbe such thatp∈Pn. Step into Vn where the
generic containsp. Fixk≥k0such thatxn(k) =y(k). Letqbe a condition
in the remainder forcing which forces ˙x(k) =xn(k). Then clearlyrˆ ˙qforces ˙
x(k) =y(k) for somer≤pinPn, as required. q.e.d.
Theorem 17 (Dichotomy for iterated Hechler forcing). Let (Pα,D˙α;α < γ) be a finite support iteration of Hechler forcing. Let x be a real in the
Pγ-generic extension. Then
(i) eitherxis dominating overV
(ii) orxis not eventually different overV.
More explicitly, ifxis not dominating overV, then there are infinite partial functions xn :ω →ω, n∈ω, such that whenever y ∈ωω∩V is infinitely
often equal to allxn, theny is infinitely often equal toxas well.
Proof. By induction onγ. The caseγ= 1 is Corollary 13. More generally,
ifγ=δ+ 1 is a successor, apply Theorem 12 withW being thePδ-generic
Theorem 18. LetGbe L-generic for theω1-stage finite support iteration
of Hechler forcing. Then inL[G],∆12(D) holds while ∆12(E) fails.
Proof. By Theorem 17, it is immediate that the finite support iteration of
Hechler forcing does not add anE-generic over the ground modelV. Thus,
if the ground model is L, then in the generic extension after adding ω1
Hechler reals, there are noE-generics over L. This implies the failure of ∆1
2(E) by Theorem 6. q.e.d.
Σ12(E) =Σ12(D)
#+ O O O O O O O O O O O O O O O O O O O O
z~~~~ ~~~~ ~~~~ ~~~~ ~~ ~~~~ ~~~~ ~~~~ ~~~~ ~~
Σ12(B) =∆12(A)
( J J J J J J J J J J J J J J J J
s{oooooooo oo oooooooo
oo
Σ12(R) =∆12(R)
Σ12(C) =∆12(D)
s{nnnnnnnn nnn nnnnnnnn
nnn ∆
1 2(E)
s{ooooooooo o oooooo
oooo
∆12(B)
jr
Σ12(L) =∆12(L)
$,
∆12(C)
Σ12(V)
s{nnnnnnnn nnn nnnnnnnn
nnn ev. diff.
go
Σ12(M) =∆12(M)
∆12(V)
s{nnnnnnnn nnn nnnnnnnn
nnn
Σ12(S) =∆12(S)
Figure 1.
6
Conclusions
Our two main results, Theorems 7 and 18, are enough to place the two statements Σ12(E) and ∆12(E) in the diagram of regularity statements, as
it has been developed by other work. In the diagram given in Figure 1, the letters A, B, C, D, E, L, M, R, S, and Vstand for Amoeba, random,
there is an eventually different real overL[x]”. All implications and non-implications not involving E have been known before this paper, and in
the following we’ll give arguments for all implications and non-implications involving∆12(E). In the arguments, we shall freely use Theorem 6 and its
analogues for other forcings:
The statement “∆12(E)⇒ev.diff.” is trivial, and “∆ 1
2(E)⇒∆ 1 2(C)” is
easy becauseEadds Cohen reals. Corollary 11 yields “Σ12(B)⇒∆12(E)”, and Theorem 18 shows that the Hechler model witnesses “∆12(D)6⇒∆12(E)”. Note that neither random nor Mathias forcing add Cohen reals, so that the random model is a model of∆12(B)∧ ¬∆1
2(C) and the Mathias model
is a model of ∆12(R)∧ ¬∆1
2(C). Since every E-generic defines a Cohen
real, these two models witness “∆12(B)6⇒∆12(E)” and “∆12(R)6⇒∆12(E)”, respectively.
Finally, the iteration of E adds neither dominating nor random reals,
and thus the eventually different model witnesses both “∆12(E)6⇒∆12(L)”
and “∆12(E)6⇒∆12(B)”.
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