aa r X i v : . [ m a t h . L O ] J un COMBINING RESURRECTION AND MAXIMALITY
KAETHE MINDEN
Abstract.
It is shown that the resurrection axiom and the maximality prin-ciple may be consistently combined for various iterable forcing classes. Theextent to which resurrection and maximality overlap is explored via the localmaximality principle. Introduction
The maximality principle ( MP ) was originally defined by Stavi and V¨a¨an¨anen[SV02] for the class of ccc forcings. Maximality principles were defined in fullgenerality by Hamkins [Ham03], and expanded upon for different classes of forcingnotions by Fuchs [Fuc08], [Fuc09], and Leibman [Lei]. The axiom MP states that ifa sentence may be forced in such a way that it remains true in every further forcingextension, then it must have been true already, in the original ground model. Theresurrection axiom ( RA ) is due to Hamkins and Johnstone [HJ14a]. Very roughlyspeaking, RA posits that no matter how you force, it is always possible to forceagain to “resurrect” the validity of certain sentences - meaning a statement maynot be true after some forcing, but there is always a further forcing which will undothis harm. In fact the axiom grants a bit more than this, and posits an amount ofelementarity between the two-step extension and the ground model.A forcing class Γ is generally meant to be definable, closed under two-step iter-ations, and to contain trivial forcing. Resurrection and maximality may hold formore forcing classes than forcing axioms can, while they tend to imply their relevantbounded forcing axiom counterparts. In order for a forcing axiom to make sense fora particular class of forcing notions, the forcings should preserve stationary subsetsof ω . However, this restriction does not exist for the resurrection axiom or themaximality principle.A question reasonable to ask about any forcing class is whether or not the res-urrection axiom and the maximality principle may consistently both hold for thatclass. I answer the question positively and show that the consistency strength of thecombined boldface principles together is that of a strongly uplifting fully reflectingcardinal.Perhaps many forcing classes are ignored in this paper. The focus is on forcingclasses containing forcing notions which may potentially either collapse cardinals(to ℵ or larger), add reals, or both. I also want to look at forcing classes whichhave a corresponding forcing axiom. Thus the focus is on proper, ccc , countably Mathematics Subject Classification.
Key words and phrases. subcomplete forcing, proper forcing, the maximality principle, theresurrection axiom.Some of the material presented here is based on the author’s doctoral thesis [Min17], writtenat the Graduate Center of CUNY under the supervision of Gunter Fuchs. closed, subcomplete, and the class of all forcing notions. The results stated here forproper forcing should work similarly for semiproper and subproper forcing. Futurework may certainly flesh out important distinctions.In section 2 the definition of the maximality principle is given for various classesof forcing, and the relevant equiconsistency result is stated. The same is done forthe resurrection axiom in section 3. In section 4, it is shown that the two mayconsistently be combined. In 5, the local maximality principle is introduced as anatural axiom similar to a bounded forcing axiom but stronger, which both themaximality principle and the resurrection axiom imply.2.
The Maximality Principle
One motivation behind maximality principles is their connection to modal logic.In modal logic, necessary ( ✷ ) and possible ( ♦ ) are modal operators. In the realm ofset theory one interprets “possible” as forceable, or true in some forcing extension,and “necessary” as true in every forcing extension. Definition 2.1.
Let Γ be a forcing class defined by a formula (to be evaluated inthe forcing extensions in cascaded modal operator uses).We say that a sentence ϕ ( ~a ) is Γ- forceable if there is P ∈ Γ such that for every q ∈ P , we have that q (cid:13) ϕ ( ~a ). In other words, a statement is Γ-forceable if it isforced to be true in an extension by a forcing from Γ.A sentence ϕ ( ~a ) is Γ- necessary if for all P ∈ Γ and all q ∈ P , we have that q (cid:13) ϕ ( ~a ). So a sentence is Γ-necessary if it holds in any forcing extension by aforcing notion from Γ. If Γ contains the trivial forcing then a statement beingΓ-necessary implies that it is true.If S is a term in the language of set theory, then the Maximality Principle forΓ with parameters from S , which we denote MP Γ ( S ), is the scheme of formulaestating that every sentence with parameters from ~a from S that is Γ-forceably Γ-necessary is true. I.e., if the sentence “ ϕ ( ~a ) is Γ-necessary” is Γ-forceable, then ϕ ( ~a ) is true. In brief, the maximality principle posits ♦ ✷ ϕ = ⇒ ϕ .Write MP sc to stand for MP Γ where Γ = { P | P is subcomplete } , MP c in thecase where Γ is countably closed forcings, MP p for the class of proper forcings,and MP ccc for the class of ccc forcings. We leave out Γ if we are considering allforcing notions. Since all of these classes of forcing notions Γ contain trivial forcing,it follows that MP Γ is equivalent to the statement that every sentence that is Γ-forceably Γ-necessary is Γ-necessary.Before moving on, it should be noted that it of course does not technically makesense to write MP Γ = ⇒ P for some proposition P in the language of set theory,since MP Γ is a scheme. When something like this is written, it should be interpretedas saying instead ZFC + MP Γ ⊢ P .First we analyze the parameter set S that may be allowed in the definition. Basedon [Ham03, Obs. 3], it is clear that S = H ω is the natural parameter set for themaximality principle for the class of all forcing notions. We will write MP for theboldface version of the maximality principle for all forcing, i.e., MP = MP ( H ω ).For the case where we consider MP c as in [Fuc08], the natural parameter set touse is H ω . The same is true for MP sc . The next lemma follows Fuchs [Fuc08,Thm. 2.4]. OMBINING RESURRECTION AND MAXIMALITY 3
Lemma 2.2.
Let Γ be a forcing class containing forcing notions which collapsearbitrarily large cardinals to ω . Then MP Γ cannot be consistently strengthened byallowing parameters that aren’t in H ω . In particular, MP Γ ( S ) = ⇒ S ⊆ H ω . Proof.
The point is that for any set a , it is Γ-forceably Γ-necessary that a ∈ H ω .Indeed, after forcing to collapse | TC( { a } ) | to ω , we have that a ∈ H ω in theforcing extension. This must remain true in every further forcing extension. So, if MP sc ( { a } ) holds, it follows that a ∈ H ω . (cid:3) Write MP sc for MP sc ( H ω ), MP c for MP c ( H ω ). Lemma 2.3.
Let Γ be a forcing class which may add an arbitrary amount of realsbut cannot collapse cardinalities. Then MP Γ ( S ) = ⇒ S ⊆ H c . Proof.
The point is that “2 ω is greater than the hereditary size of a ” is Γ-forceablyΓ-necessary. (cid:3) Thus write MP ccc for MP ccc ( H c ). Note that by Lemma 2.2, MP p ( H c ) = ⇒ H c ⊆ H ω . In this paper, we choose to have the boldface version of the maximalityprinciple for proper forcing to be MP p = MP p ( H ω ).Assuming there is a regular cardinal δ satisfying V δ ≺ V , the maximality prin-ciple is consistent. The proof of this uses a technique that adapts arguments ofHamkins [Ham03, Lemma 1.22]. Hamkins has described the proof as “runningthrough the house and turning on all the lights”, in the sense that the posets thatare forced are those that push “buttons”, sentences that can be “switched on” andstay on, in all forcing extensions. A button in our case is a sentence ϕ ( ~a ) that isΓ-forceably Γ-necessary. As Hamkins [Ham03] discusses in detail, the existence ofa regular cardinal δ such that V δ ≺ V is a scheme of formulas sometimes referredto as the “L´evy scheme.” We refer to the L´evy scheme as positing the existence ofwhat we refer to as a fully reflecting cardinal. Theorem 2.4 ([Ham03, Thm. 31]) . The following consistency results hold.(1) MP = ⇒ ℵ V is fully reflecting in L .(2) MP ccc = ⇒ c V is fully reflecting in L .(3) MP p = ⇒ ℵ V is fully reflecting in L .(4) MP c = ⇒ ℵ V is fully reflecting in L .(5) [Min17, Lemma 4.1.4] MP sc = ⇒ ℵ V is fully reflecting in L . It follows that MP Γ cannot hold in L , since then L would think that ℵ or c isinaccessible in L , a contradiction. Theorem 2.5 ([Ham03, Thm. 32]) . Let δ be a fully reflecting cardinal. Then thereare forcing extensions in which the following hold:(1) MP and δ = c = ℵ .(2) MP ccc and δ = c .(3) MP p and δ = c = ℵ .(4) MP c and δ = ℵ and CH .(5) [Min17, Thm. 4.1.3] MP sc and δ = ℵ and CH . KAETHE MINDEN
Here a least-counterexample lottery sum iteration, which takes lottery sumsat each stage, is favored. Hamkin’s proofs make use of a bookkeeping function.Overall, the two methods follow the same “running through the house” method.We define the lottery sum poset below.
Definition 2.6.
For a family P of forcing notions, the lottery sum poset is definedas follows: M P = { P } ∪ {h P , p i | P ∈ P ∧ p ∈ P } with P weaker than everything and h P , p i ≤ h P ′ , p ′ i if and only if P = P ′ and p ≤ P p ′ .For two forcing notions, P and Q , write P ⊕ Q for L { P , Q } .One major difference encountered using a lottery sum is that ccc forcing notionsare not closed under lottery sums. So in the case of the ccc forcing class, we useHamkin’s method more directly.3. The Resurrection Axiom
The idea behind the resurrection axiom is to look at the model-theoretic conceptof existential closure in the realm of forcing, because, as is pointed out by Hamkinsand Johnstone ([HJ14a]), the notions of resurrection and existential closure aretightly connected in model theory. A submodel
M ⊆ N is existentially closed in N if existential statements in N using parameters from M are already truein M , i.e., M is a Σ -elementary substructure of N . Many forcing axioms can beexpressed informally by stating that the universe is existentially closed in its forcingextensions, since forcing axioms posit that generic filters, which normally exist ina forcing extension, exist already in the ground model. Hamkins and Johnstoneconsider resurrection for forcing extensions to be a more “robust” formulation offorcing axioms for various forcing classes. Resurrection axioms imply the truth oftheir associated forcing axiom, but not the other way around. Definition 3.1.
Let Γ be a class of forcing notions closed under two-step iterations.Let τ be a term for a cardinal to be computed in various models; e.g. c , ℵ , etc.The Resurrection Axiom RA Γ ( H τ ) asserts that for every forcing notion Q ∈ Γ thereis a further forcing ˙ R with (cid:13) Q ˙ R ∈ Γ such that if g ∗ h ⊆ Q ∗ ˙ R is V -generic, then H Vτ ≺ H V [ g ∗ h ] τ . Hamkins and Johnstone [HJ14a] examined RA Γ ( H c ) for Γ such as proper, ccc ,countably closed, and the class of all forcing notions. The reason H c is required ingeneral is that if some forcing notion in Γ adds new reals, then H κ , where κ > c in V , simply cannot be existentially closed in the forcing extension; the addedreal itself is witnessing the lack of existential closure. So certainly for κ > c andany class of forcing notions Γ which potentially add new reals, RA Γ ( H κ ) cannothold. For proper forcing and ccc forcing, we write RA p for RA p ( H c ) and RA ccc for RA ccc ( H c ). However, by the following result we see that RA c ( H c ) and RA sc ( H c ) areboth equivalent to CH . Proposition 3.2. [HJ14a, Thm. 6]
Suppose Γ contains a forcing which forces CH but no forcing in Γ adds new reals. Then CH ⇐⇒ RA Γ ( H c ) . OMBINING RESURRECTION AND MAXIMALITY 5
Proof.
For the forward implication, suppose that CH holds. Then since no newreals are added, H ω is unaffected by each forcing in Γ, and moreover, c remains ω in every extension by a forcing in Γ.For the backward direction, assume RA Γ ( H c ) holds. Let P force CH . Then thereis a further forcing ˙ R satisfying (cid:13) P ˙ R ∈ ˙Γ such that letting g ∗ h ⊆ P ∗ ˙ R begeneric, we have H c ≺ H V [ g ∗ h ] c . We know that CH has to hold still in V [ g ∗ h ] sinceno new reals are added to make c larger. Thus CH holds in V by elementarity,as desired. Indeed, CH is equivalent to the statement that H c contains only oneinfinite cardinal, which can be expressed in H c . (cid:3) Perhaps RA sc ( H c ), or indeed RA c ( H c ), is not necessarily the right axiom tolook at. So what is the correct axiom to examine? I will discuss two reasonablepossibilities for the hereditary sets: H ω and H ω . Let’s see what RA sc ( H ω ) and RA c ( H ω ) imply about the size of 2 ω . Proposition 3.3.
Suppose Γ contains forcing to collapse to ω . Then RA Γ ( H ω ) = ⇒ ω = ω .Proof. We show the contrapositive. Let 2 ω ≥ ω . Let κ = ω V . Then H ω | =“ κ = ω ”. But after forcing with C oll ( ω , κ ) we have that κ < ω in the extension.Moreover, if R is any further forcing in Γ, we will still have that for h ⊆ R generic, H V [ g ][ h ]2 ω | = “ κ < ω ”. So RA Γ ( H ω ) must fail. (cid:3) The next proposition gives a relationship between RA c ( H ω ) and RA c ( H ω ) aswell as between RA sc ( H ω ) and RA sc ( H ω ). However, the answer to the followingquestion is unknown. Question 3.4.
Is it the case that RA sc ( H ω ) ⇐⇒ RA sc ( H ω )? Indeed, is it thecase that RA c ( H ω ) ⇐⇒ RA c ( H ω )? Proposition 3.5.
Let Γ be a forcing class containing forcing notions which collapsearbitrarily large cardinals to ω . Then RA Γ ( H ω ) ⇐⇒ ω = ω + RA Γ ( H ω ) .Proof. For the forward direction, we already have that RA Γ ( H ω ) = ⇒ ω = ω by the previous proposition. Moreover, if RA Γ ( H ω ) holds, so does RA Γ ( H ω ),since H ω = H ω in the extension by elementarity.For the backward direction, suppose that RA Γ ( H ω ) holds and 2 ω = ω . Wewould like to show that RA Γ ( H ω ) holds. Toward that end, suppose that Q is inΓ and let g ⊆ Q be generic. Then we have that there is some forcing R with h ⊆ R generic over V [ g ], such that H V ω = H Vω ≺ H V [ g ∗ h ] ω . So if in V [ g ∗ h ] we have that2 ω = ω , then we are done. If not, let G collapse 2 ω to be ω over V [ g ][ h ]. Then H V [ g ∗ h ] ω = H V [ g ∗ h ∗ G ] ω = H V [ g ∗ h ∗ G ]2 ω , so we are done. (cid:3) In comparison to Proposition 3.2 and [HJ14a, Thm. 6], the lack of any obviousrestraints for the size of 2 ω lend credibility to H ω being the right parameter setto consider for forcing notions which do not add reals, but do contain the relevantcollapses. So this is what we will be using. Write RA sc for RA sc ( H ω ) and RA c for RA c ( H ω ).Since we are focusing our attention to the boldface maximality principle in thispaper, we should also look at the boldface version of resurrection, in which we carryaround a kind of parameter set. KAETHE MINDEN
Definition 3.6.
Let Γ be a fixed, definable class of forcing notions. Let τ be aterm for a cardinality to be computed in various models; e.g. c , ω , etc. The Boldface Resurrection Axiom RA Γ ( H τ ) asserts that for every forcing notion Q ∈ Γand A ⊆ H τ there is a further forcing ˙ R with (cid:13) Q ˙ R ∈ Γ such that if g ∗ h ⊆ Q ∗ ˙ R is V -generic, then there is an A ∗ ∈ V [ g ∗ h ] such that h H Vτ , ∈ , A i ≺ h H V [ g ∗ h ] τ , ∈ , A ∗ i . Again, it doesn’t make too much sense to talk about the resurrection axiom at thecontinuum for forcing classes which can’t add new reals. Thus as in the lightfaceversions, the notion we will be looking at for the boldface version is RA sc ( H ω )which we will just refer to as RA sc , RA p for RA p ( H c ), RA ccc for RA ccc ( H c ), and RA for RA ( H c ). Hamkins and Johnstone determined that the consistency strengthof boldface resurrection is a strongly uplifting cardinal. Definition 3.7.
We say that κ is strongly uplifting so long as κ is θ -strongly uplift-ing for every ordinal θ . This means that for every A ⊆ V κ there is an inaccessiblecardinal γ ≥ θ and a set A ∗ ⊆ V γ such that h V κ , ∈ , A i ≺ h V γ , ∈ , A ∗ i is a properelementary extension. If κ is strongly uplifting then κ is inaccessible. Theorem 3.8 ([HJ14b, Thm. 19]) . Let κ be strongly uplifting. Then there areforcing extensions in which the following hold:(1) RA and κ = c = ℵ .(2) RA ccc and κ = c .(3) RA p and κ = c = ℵ .(4) RA c and κ = ℵ and CH .(5) [Min17, Thm. 4.2.12] RA sc and κ = ℵ and CH . Theorem 3.9 ([HJ14b, Thm. 19]) . We have the following implications:(1) RA = ⇒ ℵ V is strongly uplifting in L .(2) RA ccc = ⇒ c V is strongly uplifting in L .(3) RA p = ⇒ c V = ℵ V is strongly uplifting in L .(4) RA c = ⇒ ℵ V is strongly uplifting in L .(5) [Min17, Thm 4.2.13] RA sc = ⇒ ℵ V is strongly uplifting in L . Combining Resurrection and Maximality
Hamkins and Johnstone [HJ14a, Section 6] combine the resurrection axiom withforcing axioms, like
PFA for example, and show that they hold after a forcingiteration. Fuchs explores combinations of maximality principles for closed forcingnotions [Fuc08] combined with those for collapse and directed closed forcing notions[Fuc09] and hierarchies of resurrection axioms with an emphasis on subcompleteforcing [Fuc18a]. Trang and Ikegami studied classes of maximality principles com-bined with forcing axioms [IT18]. This section focuses on a different question in asimilar vein: is it possible for the resurrection axiom and the maximality principleto hold at the same time? The axioms do not imply each other directly, MP surelydoes not imply RA , since the consistency strength of MP is that of a fully reflectingcardinal, while RA has the consistency strength a strongly uplifting cardinal. These As described by Hamkins and Johnstone in the comments on page 5 of their paper, we maylet γ be regular, uplifting, weakly compact, etc. OMBINING RESURRECTION AND MAXIMALITY 7 two cardinals are consistently different, they certainly don’t imply each other. If κ is fully reflecting, take the least γ such that V κ ≺ V γ . If there isn’t such a γ , then κ isn’t uplifting anyway. But in V γ , we have that κ is not uplifting. There is noimplication in the other direction as well. This is because working in a minimalmodel of ZFC + “ V = L ” + “there is an uplifting cardinal”(i.e., no initial segment of the model satisfies this theory), we may force over thisminimal model to obtain RA . Now MP can’t hold in the extension, since letting κ be the ω of the extension, if MP were true, then that would imply that L κ iselementary in L – contradicting the minimality of the model we started with.This section is dedicated to showing that it is possible for maximality and resur-rection to both hold, by combining the techniques showing the consistency of eachprinciple, all in one minimal counterexample iteration.An inaccessible cardinal κ is strongly uplifting fully reflecting so long as it is bothstrongly uplifting and fully reflecting.Combining these two large cardinal notions is almost natural. If κ is upliftingthen there are unboundedly many γ such that V κ ≺ V γ , and if on top of that κ isfully reflecting, we add that V κ ≺ V as well, where V is in some sense the limit ofthe V γ ’s. Moreover, strongly uplifting fully reflecting cardinals are guaranteed toexist if there are subtle cardinals. Definition 4.1.
A cardinal δ is subtle so long as for any club C ⊆ δ and for anysequence A = h A α | α ∈ C i with A α ⊆ α , there is a pair of ordinals α < β in C such that A α = A β ∩ α . Fact 4.2.
If a cardinal δ is subtle, then δ is inaccessible. Proposition 4.3. If δ is subtle, then it is consistent that there is a strongly upliftingfully reflecting cardinal. Namely, the set { κ < δ | V δ | = “ κ is strongly uplifting and V κ ≺ V ” } is stationary in δ .Proof. Hamkins and Johnstone [HJ14b, Thm. 7] show that if δ is subtle, then theset of cardinals κ below δ that are strongly uplifting in V δ is stationary. But since δ is subtle, it must also be inaccessible by Fact 4.2. Thus in V δ , by the proof of thedownward Lowenheim-Skolem theorem, there is a club C ⊆ δ of cardinals κ suchthat V κ ≺ V δ ; meaning that κ is fully reflecting in V δ . This means that there issome α < κ that is both strongly uplifting and fully reflecting in V δ , giving us therequired consistency. (cid:3) We can immediately see that if resurrection and maximality both hold, we musthave a strongly uplifting fully reflecting cardinal, by combining the results of 2.4and 3.9.
Observation 4.4.
The following consistency results hold.(1) MP + RA = ⇒ ℵ V is strongly uplifting fully reflecting in L .(2) MP ccc + RA ccc = ⇒ c V is strongly uplifting fully reflecting in L .(3) MP p + RA p = ⇒ ℵ V is strongly uplifting fully reflecting in L .(4) MP c + RA c = ⇒ ℵ V is strongly uplifting fully reflecting in L .(5) MP sc + RA sc = ⇒ ℵ V is strongly uplifting fully reflecting in L . KAETHE MINDEN
For the other direction of the consistency result, we restate here a restrictedversion of the lifting lemma [HJ14b, Lemma 17], using the comments following it,to allow the lifting of certain embeddings to generic extensions.
Fact 4.5 (Lifting Lemma) . Suppose that h M, ∈ A i ≺ h M ∗ , ∈ , A ∗ i are transitivemodels of ZFC and P is an Ord -length forcing iteration without any condition withfull support in M . If G ⊆ P is an M -generic filter and G ∗ ⊆ P ∗ is M ∗ -generic with G = G ∗ ∩ P , then h M [ G ] , ∈ , A, G i ≺ h M ∗ [ G ∗ ] , ∈ , A ∗ , G ∗ i . Theorem 4.6.
Let κ be a strongly uplifting fully reflecting cardinal. Then there isa forcing extension in which both RA and MP hold, and κ = c = ℵ .Proof. Let κ be strongly uplifting fully reflecting. Below we define P to be the least-counterexample to RA + MP lottery sum finite support iteration of length κ . Wegenerically pick, using the lottery sum, whether at each stage to force with a least-rank counterexample to the maximality principle or a least-rank counterexampleto the boldface resurrection axiom.In particular, we are defining the poset P = P κ = h ( P α , ˙ Q α ) | α < κ i as follows:At stage α , consider all of the sentences with parameters having names in V P α κ that are not true in V P α κ , but can be forced by some poset ˙ Q to be necessary. Let M be the collection of such possible forcing notions ˙ Q in V P α κ of minimal rank in V P α κ . In other words, M contains the current minimal rank counterexamples to theboldface maximality principle.Similarly we take R to contain all of the current minimal rank counterexamplesto the boldface resurrection axiom. In particular, let R be the collection of forcingnotions ˙ Q of minimal rank such that there is ˙ A ⊆ H V P α c where after any furtherforcing ¨ R , it is not the case that h H V P α c , ∈ , ˙ A i ≺ h H V P α ∗ ˙ Q ∗ ¨ R c , ∈ , ˙ A ∗ i for any ˙ A ∗ .Then take P α +1 = P α ∗ ˙ Q α where ˙ Q α is a term for the lottery sum L R ⊕ L M .Limit stages are taken care of via finite support.Let G ⊆ P be generic. We need to show that both RA and MP hold in V [ G ]. Claim . MP holds in V [ G ]. Pf.
Assume it fails; the sentence ϕ ( ~a ), where ~a ∈ H V [ G ] ω is a parameter set, has theproperty that: V [ G ] | = “ ϕ ( ~a ) is forceably necessary but ϕ ( ~a ) is false.”Choose a condition p ∈ G that forces the above statement. P has the κ - cc , sinceall of the forcing notions are small– they are all elements of least rank in V κ – soat no stage in the iteration is κ collapsed. This means that there has to be somestage where ~a appears. So there is some stage in the iteration beyond the supportof p , say α < κ , where ~a ∈ V κ [ G α ]. Specifically ϕ ( ~a ) is an available button at stage α , since after the rest of the iteration, P tail where P = P α ∗ P tail , we have that ϕ ( ~a ) is forceably necessary. Indeed, this is reasoning available in V [ G α ], which thussees that ϕ ( ~a ) is a button. By elementarity, as κ is fully reflecting, it follows that V κ [ G α ] | = “ ϕ ( ~a ) is forceably necessary” . From that point on, ϕ ( ~a ) continues to bea button, since we have that α is beyond the support of p ∈ G . Thus it is dense, in P , for ϕ ( ~a ) to be “pushed” at some point after stage α , say β . So we have β < κ such that there is some Q forcing ϕ ( ~a ) to be necessary in V κ [ G β ]. Let H ⊆ Q begeneric over V [ G β ] so that there is some G tail generic for the rest of P satisfying V [ G β ][ H ][ G tail ] = V [ G ]. The sentence ϕ ( ~a ) is now necessary in V κ [ G β ][ H ]. But OMBINING RESURRECTION AND MAXIMALITY 9 then since V κ [ G β ][ H ] ≺ V [ G β ][ H ], as we are still in an initial segment of thefull iteration, we have that ϕ ( ~a ) is necessary in V [ G β ][ H ], by elementarity. Thussince the rest of the iteration is a forcing notion in its own right, ϕ ( ~a ) is true in V [ G β ][ H ][ G tail ] = V [ G ], contradicting our assumption that ϕ ( ~a ) is false in V [ G ].Thus MP holds in V [ G ]. (cid:3) Claim . RA holds in V [ G ] Pf.
Assume toward a contradiction that RA fails. This means we can choose a leastrank counterexample, a forcing Q in V [ G ] that supposedly cannot be resurrected.Let A ⊆ H V [ G ] c , be its associated predicate. Let ˙ Q be a name for Q of minimalrank. Since P has the κ - cc , there must be a name for the predicate in the extensionsuch that ˙ A ⊆ H κ with A = ˙ A G .We will argue that ˙ Q appears at stage κ of the same exact iteration, except de-fined in some larger V γ [ G ] = V V [ G ] γ where γ is inaccessible. Use the strong upliftingproperty of κ , and code the iteration P as a subset of κ , to find a sufficiently largeinaccessible cardinal γ so that h V κ , ∈ , P , ˙ A i ≺ h V γ , ∈ , P ∗ , ˙ A ∗ i , where P ∗ is the least-counterexample to RA lottery sum iteration of length γ as defined in V γ . Obtaininga large enough γ involves a process of closing under least-rank counterexamples.Not only does V γ need to agree with V about the rank of least-rank counterexam-ples to the resurrection axiom throughout the iteration P , it must also computethe least-rank counterexamples to the maximality principle appropriately as well.Since V κ ∈ V γ , and the ranks throughout the maximality iteration were computedin V κ , the minimal ranks are guaranteed to be computed properly in V γ . Indeedwe have argued above that P ∗ is defined the same way as P below stage κ , so wemay assume below a condition in G that ˙ Q may be picked at stage κ . So belowa condition that opts for ˙ Q at the stage κ lottery we may say that P ∗ factors as P ∗ ˙ Q ∗ ˙ P ∗ tail . Let H ∗ G ∗ tail ⊆ Q ∗ P ∗ tail be V [ G ]-generic. Letting G ∗ = G ∗ H ∗ G ∗ tail ,this means that G ∗ ⊆ P ∗ is generic over V .Thus by the lifting lemma (Fact 4.5) the strongly uplifting embedding h V κ , ∈ , P , ˙ A i ≺ h V γ , ∈ , P ∗ , ˙ A ∗ i lifts to h V κ [ G ] , ∈ , P , ˙ A, G i ≺ h V γ [ G ∗ ] , ∈ , P ∗ , ˙ A ∗ , G ∗ i in V [ G ∗ ]. Since A is definable from ˙ A and G , we may say that h V κ [ G ] , ∈ , P , A i ≺ h V γ [ G ∗ ] , ∈ , P ∗ , A ∗ i . We have that V κ [ G ] = H V [ G ] κ = H V [ G ] c , since κ is inaccessible and P has the κ - cc We can argue the same way as above, replacing κ with γ , to get that V γ [ G ∗ ] = H V [ G ∗ ] γ = H V [ G ∗ ] c . This establishes h H V [ G ] c , ∈ , A i ≺ h H V [ G ∗ ] c , ∈ , A ∗ i , so RA in factholds as desired. (cid:3)(cid:3) Theorem 4.7.
Let κ be a strongly uplifting fully reflecting cardinal. Then thereare forcing extensions in which we have the following:(1) RA p + MP p + κ = c = ℵ .(2) RA c + MP c + κ = ℵ + CH .(3) RA sc + MP sc + κ = ℵ + CH . Proof.
For all of these arguments, use Theorem 4.6 as a blueprint. The definition ofthe iteration is always the same, with the caveat that the forcing notions are alwaystaken to be in the relevant class we are thinking about (where Γ is proper whenwe show (1), and so on). Let’s repeat the description of the iteration as before,relativized to a forcing class Γ.Each time we define an iteration P = P κ = h ( P α , ˙ Q α ) | α < κ i as follows:At stage α , consider all of the sentences with parameters having names in V P α κ that are not true in V P α κ , but can be forced by some poset ˙ Q ∈ Γ V P ακ to be necessary.Let M be the collection of such possible forcing notions of minimal rank in V P α κ forwhich the above holds. So M contains the minimal rank counterexamples to MP Γ .Additionally, let R be the collection of forcing notions ˙ Q ∈ Γ V P α of minimal ranksuch that there is ˙ A ⊆ H V P α c where after any further forcing ¨ R ∈ Γ V P α ∗ ˙ Q and forall ˙ A ∗ , it is not the case that h H V P α c , ∈ , ˙ A i ≺ h H V P α ∗ ˙ Q ∗ ¨ R c , ∈ , ˙ A ∗ i . So R contains theminimal rank counterexamples to RA Γ .Then take P α +1 = P α ∗ ˙ Q α where ˙ Q α is a term for the lottery sum L R ⊕ L M .The argument for each forcing class follows the above template, granted a usefulsupport is used. For (1) and (2), use countable support. For (3), revised count-able support works, after seeing that such iterations of subcomplete forcing aresubcomplete [Jen14]. (cid:3) The iteration as defined is clearly ill-suited for ccc forcings, since ccc forcings arenot closed under lottery sums. To resolve this let’s combine the arguments from[HJ14b] and [Ham03]. This method could also be used for other forcing classes.
Definition 4.8. [HJ14b] A
Laver function ℓ for a strongly uplifting cardinal κ is apartial function from κ to V κ satisfying that for every A ⊆ κ , every ordinal θ , andevery set x , there is a proper elementary extension h V κ , ∈ , A, ℓ i ≺ h V γ , ∈ , A ∗ , ℓ ∗ i where γ ≥ θ is inaccessible and ℓ ∗ ( κ ) = x .By [HJ14b, Thm. 11] every strongly uplifting cardinal has such a Laver functiondefinable in h V κ , ∈i . Theorem 4.9.
Let κ be a strongly uplifting fully reflecting cardinal. Then there isa forcing extension in which RA ccc + MP ccc holds, and κ = c .Proof. Let ℓ be a strongly uplifting Laver function for κ . Let ~ϕ = h ϕ α ( ˙ ~a ) | α < κ i enumerate, with unbounded repetition, all sentences in the language of set theorywith names for parameters in V κ coming from a forcing extension by a forcing ofsize less than κ .Define a finite support κ -iteration of ccc forcing so that at successor stages α = β + 1, the forcing Q α is least rank forcing ϕ β to be necessary over V P α κ , if possible.Otherwise, do trivial forcing at that stage. At limit stages, force with ℓ ( α ), providedthat this is a P α -name for a ccc forcing. Otherwise, again do trivial forcing at thatstage.Unboundedly often, the forcing will not be trivial in this iteration. Forcing tomake c arbitrarily large below κ will happen periodically at successor stages. Atlimits, the Laver function will choose Cohen forcing unboundedly often.Let G ⊆ P be generic. We have that κ = c V [ G ] , which follows from the above, andalso since P is a finite support iteration of ccc forcings, a Cohen real will be addedin the forcing up to limit stages of countable cofinality. Since there are κ -manysuch stages, κ -many Cohen reals are added. OMBINING RESURRECTION AND MAXIMALITY 11
Claim . MP ccc holds in V [ G ]. Pf.
Let ϕ ( ~a ) be ccc -forceably ccc -necessary over V [ G ] with ~a ∈ H V [ G ] c = H V [ G ] κ .Since P is ccc , ~a has a P α -name ˙ ~a , where β +1 = α < κ is some successor ordinal and ϕ ( ˙ ~a ) = ϕ β ( ˙ ~a ). Since ϕ ( ~a ) is ccc -forceably ccc -necessary in V [ G ] it must have alsobeen ccc -forceably ccc -necessary in such a V [ G α ] (by adding the rest of the iterationto the beginning of whatever forcing notion makes the sentence ccc -necessary in V [ G ]). Since V κ [ G α ] ≺ V [ G α ], the sentence must have been ccc -forceably ccc -necessary in V κ [ G α ]. Thus ϕ ( ~a ) was forced to be ccc -necessary; and so it is ccc -necessary in V [ G α +1 ]. Since the rest of the iteration is ccc , it is thus true in V [ G ]. (cid:3) Claim . RA ccc holds in V [ G ]. Pf.
Suppose that A ⊆ κ and Q is a ccc forcing in V [ G ]. Let ˙ A and ˙ Q be P -namesfor A and Q respectively. Let θ be an ordinal. Since ℓ is a strongly uplifting Laverfunction for κ , there is an extension h V κ , ∈ , ˙ A, P , ~ϕ, ℓ i ≺ h V γ , ∈ , ˙ A ∗ , P ∗ , ~ϕ ∗ , ℓ ∗ i with γ ≥ θ inaccessible and ℓ ∗ ( κ ) = ˙ Q . Note that P is definable from ~ϕ and ℓ . Thus P ∗ isthe corresponding γ -iteration defined from ℓ ∗ and ~ϕ ∗ . Furthermore P ∗ = P ∗ ˙ Q ∗ ˙ P ∗ tail ,where ˙ P ∗ tail is the rest of the iteration after stage κ up to γ , which is ccc in V [ G ][ H ]( H ⊆ Q generic over V [ G ]) since it is a finite support iteration of ccc forcing.Let G ∗ ⊆ P ∗ be generic over V , containing G ∗ H . By Fact 4.5, we may lift theelementary extension to h V κ [ G ] , ∈ , ˙ A, P , ~ϕ, ℓ, G i ≺ h V γ [ G ∗ ] , ∈ , ˙ A ∗ , P ∗ , ~ϕ ∗ , ℓ ∗ , G ∗ i . As γ = c V [ G ∗ ] , the desired result follows. (cid:3)(cid:3) Local Maximality
The local maximality principle is a natural axiom which elucidates somewhathow the resurrection axiom and the maximality principle intersect. It is one kindof intermediate step between the maximality principle and bounded forcing axiom.In the local version of the maximality principle, the truth of a forceably necessarysentence will be checked not in V but in a much smaller structure. This shouldbe compared to one of the equivalent ways of defining the bounded forcing axiom,namely generic absoluteness.For the appropriate definition of the bounded forcing axiom BFA κ (Γ) refer to[Bag00, Definition 2]. Definition 5.1.
Let n, m be natural numbers, Γ be a class of forcing notions,and M be a transitive set (usually either ω or H ω ). Then Γ -generic Σ mn ( M ) -absoluteness with parameters in S ⊆ P ( M ) is the statement that for any Σ mn -sentence ϕ ( ~a ) where ~a ∈ S ∩ M and predicate symbols ~ ˙ A , the following holds:Whenever ~A ∈ S ∩ P ( M ), P ∈ Γ, and G is P -generic over V , then( h M, ∈ , ~A i | = ϕ ( ~a )) V ⇐⇒ ( h M, ∈ , ~A i | = ϕ ( ~a )) V [ G ] , where ~ ˙ A is meant to be interpreted in M as ~A , and the satisfaction is order m + 1. Fact 5.2 ([Bag00]) . Let κ be an infinite cardinal of uncountable cofinality and let Γ be a class of forcing notions. Then the following are equivalent:(1) BFA κ (Γ) (2) For every P ∈ Γ and generic G ⊆ P , H κ + ≺ Σ H V [ G ] κ + . (3) Γ -generic Σ ( H κ + ) -absoluteness. The bounded forcing axiom
BFA κ (Γ) where κ = 2 ω and Γ is the class of ccc forcing notions is equivalent to MA κ . We look at κ = ω . The bounded forcingaxiom for the class of countably closed forcing is a theorem of ZFC . We write
BSCFA for when Γ = { P | P is subcomplete } , and BPFA for proper forcing. Theconsistency strength of each of these bounded forcing axioms is exactly that ofa reflecting cardinal (as is shown by Goldstern [GS95] for proper forcing and thesubcomplete version is shown by Fuchs [Fuc18b]).In the following, we write ϕ M ( ~a ) for the sentence M | = ϕ ( ~a ). Definition 5.3.
Let Γ be a class of forcing notions, and let S be a set of parameters.Let M be a defined term for a structure to be reinterpreted in forcing extensions,and S ⊆ M . The Local Maximality Principle relative to M ( MP M Γ ( S )) is thestatement that for every parameter set ~a ∈ S and every sentence ϕ ( ~a ), if ϕ M ( ~a ) isΓ-forceably Γ-necessary, then ϕ M ( ~a ) is true.Write LMP for the local version of MP , and LMP Γ for the local version of MP Γ with forcing classes Γ. As before and as discussed in Lemmas 2.2 and 2.3, the choiceof H ω makes sense for the parameter set for the boldface subcomplete, countablyclosed, and proper maximality principles, and the choice of H c makes sense for theboldface ccc maximality principle. Additionally, the smallest model M that makessense to use for the local version has to at least contain the parameter set, so S = M is what we will work with here.Clearly for forcing classes Γ, MP Γ ( H κ ) = ⇒ MP H κ Γ ( H κ ). Proposition 5.4.
For Γ a class of forcing notions, RA Γ ( H κ ) = ⇒ MP H κ Γ ( H κ ) .Proof. Suppose that RA Γ ( H κ ) holds. To see that the local maximality principleholds, suppose that ϕ ( ~a ) is a sentence such that the sentence “ H κ | = ϕ ( ~a )” is Γ-forceably Γ-necessary. So there is a forcing notion P ∈ Γ such that after any furtherforcing, we have that “ H κ | = ϕ ( ~a )” holds in the two-step extension. By resurrection,there is a further ˙ R such that (cid:13) P “ ˙ R ∈ Γ” such that letting G ∗ h ⊆ P ∗ ˙ R be genericwe have H κ ≺ H V [ G ∗ h ] κ . Since “ H κ | = ϕ ( ~a )” in the two-step extension V [ G ][ h ] byour assumption, this means that H κ | = ϕ ( ~a ) holds by elementarity, so MP H κ Γ ( H κ )holds as desired. (cid:3) Proposition 5.5.
For Γ a class of forcing notions, MP H κ + Γ ( H κ + ) = ⇒ BFA κ (Γ) .Proof. Assume that MP H κ + Γ ( H κ + ) holds. We use characterization 3 of BFA κ (Γ)from Fact 5.2. To show that Γ-generic Σ ( H κ + )-absoluteness holds, let ϕ ( ~x ) be aΣ -formula, ~a ∈ H κ + , and P ∈ Γ, satisfying (cid:13) P ϕ (ˇ ~a ). Let G ⊆ P be generic. Since ϕ ( ~x ) is Σ and H κ + ≺ Σ V as κ + is regular, we have that ϕ H κ + ( ~a ) = ϕ H κ + ((ˇ ~a ) G )holds in all future forcing extensions. Thus ϕ H κ + ( ~a ) is Γ-forceably Γ-necessary,which means that ϕ H κ + ( ~a ) is true (in V ) by the local maximality principle. Thus ϕ ( ~a ) holds in V as desired. (cid:3) Proposition 5.6.
LMP sc (and LMP c ) implies:(1) There is a Suslin tree.(2) ♦ holds.(3) CH holds. OMBINING RESURRECTION AND MAXIMALITY 13
Proof.
Firstly, H ω is enough to verify each of these properties.For 1, note that the forcing to add a Suslin tree is countably closed and thusis subcomplete as well. But any particular Suslin tree will continue to be a Suslintree after any subcomplete (or countably closed) forcing [Jen09, Ch. 3 p. 10]. Thusthe existence of a Suslin tree is sc -forceably sc -necessary, and hence true by LMP sc (and likewise for countably closed forcing).For 2 note that Jensen [Jen09, Ch. 3 p. 7] shows that ♦ will hold after performingsubcomplete forcing if it held in the ground model (although we haven’t yet seenthat any particular instance of a ♦ -sequence will be preserved). Since forcing to adda ♦ -sequence is countably closed, and ♦ will continue to hold after any subcomplete(or countably closed) forcing, so must be true by the relevant boldface maximality.Of course then 3 follows, since ♦ implies CH . (cid:3) Proposition 5.7.
LMP p (and LMP ccc ) implies:(1) There are no Suslin trees.(2) All Aronszajn trees are special.Proof.
Forcing with a Suslin tree is proper (indeed, ccc ), and adds a branch throughthe tree making it fail to be Suslin. Additionally specializing Aronszajn trees isproper, in fact ccc , and kills Suslin trees, but also once a specializing function isadded it can’t be removed by further proper (or ccc ) forcing. (cid:3)
Consistency of the Local Maximality Principle.
We will now introducethe large cardinal property that is equiconsistent with the local maximality prin-ciple. When showing the consistency of the resurrection axiom in section 3, wedefined the notion of an uplifting cardinal , of which the following property is thesuitable “local” version.
Definition 5.8.
An inaccessible cardinal δ is locally uplifting so long as for everyformula ϕ ( x ) and a ∈ V δ , for every θ we have that θ -locally uplifting, meaning thatthere is an inaccessible γ > θ such that V δ | = ϕ ( a ) ⇐⇒ V γ | = ϕ ( a ) . Note that if a regular cardinal δ has the property of being locally uplifting,without necessarily being inaccessible, then δ must be inaccessible, since otherwiseif 2 α ≥ δ for some α < δ , this is seen by some larger V γ , i.e., V γ | = ∃ β [2 α = β ]. Soby elementarity there is some β ′ = 2 α in V δ , a contradiction.Clearly if κ uplifting then κ is locally uplifting, and if κ is fully reflecting, it islocally uplifting. We have the following relationship between locally uplifting andΣ -reflecting cardinals. Proposition 5.9. If κ is locally uplifting then κ is Σ -reflecting.Proof. Suppose that κ is locally uplifting. To show that κ is Σ -reflecting, let ϕ ( x )be a formula and let a ∈ H κ , and assume that there is θ > κ where H θ | = ϕ ( ~a ).Define ψ as follows: ψ ( a ) : ∃ δ [ H δ | = ϕ ( ~a )] . Then if we take γ > θ satisfying H θ ∈ H γ , we have that H γ | = ψ ( ~a ). As κ is locallyuplifting, this implies that H κ | = ψ ( ~a ). Thus there is δ < κ such that H δ | = ϕ ( ~a )as desired. (cid:3) The local maximality principle is equiconsistent with the existence of a locallyuplifting cardinal, using the same method as with the proof of the maximalityprinciple but with some care in relativizing to H ω . Theorem 5.10. If LMP holds, then ℵ V is locally uplifting in L .Proof. Let κ = ℵ V and suppose that the local maximality principle holds.Firstly, κ is a limit cardinal in L , since for γ < κ , the statement H ω | = “thereis a cardinal in L greater than γ ” is forceably necessary (by taking C oll ( ω, κ )) andthus true in H ω . So we have that κ is inaccessible.Assume L κ | = ϕ ( ~a ). In other words, H Vω | = ϕ L ( ~a ). We need to show that thereis a larger γ such that L γ | = ϕ ( ~a ). In order to do this, let’s work in L and first seethat the following is necessarily forceable:(1) H ω | = ( ϕ L ( ~a ) ∧ “there are unboundedly many cardinals in L ”) . This holds since otherwise it is forceably necessary that H ω | = ¬ ϕ L ( ~a ), so H Vω | = ¬ ϕ L ( ~a ) holds, a contradiction.So, given some θ > κ , we may force over L to collapse θ to ω . Then as (1) isnecessarily forceable, there is further forcing to reach a model V [ G ][ H ] such that H V [ G ][ H ] ω | = ϕ L ( ~a ) . Thus in V [ G ][ H ], ϕ L ω ( ~a ) holds. Since ω V [ G ][ H ]1 = γ > θ > κ inthis extension now, and furthermore by (1) L γ | = ϕ ( ~a ) ∧ “there are unboundedly many cardinals” , we now have a suitable γ that is inaccessible in L and L γ | = ϕ ( ~a ) as desired. (cid:3) Observation 5.11.
The following consistency results hold.(1)
LMP ccc = ⇒ c V is locally uplifting in L .(2) LMP p = ⇒ ℵ V is locally uplifting in L .(3) LMP c = ⇒ ℵ V is locally uplifting in L .(4) LMP sc = ⇒ ℵ V is locally uplifting in L .Proof. (2)-(4) hold by collapsing to ω instead of ω in the proof of 5.10. For (1)instead of collapsing κ to be as small as desired, blow up the continuum as needed,like in the comparable proof for MP ccc in [Ham03, Thm. 31.2]. (cid:3) Theorem 5.12. If δ is locally uplifting, then there is a forcing extension in which LMP holds and δ = ℵ .Proof. Let δ be locally uplifting. Define the δ -length lottery sum finite supportiteration P = P δ as follows: for α < δ let P α +1 = P α ∗ ˙ Q α where ˙ Q α is a P α -namefor the lottery sum of all minimal rank posets that force some sentence relativizedto H V P αδ ω to be necessary. In particular, let Φ be the collection of sentences ϕ ( ~a )with parameter ~a ∈ H V P αδ ω such that: V P α δ | = “ ϕ H ω ( ~a ) is forceably necessary.”So Φ is the set of all possible H ω -buttons available at this point in the iteration.Then we let˙ Q α = M ϕ ∈ Φ n ˙ Q ∈ V P α δ | ˙ Q is least rank, V P α δ | = “ ˙ Q forces ‘ ϕ ( ~a ) H ω is necessary.’” o We shall refer to this definition as the least-rank
LMP lottery sum iteration oflength δ .Since we will want the full iteration P to remain relatively small in size andto have the δ - cc , notice that here we insist that the parameters for our sentencescome from H V P αδ ω . As δ is inaccessible, it is large enough so that H Vω = H V δ ω , and OMBINING RESURRECTION AND MAXIMALITY 15 moreover this remains true in each subsequent extension in the iteration so H ω in the subsequent extensions gets interpreted the same in V P α δ as in V P α . Thisis because δ is locally uplifting: if ˙ Q ′ ∈ V P α forces that a sentence ϕ ( ~a ) H ω isnecessary, where ~a ∈ H ω , then take θ large enough so that ˙ Q ′ ∈ V P α θ . Then as δ islocally uplifting we have an inaccessible γ > θ such that V P α γ | = “There is a forcing notion ˙ Q which forces ‘ ϕ ( ~a ) H ω is necessary.’”and V P α δ models the above sentence as well. So since each of the iterands of theforcing P are taken to be of least rank, they are all in V δ anyway. If on the otherhand at stage α + 1 we have that ˙ Q ′ ∈ V P α δ is of least rank forcing that a sentence ϕ ( ~a ) H ω is necessary, then the only way it could be wrong is that there is somefurther forcing ˙ R ′ that is not in V P α ∗ ˙ Q ′ δ that forces the sentence to be false. Butthen we may take γ larger than the verification of this forcing ˙ R ′ , and use the factthat δ is locally uplifting to see that V P α ∗ ˙ Q ′ δ | = “There is a forcing notion ˙ R which forces ¬ ϕ ( ~a ) H ω .”This contradicts the choice of ˙ Q ′ in V δ , which means that V δ is correct. Thus theiteration is the same as if it were defined over V .Now suppose that G ⊆ P is generic over V . Let’s see that V [ G ] | = LMP . Assumetoward a contradiction that it fails: namely ϕ ( ~a ) is a sentence with ~a ∈ H V [ G ] ω suchthat in V [ G ] it is a “local button”, i.e., ϕ H ω ( ~a ) is forceably necessary, and alsothat ϕ ( ~a ) H ω is not true in V [ G ]. Let us also take p ∈ G forcing the above to bethe case.Note that H V [ G ] δ = H V [ G ] ω , since δ is regular and P has the δ -cc, so the length ofthe iteration is collapsed to ω .Let ˙ Q be a name for Q , a least rank poset in V P and ˙ ~a be a name for ~a suchthat in V P , we have that “ ϕ ( ˙ ~a ) H ω is necessary.”Since P has the δ - cc , at no stage in the iteration could δ be collapsed. Thismeans that there is some stage where the parameters ˙ ~a appear. Thus we may finda stage in the iteration where the parameters ~a are available, past the support of p , say ~a ∈ V δ [ G α ].Now we let θ satisfy P ∈ V θ and ˙ Q ∈ V P θ . Then as δ is locally uplifting, we havethat there is an inaccessible γ > θ satisfying V γ [ G α ] | = “ ϕ ( ~a ) H ω is forceably necessary.”Namely, P tail ∗ ˙ Q makes ϕ ( ~a ) H ω necessary. So by the fact that δ is uplifting, wehave that V δ [ G α ] | = “ ϕ ( ~a ) H ω is forceably necessary.”Moreover, ϕ ( ~a ) H ω must continue to be a button in later stages, since it is a buttonin V [ G ]. So it is dense for the button to be pushed – ϕ ( ~a ) H ω is necessary in V δ [ G β ]for some β < δ . Thus ϕ ( ~a ) H ω is true in V [ G ], after the rest of the iteration; acontradiction. (cid:3) Theorem 5.13.
Let κ be a locally uplifting cardinal. Then there are forcing exten-sions in which we have the following:(1) LMP ccc + κ = c .(2) LMP p + κ = c = ℵ .(3) LMP c + κ = ℵ + CH . (4) LMP sc + κ = ℵ + CH .Proof. For (2)-(4), follow the same blueprint as (5.12), by defining a least-rank
LMP Γ lottery sum iteration of length δ . Use H ω instead of H ω and relativize tothe particular forcing class, modifying the support of the iteration as in Theorem4.7 for each forcing class. Again more should be said about (1), since ccc forcingnotions are not closed under lottery sums. We use the same adjustment as inTheorem 4.9. (cid:3) References [Bag00] Joan Bagaria. Bounded forcing axioms as principles of generic absoluteness.
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