Notes to The Resurrection Axioms

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1 Notes to The Resurrection Axioms Thomas Johnstone Talk in the Logic Workshop CUNY Graduate Center September 11, 009 Abstract I will discuss a new class of forcing axioms, the Resurrection Axioms (RA), and the Weak Resurrection Axioms (wra). While Cohen s method of forcing has been designed to change truths about the set-theoretic universe you live in, the point of Resurrection is that some truths that have been changed by forcing can in fact be resurrected, i.e. forced to hold again. In this talk, I will illustrate how RA is tied to forcing axioms such as MA and BPFA, and how it affects the size of the continuum. The main theorem will show that RA and a particular instance of wra are equiconsistent with the existence of an uplifting cardinal, a large cardinal notion consistent with V = L. This is joint work with Joel David Hamkins. 1 Notation For any infinite cardinal δ, we write H δ for the set of all sets hereditarily of size less than δ. If δ is the description of a particular such cardinal, such as the continuum c or the third infinite cardinal ω, then this concept relativizes. More specifically, if W is a model of set theory to a particular model of set theory, then we write Hc W to mean the collection of all those sets in W that are hereditarily of size less than c W in W. Correspondingly, we write Hω W to mean the collection of all those sets in W that have in W hereditary size less than ω W. We omit the exponent W if the model W equals the universe V. Motivation Recall the Bounded Proper Forcing Axiom (BPFA), introduced by Goldstern and Shelah in The axiom BPFA holds if for every proper poset Q, if g Q is a V -generic filter, then H ω Σ1 Hω V [g]. Justin Moore showed that BPFA implies that the continuum c = ℵ. It is easy to see that we cannot hope for more than Σ 1 -elementarity in the definition of BPFA. For if Q = Add(ω 1, 1) is the canonical poset to force CH, 1

2 using countable conditions, then Q is countably closed and therefore proper. Suppose that g Q is V -generic. Then c = ℵ in V, but c = ℵ 1 in V [g]. Consequently, Hω V [g] contains P(ω) as an element, while H ω thinks that P(ω) does not exist. This means that H ω and Hω V [g] disagree on a Σ -expressible statement. It is also easy to see that we cannot hope to replace ω by the continuum c and require H c Σ1 Hc V [g] in the definition of BPFA. For, if Q = Add(ω 1, 1) is the same poset as above, then ℵ V 1 is an element of H c, but ℵ V 1 is not an element of Hc V [g]. This shows that H c Hc V [g]. This motivates the idea of resurrection. We know that we cannot require that H c Σ1 Hc V [g]. Instead, we would like to require the existence of a poset R V [g] such that if h R is V [g]-generic, we then have full elementarity for the sets hereditarily of size less than the continuum, namely that H c Hc V [g][h]. We view R as a poset that resurrects the truths of H c. For example, given the terminology of the next definition, we shall say that wra(proper) holds if such a resurrecting poset R exists for every proper poset Q. Among other results, we shall show that wra(proper) is consistent relative to the existence of a Mahlo cardinal. 3 The Resurrection Axioms Definition The Resurrection Axiom RA is the assertion that for every forcing notion Q there is further forcing Ṙ such that whenever g h Q Ṙ is V -generic, then H c Hc V [g h].. More generally, for any class Γ of forcing notions, the Resurrection Axiom RA(Γ) asserts that for any Q Γ, there is Ṙ ΓV Q such that whenever g h Q Ṙ is V -generic, then H c Hc V [g h]. 3. The weak Resurrection Axiom wra(γ) is the assertion that for every Q Γ there is Ṙ such that if g h Q Ṙ is V -generic, then H c H V [g h] c. It is clear that RA implies wra(γ) for any class Γ of forcing notions. But in general, if RA is consistent, then RA does not imply RA(Γ). (As an example, consider the model V [G] of Theorem 17 in which RA + CH holds. We know by assertion of Theorem 6 that V [G] does not satisfy RA(ccc)). It is clear that RA(Γ) implies wra(γ). Suppose that Γ 1 is a subclass of Γ, a class of forcing notions. Then wra(γ ) implies wra(γ 1 ). But in general, if RA(Γ ) is consistent, then RA(Γ ) does not imply RA(Γ 1 ). (As an example, consider the model V [G] of Theorem 15, in which RA(proper) + c = ℵ holds. We again know by Theorem 6 that V [G] does not satisfy RA(ccc)).

3 4 Effect on Forcing Axioms and the Size of the Continuum Let s first investigate the effect of the Resurrection Axioms on forcing axioms such as Martin s Axiom MA or the Bounded Proper Forcing Axiom BPFA. Recall that for an infinite cardinal κ the axiom MA(κ) is the assertion that for every ccc poset Q and for every collection D of κ many dense subsets of Q, there is a D-generic filter on Q. Martin s Axiom MA is then the assertion MA(κ) for all infinite κ< c. It is a standard Skolem-Loewenheim argument that shows that MA(κ) is equivalent to only requiring the existence of the D-generic filter for ccc posets of hereditary size at most κ. Consequently, in order to establish MA, it suffices to show that for every ccc poset Q H c and for every collection D H c of dense subsets of Q, there is a D-generic filter on Q. Theorem. The weak Resurrection Axiom wra(ccc) implies MA. Proof. Assume wra(ccc). Fix any ccc poset Q H c and any collection D H c of dense subsets of Q. By wra(ccc), there is Ṙ such that if g h Q Ṙ is V -generic, then H c Hc V [g h]. Note that Q and D are elements of H c, and the filter g Q is an element of Hc V [g h]. Since g Q is fully V -generic, it is also D-generic, and so the structure Hc V [g h] has a D-generic filter on Q. Thus, by elementarity, there is such a filter in Hc V, and thus in V, as desired. Before we show the connection of wra(proper) to BPFA, let s first discuss some implications of the weak resurrection axioms on the size of the continuum. Theorem 3. Assume wra(γ). If some forcing Q Γ collapses a cardinal δ, then c δ. Proof. Fix some cardinal δ and a poset Q Γ such that Q collapses the cardinal δ. Suppose for contradiction that δ< c. By wra(γ) there is Ṙ such that if g h Q Ṙ is V -generic, then H c Hc V [g h]. But δ is a cardinal in the former structure and not in the latter structure, a contradiction. Corollary RA implies CH.. wra(proper) implies c ℵ. Proof. For assertion 1, consider the poset Q = Coll(ℵ 0, ℵ 1 ), the poset that uses finite conditions to collapse ℵ 1 to ℵ 0. The axiom RA allows for the poset Q, and thus implies by the previous theorem that c ℵ 1. For assertion, consider the poset Q = Coll(ℵ 1, ℵ ), the poset that uses countable conditions to collapse ℵ 1 to ℵ 0. The axiom wra(proper) allows for the poset Q, since Q is proper, and thus implies by the previous theorem that c ℵ. 3

4 Here is the theorem that connects wra(proper) to BPFA, the axiom that motivated us to introduce the Resurrection Axioms. Theorem 5. wra(proper)+ CH implies BPFA. Proof. Assume wra(proper) + CH. We want to verify BPFA, namely that for every proper poset Q, if g Q is V -generic, then H ω Σ1 Hω V [g]. Fix therefore any proper poset Q and suppose that g Q is V -generic. By wra(proper) we know that c ℵ, and thus by CH, we see that c = ℵ in V. By wra(proper), let Ṙ such that if h Ṙ is V [g]-generic, then H c Hc V [g h]. Since ℵ V 1 is the largest cardinal in H c, it is also the largest cardinal in Hc V [g h], and we see that c = ℵ in V [g h]. Consequently, we have that H ω Hω V [g h]. It is clear that ℵ V ℵ V [g] ℵ V [g h] and so H ω Hω V [g] Hω V [g h]. Since a Σ 1 -expressible statement that holds in Hω V [g] is absolute upwards to Hω V [g h], it also holds in H ω by elementarity. This shows that H ω Σ1 Hω V [g], as desired. Note that it is necessary in the theorem above to include the additional assumption that CH fails. For we shall find in Theorem 18 a model of set theory that satisfies RA and thus CH. It is clear that this model therefore satisfies wra(proper), but it cannot satisfy BPFA, as BPFA implies that c = ℵ. Let me state without proof a few more of our results that are concerned with the effect of the Resurrection Axioms on the size of the continuum. Theorem wra(ℵ 1 -preserving) implies CH.. RA(ccc) implies that c is a weakly inaccessible cardinal. 3. RA(proper) is relatively consistent with CH. We have found several interesting consequences of the Resurrection Axioms, but the crucial question that we haven t addressed yet is of course. Question 1. Are any of these resurrection axioms consistent? If so, what is their large cardinal strength? Can we find the exact large cardinal strength? The next two sections will answer this question positively, as long as we allow ourselves the existence of a Mahlo cardinal. 5 The Large Cardinal Strength of the Resurrection Axioms Definition 7. An inaccessible cardinal κ is an uplifting cardinal if V κ V γ for arbitrarily large inaccessible cardinals γ. Uplifting cardinals are downward absolute to L, and are consistent relative to a Mahlo cardinal: 4

5 Theorem If κ is uplifting, then κ is uplifting in L.. If κ is a Mahlo cardinal, then V κ has a proper class of uplifting cardinals Proof. For assertion 1, fix any uplifting cardinal κ in V. The cardinal κ is clearly inaccessible in L. Suppose that γ is any inaccessible cardinal with V κ V γ. Then γ is inaccessible in L and, by relativizing formulas to the constructible sets, we see that Vκ L =(V κ L) (V γ L) =Vγ L, as desired. For assertion, suppose that κ is a Mahlo cardinal. Since κ is inaccessible, it follows from the Loewenheim-Skolem theorem, that there is a closed unbounded set C κof cardinals γ with V γ V κ. If γ 1 <γ are cardinals that are both in C, then V γ1 V γ. But since κ is Mahlo, we know that the club C contains unboundedly many inaccessible cardinals. It thus follows that every inaccessible cardinal γ C is uplifting in V κ, as desired. Uplifting cardinals can be equivalently characterized as follows. Theorem 9. The following are equivalent. 1. κ is an uplifting cardinal.. κ is regular and H κ H γ for arbitrarily large regular cardinals γ. Proof. The implication 1 is trivial. To prove that that 1, assume that κ is regular and H κ H γ for arbitrarily large regular cardinals γ. To see that κ is inaccessible, it suffices to show that κ is an uncountable strong limit cardinal. Fix thus any cardinal α < κ. By assumption, we may find a regular uncountable cardinal γ above P(α) such that H κ H γ. The set P(α) exists in H γ, and it must thus by elementarity exist in H κ also. This shows that α <κ and thus that κ is a strong limit cardinal. Since H γ contains an infinite set, it follows by elementarity that κ > ω, and thus that κ is inaccessible. To see that V κ V γ for arbitrarily large inaccessible cardinals, fix any regular γ such that H κ H γ. By assumption, we know that there are arbitrarily large such cardinals γ. Since κ is inaccessible, it follows that for all cardinals α < κ the cardinal α exists in H κ = V κ. Using the elementarity H κ H γ it follows that for all α < γ the cardinal α exists in H γ. The cardinal γ is thus a strong limit cardinal and therefore inaccessible also. This implies that H γ = V γ and V κ V γ as desired. Theorem 10. RA implies that c = ℵ 1 is uplifting in L. Proof. Assume the Resurrection Axiom RA. We saw in Corollary 4 that this implies that the CH holds. Let κ = c = ℵ 1. The cardinal κ is regular in L. To see that κ is uplifting in L, we will verify characterization of Theorem 9. Fix thus any cardinal α > κ. The idea now is to use RA and find a suitable forcing extension of V in which the size of the continuum is above α. So let Q be any forcing that collapses the cardinal α to ℵ 0. By RA there is Ṙ such 5

6 that if g h Q Ṙ is V -generic, then H c Hc V [g h]. Since every object in the former structure in countable there, this is also true by elementarity in the latter structure, and so c = ℵ 1 in V [g h] also. As α is countable in V [g] it follows that c V [g h] >α. Let γ = c V [g h] = ℵ V [g h] 1. So γ is a regular cardinal in V [g h] with κ < α < γ and Hκ V Hγ V [g h]. It is clear that γ is regular in L, and by relativizing formulas to the constructible sets, we see that Hκ L =(Hκ V L) (Hγ V [g h] L) =Hγ L, as desired. Following the idea of the proof of Theorem 10 closely, but by replacing ℵ 1 by ℵ when appropriate, we get the following. Theorem 11. wra(proper)+ CH implies that c = ℵ is uplifting in L. Proof. Assume that wra(proper) + CH. We saw in Corollary 4 that this implies that c = ℵ. Let κ = c = ℵ. The cardinal κ is regular in L. To see that κ is uplifting in L, we will verify characterization of Theorem 9. Fix thus any cardinal α > κ. The idea now is to use wra(proper) and find a suitable forcing extension of V in which the size of the continuum is above α. So let Q be the forcing that collapses the cardinal α to ℵ 1 using countable conditions. The poset Q is proper. By wra(proper) there is Ṙ such that if g h Q Ṙ is V -generic, then H c Hc V [g h]. Since ℵ V 1 is the largest cardinal in the former structure, this is also true by elementarity in the latter structure, and so c = ℵ in V [g h] also. As α is an ordinal of size ℵ 1 in V [g] and thus in V [g h], it follows that c V [g h] >α. Let γ = c V [g h] = ℵ V [g h]. So γ is a regular cardinal in V [g h] with κ < α < γ and Hκ V Hγ V [g h]. As before, it is clear that γ is regular in L, and by relativizing formulas to the constructible sets, we see that Hκ L =(Hκ V L) (Hγ V [g h] L) =Hγ L, as desired. Note that it is necessary in the theorem above to include the additional assumption that CH fails. For we shall find in the Theorem 17 a model of set theory that satisfies RA and thus CH. 6 The Relative Consistency of the Resurrection Axioms In this section we first show how one can use an uplifting cardinal κ to produce a model of set theory which satisfies RA(proper) + c = ℵ. Second, we will also show how to use an uplifting cardinal κ to produce a model which satisfies RA. But before we can provide the desired relative consistency theorems, we need one more definition and two crucial lemmas which I state without proof. Definition 1. Let κ be an uplifting cardinal. We say that a function f.κ κ has the Menas property for κ, if for every ordinal β there are arbitrarily large inaccessible cardinals γ with V κ,f V γ,f for some function f. γ γ having f (κ) β. 6

7 Lemma 13. If κ is uplifting in V, then there is a model W = ZFC such that W = κ is uplifting and κ there exists a Menas function f. κ κ Lemma 14. Suppose that M, N are transitive set models of ZFC, that A is a subset of M, and that M, A N, A for some subset A N. Suppose that P M is a class forcing iteration that is definable in M, A, with P being the analogously defined class forcing iteration in N, A. Suppose also that G P is M-generic and G P is N-generic, with G = G P. Then M[G] N[G ]. Here is the rough idea that underlies Theorems 15 and 17. We will use an uplifting cardinal κ and a function f. κ κ with the Menas property to define a κ-iteration P V κ. If G P is V -generic, we will aim to show that V [G] satisfies the particular resurrection axiom we are interested in. Since f has the Menas property for κ, we may find some large enough inaccessible γ such that V κ,f V γ,f for some function f. γ γ. The lifting lemma above will allow us to lift this elementarity to V κ [G] V γ [G ], where G = G g h is V -generic for the poset P V γ, the iteration corresponding to P as defined in V γ from f. Since κ is inaccessible in V and the poset P happens to be κ-cc, it will then follow from c = κ in V [G] that H V [G] c Hc V [G][g h], as desired. Theorem 15. Suppose that κ is uplifting and that f. κ κ has the Menas property for κ. If P is the PFA lottery preparation of κ relative to f, and G P is V -generic, then V [G] = RA(proper)+c = ℵ. Proof. Fix the uplifting cardinal κ and the corresponding function f. κ κ. Let P be the PFA lottery preparation of κ relative to f. That is, P is a countable support κ-iteration of proper forcing, where the forcing at stage β dom(f) is the lottery sum of all proper posets in H V [G β] f(β). Suppose that G P is V -generic. + My first claim is that V [G] satisfies κ = c = ℵ. Since P is a countable iteration of proper posets, it follows that P is proper and consequently that ℵ 1 is preserved as a cardinal. If α is a cardinal with ℵ 1 < α < κ, then it is dense that the generic filter opts to collapse α to ℵ 1. So α is an ordinal of size ℵ 1 in V [G]. The cardinal κ is preserved since P is κ-cc, and thus κ = ℵ in V [G]. Moreover, since the generic filter opts unboundedly often in κ to add a Cohen real, it follows that c κ in V [G]. Lastly, as P is κ-cc, we may count the nice names for subsets of ω to see that c κ in V [G]. In summary, we have that V [G] satisfies κ = c = ℵ, which proves the first claim. My second claim is that V [G] satisfies RA(proper). To prove this claim, fix any proper poset Q V [G]. Let Q be a name for Q that necessarily yields a proper poset. Since κ is uplifting and f has the Menas property for κ, there is an inaccessible cardinal γ such that V κ,f V γ,f for some function f. γ γ having f (κ) trcl( Q). Let P be the corresponding iteration, as defined in V γ from f. Notice that P and P agree on the stages below κ. Also, since Q is proper in V γ [G] and f (κ) is sufficiently large, it follows that the poset Q appears in the stage κ lottery of P. Below a condition opting for Q at stage κ, we may thus factor P as P Q Ṗtail. Force to add a V [G]- generic filter g h Q Ṗtail. It follows that G g h generates a V -generic 7

8 filter G P. Using the same arguments as in the first claim, but now for the corresponding iteration P, we see that γ = c = ℵ in V [G ]. Since G and G agree on the first κ many stages, it follows by Lemma 14 that V κ V γ lifts to V κ [G] V γ [G ]. As κ is inaccessible in V we have that V κ = H κ, and consequently that V κ [G] =H κ [G]. Since P is κ-cc, we may use nice names for bounded subsets of κ to see that H κ [G] =Hκ V [G]. Since κ = c in V [G], it follows in summary that V κ [G] =Hc V [G]. Arguing correspondingly for the cardinal γ, we also see that V γ [G] =H V [G ] c, which clearly equals Hc V [G][g h]. In other words, we have the desired elementarity Hc V [G] Hc V [G][g h]. Lasty, note that P tail is a countable support iteration of proper posets, and thus itself a proper poset in V [G][g]. This shows that V [G] satisfies RA(proper), which proves the second claim. In the theorem above, we had the κ-iteration P which forced at stage β with the lottery sum of all proper posets in H V [G β] f(β). This lottery sum was rich enough + to anticipate any given proper poset Q V [G]. But if we want to produce a model V [G] that satisfies RA, we must be much more generous at each stage β lottery, since we have to anticipate any given poset Q. This motivates the following definition. Definition 16. Suppose that κ is an inaccessible cardinal and f. κ κ is a partial function. We say that P is the exhaustive lottery preparation of κ relative to f, if P is a finite support iteration of length κ, where the forcing at stage β dom(f) is the lottery sum of all posets in H V [G β] f(β) +. Theorem 17. Suppose that κ is uplifting and that f. κ κ has the Menas property for κ. If P is the exhaustive lottery preparation, and G P is V - generic, then V [G] = RA + c = ℵ 1. Proof. The proof is quite similar to the proof of Theorem 15. Fix the uplifting cardinal κ and the corresponding function f. κ κ. Let P be the exhaustive lottery preparation of κ relative to f, as specified in Definition 16. Note that for each stage β < κ, the lottery sum at stage β is κ-cc, and since P is a finite support iteration it follows that P is κ-cc. Suppose that G P is V -generic. My first claim is that V [G] satisfies κ = c = ℵ 1. If α < κ is an uncountable cardinal, then it is dense that the generic filter opts to collpase α to ℵ 0. So α is a countable ordinal in V [G]. The cardinal κ is preserved since P is κ-cc, and thus κ = ℵ 1 in V [G]. As in the proof of Theorem 15 we also see that V [G] satisfies c = κ, which proves the first claim. My second claim is that V [G] satisfies RA. To prove this claim, fix any poset Q V [G], and let Q be any name for Q. Since κ is uplifting and f has the Menas property for κ, there is an inaccessible cardinal γ such that V κ,f V γ,f for some function f. γ γ having f (κ) trcl( Q). Let P be the corresponding iteration, as defined in V γ from f. As in the proof of Theorem 15, it follows that the poset Q appears in the stage κ lottery of P, and we may thus factor P as P Q Ṗtail below a condition opting for Q at stage κ. Again, we force to 8

9 add a V [G]-generic filter g h Q Ṗtail, so that G g h generates a V -generic filter G P. Using the same arguments as in the first claim, we see that γ = c = ℵ 1 in V [G ]. Again, we use Lemma 14 to see that V κ V γ lifts to V κ [G] V γ [G ]. Again, this implies that Hc V [G] Hc V [G][g h]. This shows that V [G] satisfies RA, which proves the second claim. 7 The Equiconsistency Result We may sum up the results of the previous two sections to get the following equiconsistency result. Theorem 18. The following are equiconsistent. 1. The Resurrection Axiom RA.. RA(proper)+ CH. 3. wra(proper)+ CH. 4. There is an uplifting cardinal. Proof. Statements 1 and 4 are equiconsistent by Theorem 10, Lemma 13, and Theorem 17. Statement implies statement 3 directly. It follows that statements, 3, and 4 are equiconsistent by Theorem 11, Lemma 13, and Theorem 15. 9