aa r X i v : . [ m a t h . L O ] A p r A NOTE ON L ¨OWENHEIM-SKOLEM CARDINALS
TOSHIMICHI USUBA
Abstract.
In this note we provide some applications of L¨owenheim-Skolem cardinals introduced in [3]. Introduction
Throughout this note, our base theory is ZF unless otherwise specified. InUsuba [3], we introduced the notion of L¨owenheim-Skolem cardinal , whichcorresponds to the downward L¨owenheim-Skolem theorem in the context of
ZFC . Definition 1.1.
Let κ be an uncountable cardinal.(1) κ is weakly L¨owenheim-Skolem (weakly LS, for short) if for every γ < κ , α ≥ κ , and x ∈ V α , there is X ≺ V α such that V γ ⊆ X , x ∈ X , and the transitive collapse of X belongs to V κ .(2) κ is L¨owenheim-Skolem (LS, for short) if for every γ < κ , α ≥ κ ,and x ∈ V α , there is β ≥ α and X ≺ V β such that V γ ⊆ X , x ∈ X , V γ ( X ∩ V α ) ⊆ X , and the transitive collapse of X belongs to V κ .Note that if κ is a limit of LS (weakly LS, respectively) cardinals, then κ is LS (weakly LS, respectively). In [3], we proved that some non-trivialconsequences of the Axiom of Choice AC holds on the successor of a singularLS cardinal: Theorem 1.2.
Suppose κ is a singular weakly LS cardinal (e.g., a singularlimit of weakly LS cardinals).(1) There is no cofinal map from V κ into κ + , hence κ + is regular.(2) For every function f from V κ into the club filter over κ + , the in-tersection T f “ V κ contains a club in κ + . Hence the club filter is κ + -complete.(3) For every regressive function f : κ + → κ + , there is α < κ + suchthat the set { η < κ + | f ( η ) = α } is stationary. Date : April 6, 2020.2010
Mathematics Subject Classification.
Primary 03E10, 03E25.
Key words and phrases.
Axiom of Choice, L¨owenheim-Skolem cardinal. ee [3] for more information and applications to choiceless set-theoreticgeology.In this note, we provide related results and other applications of LS cardi-nals, and we show that various consequences of AC follows from LS cardinals.We will use the following notion in this note. For a set x , let k x k be theleast ordinal α such that there is a surjection from V α onto x . Note that: • k V α k = α for every infinite ordinal α . • If x ⊆ y , or there is a surjection from y onto x , then k x k ≤ k y k . • For every ordinal α , there is a cardinal κ with k κ k > α . • If κ is weakly LS, then for every γ < κ there is no surjection from V γ onto κ ([3]), so k κ k = κ . • Suppose κ = k κ k . Let α ≥ κ and X ≺ V α be with k X k < κ .Then the transitive collapse of X is in V κ ; Let X be the transitivecollapse of X , and δ the rank of X . There is a surjection from X onto δ , so k δ k ≤ (cid:13)(cid:13) X (cid:13)(cid:13) . Since the collapsing map is a bijection from X onto X , we have k δ k ≤ (cid:13)(cid:13) X (cid:13)(cid:13) = k X k < κ . If δ ≥ κ , then we know k δ k ≥ k κ k = κ , but this is impossible and we obtain δ < κ .Using this notion, we can reformulate LS and weakly LS cardinals as follows: • κ is weakly LS if and only if k κ k = κ , and for every α ≥ κ andset A ⊆ V α with k A k < κ , there is X ≺ V α such that A ⊆ X and k X k < κ . • κ is LS if and only if k κ k = κ , and for every γ < κ , α ≥ κ and set A ⊆ V α with k A k < κ , there is β ≥ α and X ≺ V β such that A ⊆ X , k X k < κ , and for every y ⊆ X ∩ V α , if k y k ≤ γ then y ∈ X .2. On elementary submodels
First let us present somewhat trivial characterization.
Proposition 2.1.
Let κ be an uncountable cardinal. Then the following areequivalent:(1) κ is weakly LS.(2) k κ k = κ and for every first order structure M = h M ; . . . i withcountable language and A ⊆ M with k A k < κ , there is an elementarysubmodel N = h N ; . . . i ≺ M such that A ⊆ N and k N k < κ .Proof. (2) ⇒ (1). Take γ < κ , α ≥ κ , and x ∈ V α . Identifying x as a con-stant, consider the structure h V α ; ∈ , x i . By (2), we can find an elementarysubmodel X ≺ V α such that x ∈ X , V γ ⊆ X , and k X k < κ . If Y is thetransitive collapse of X , then Y must be in V κ .(1) ⇒ (2). k κ k = κ is already noted above. Fix a structure M = h M ; . . . i and A ⊆ M such that there is a surjection f from some x ∈ V κ onto A . ake a large α > κ with f, M ∈ V α , and take also a large γ < κ with x ∈ V γ . By (1), we can find X ≺ V α such that f, M ∈ V α , V γ ⊆ X ,and the transitive collapse of X is in V κ . Since x ∈ V γ ⊆ X , we have A = f “ x ⊆ X . Let N = h M ∩ X ; . . . i . Since A ⊆ X ≺ V α , it is easy to seethat N ≺ M and A ⊆ M ∩ X . If π : X → Y is the collapsing map, then π ↾ M ∩ X : M ∩ X → π ( M ) is a bijection and π ( M ) ∈ V κ . Then the inversemap gives a surjection from π ( M ) onto M ∩ X . Hence k M ∩ X k < κ . (cid:3) On successors of weakly LS cardinals In ZFC , for every cardinal κ and α ≥ κ + , we can find an elementarysubmodel X ≺ V α with | X | = κ ⊆ X and X ∩ κ + ∈ κ + . We can prove asimilar result for singular weakly LS cardinals. Lemma 3.1.
Let κ be a singular weakly LS cardinal. Then for every α ≥ κ + and x ∈ V α , there is X ≺ V α such that x ∈ X , V κ ⊆ X , X ∩ κ + ∈ κ + , and k X k = κ .Proof. Fix a large β > α , and take Y ≺ V β such that α, κ, x ∈ Y , cf( κ ) ⊆ Y ,and the transitive collapse of Y belongs to V κ .Let F = { Z ≺ V α | x ∈ Z and the transitive collapse of Z is in V κ } . Wehave F ∈ Y . Claim 3.2. (1)
F ∩ Y is upward directed, that is, for every Z , Z ∈F ∩ Y , there is Z ∈ F ∩ Y with Z ∪ Z ⊆ Z .(2) For every γ < κ , there is Z ∈ F ∩ Y with V γ ⊆ Z .Proof of Claim. (1) Take a large γ < κ such that the transitive collapses of Z and Z are in V γ , and take surjections f : V γ → Z and f : V γ → Z .Since Z , Z ∈ Y ≺ V β , we may assume that γ, f , f ∈ Y . In Y , we canchoose Z ′ ≺ V α + ω such that V γ ⊆ Z ′ , x, α, f , f ∈ Z ′ , and the transitivecollapse of Z ′ is in V κ . Let Z = Z ′ ∩ V α ∈ Y . We know Z ≺ V α , so Z ∈ F ∩ Y . Since V γ ⊆ Z , we also have Z ∪ Z ⊆ Z .(2) Since cf( κ ) ⊆ Y , we have that Y ∩ κ is cofinal in κ . Hence for a given γ < κ , there is δ ∈ Y ∩ κ with γ ≤ δ . Then in Y we can choose Z ∈ F with V δ ⊆ Z . (cid:3) Let X = S ( F ∩ Y ). By the claim above, we have that x ∈ X ≺ V α and V κ ⊆ X . Claim 3.3.
There is a surjection from V κ onto X .Proof of Claim. Let Y be the transitive collapse of Y , and π : Y → Y thecollapsing map. For each Z ∈ F ∩ Y , let Z be the transitive collapse of Z ,and π z be the collapsing map. efine f : V κ × V κ → X as follows: For h a, b i ∈ V κ × V κ , if π − ( a ) ∈F ∩ Y and b ∈ Z (where Z = π − ( a )), then f ( a, b ) = π − Z ( b ). Otherwise, f ( a, b ) = ∅ . This f is an surjection; Take c ∈ X . Then c ∈ Z for some Z ∈ F ∩ Y , and f ( π ( Z ) , π Z ( c )) = c . We can easily take a surjection from V κ onto V κ × V κ , hence we obtain a surjection from V κ onto X . (cid:3) Finally, since there is no cofinal map from V κ into κ + by Theorem 1.2,we have that sup( X ∩ κ + ) < κ + . In addition, since κ ⊆ X , we havesup( X ∩ κ + ) ⊆ X and sup( X ∩ κ + ) = X ∩ κ + ∈ κ + . (cid:3) Let κ be a singular weakly LS cardinal. While we already knew thatthe club filter over κ + is κ + -complete, we do not know if it is normal.Among this, we can construct a normal filter over κ + which is definablewith parameter κ + . Note 3.4.
Let κ be a cardinal, and F a filter over κ . Then the followingare equivalent:(1) For every X α ∈ F ( α < κ ), the diagonal intersection △ α<κ X α = { β < κ | β ∈ X α for all α < β } is in F .(2) For every X ∈ F + and regressive function f : X → κ , there is α < κ with { β ∈ X | f ( β ) = α } ∈ F + .Where F + = { X ∈ P ( κ ) | X ∩ C = ∅ for every C ∈ F } . An element of F + is an F -positive set . We say that a filter F is normal if F is proper,contains all co-bounded subsets of κ , and satisfies the above conditions (1)and/or (2). Proposition 3.5.
Let κ be a singular weakly LS cardinal. Let F ⊆ P ( κ + ) be the set such that: D ∈ F ⇐⇒ there is α ≥ κ + and x ∈ V α such that D contains the set { η < κ + | there is X ≺ V α with x ∈ X and η = X ∩ κ + } .Then F is a normal filter over κ + .Proof. By Lemma 3.1, we have that ∅ / ∈ F and κ \ η ∈ F for every η < κ .One can check that F is a filter over κ . For the normality, take a family { D γ ∈ F | γ < κ + } , and let D = △ γ<κ + D γ . Suppose to the contrarythat D / ∈ F . Now fix a large β > κ + such that for every γ < κ + , there is α < β and x ∈ V α such that { η < κ + | there is X ≺ V α with x ∈ X and η = X ∩ κ + } ⊆ D γ . Since D / ∈ F , we can find Y ≺ V β such that Y containsall relevant objects, Y ∩ κ + ∈ κ + , and Y ∩ κ + / ∈ D . Let δ = Y ∩ κ + . Wesee that δ ∈ D γ for every γ < δ , this is a contradiction.Take γ < δ . Then γ ∈ Y . Hence we can find α ∈ Y and x ∈ Y ∩ V α such that { η < κ + | there is X ≺ V α with x ∈ X and η = X ∩ κ + } ⊆ D γ .However, since α, x ∈ Y , we have x ∈ Y ∩ V α ≺ V α and ( Y ∩ V α ) ∩ κ + = δ ,hence δ ∈ D γ . (cid:3) or a non-empty set S , let Col( S ) be the poset of all finite partial func-tions from ω to S with the reverse inclusion order. The forcing with Col( S )adds a surjection from ω onto S . Proposition 3.6.
Let κ be a singular weakly LS cardinal. Then Col( V κ ) forces ( κ + ) V = ω and the Dependent Choice DC .Proof. For the equality ( κ + ) V = ω , take p ∈ Col( V κ ) and a name ˙ f fora function from κ to κ + . Define F : Col( V κ ) × κ → κ + as follows: For h q, α i ∈ Col( V κ ) × κ , if q ≤ p and q (cid:13) ˙ f ( α ) = η ” for η < κ + , then F ( q, α ) = η . Otherwise, let F ( q, α ) = 0. By Theorem 1.2, F is not acofinal map, hence γ = sup( F “(Col( V κ ) × κ )) < κ + . Then it is clear that p (cid:13) “ ˙ f “ κ ⊆ γ ”, so p (cid:13) “( κ + ) V is regular”.To show that Col( V κ ) forces DC , take p ∈ Col( V κ ), Col( V κ )-names ˙ S and˙ R such that p (cid:13) “ ˙ S is non-empty, ˙ R ⊆ ˙ S , and for every x ∈ ˙ S there is y ∈ ˙ S with h x, y i ∈ ˙ R ”.Fix a large limit α > κ + with ˙ S, ˙ R ∈ V α . By Lemma 3.1, we can find X ≺ V α such that V κ ⊆ X , X contains all relevant objects, X ∩ κ + ∈ κ + ,and k X k = κ . Note that Col( V κ ) ⊆ X .Take a ( V, Col( V κ ))-generic G , and let X [ G ] = { ˙ x G | ˙ x ∈ X is a Col( V κ )-name } (where ˙ x G is the interpretation of ˙ x by G ). Since Col( V κ ) ⊆ X , wemay assume X [ G ] ≺ V α [ G ] = V [ G ] α . There is a canonical surjection from X onto X [ G ], namely ˙ x ˙ x G , hence we can take a surjection from V κ onto X [ G ]. Since V κ is countable in V [ G ], we have that X [ G ] is countable aswell. Let S = ˙ S G and R = ˙ R G . We know S, R ∈ X [ G ]. By the elementarityof X [ G ], for every x ∈ S ∩ X [ G ], there is y ∈ S ∩ X [ G ] with h x, y i ∈ R .Now X [ G ] is well-orderable, thus we can take a map f : ω → S ∩ X [ G ] suchthat h f ( n ) , f ( n + 1) i ∈ R for every n < ω . (cid:3) When κ is regular weakly LS, we can obtain a similar result to Lemma3.1. Note that in ZFC , κ is regular weakly LS if and only if κ is inaccessible. Lemma 3.7.
Let κ be a regular weakly LS cardinal. Then for every γ < κ , α > κ , and x ∈ V α , there is X ≺ V α such that x ∈ X , γ < X ∩ κ ∈ κ , V X ∩ κ ⊆ X , and the transitive collapse of X is in V κ .Proof. Similar to Lemma 3.1. Take a large β > α and Y ≺ X β such that Y contains all relevant objects, and the transitive collapse of Y is in V κ . Weknow sup( Y ∩ κ ) < κ ; If sup( Y ∩ κ ) = κ , then ot( Y ∩ κ ) = κ since κ isregular. However then the transitive collapse of Y cannot be in V κ .Let F = { Z ≺ V α | x ∈ Z , the transitive collapse of Z is in V κ } , and X = S ( F ∩ Y ). We have X ≺ V α . For every Z ∈ F ∩ Y , we have that Z ∩ κ ≤ sup( Y ∩ κ ), and for every δ ∈ Y ∩ κ , there is Z ∈ F ∩ Y with V δ ⊆ Z . Hence X ∩ κ = sup( Y ∩ κ ) and V X ∩ κ ⊆ X . ake a large δ < κ such that the transitive collapse of Y is in V δ . Notethat sup( Y ∩ κ ) ≤ δ . For every Z ∈ F ∩ Y , the transitive collapse of Z isalso in V δ . Then as in Lemma 3.1 we can define a surjection from V δ onto X . Therefore k X k ≤ δ < κ , and the transitive collapse of X is in V κ . (cid:3) On successors of LS cardinals If κ is a singular LS cardinal and there is a weakly LS cardinal ≤ cf( κ ),we can take a small elementary submodel with certain closure property. Weuse the following fact: Theorem 4.1 ([3]) . Let κ be a weakly LS cardinal. Then for every cardinal λ ≥ κ and x ∈ V κ , there is no cofinal map from x into λ + . Lemma 4.2.
Let α be a limit ordinal α and a set x , if cf( α ) > k x k andthere is a weakly LS cardinal κ with cf( α ) ≥ κ ≥ k x k , then there is nocofinal map from x into α .Proof. First suppose k x k < κ < cf( α ). In this case we may assume x ∈ V κ .If cf( α ) is a successor cardinal, then we have done by Theorem 4.1. Supposecf( α ) is a limit cardinal. Then cf( α ) > κ + , so { η < α | cf( η ) = cf( κ + ) } isstationary in α . If f : x → α is a cofinal map, then C = { η < α | η ∩ f “ x iscofinal in η } is a club in α . Hence we can find η < α such that η ∩ f “ x iscofinal in η and cf( η ) = cf( κ + ). However then there is a cofinal map from x into κ + , This is a contradiction.Next suppose k x k = κ < cf( α ). Then we may assume x = V κ . Take f : V κ → α . By the previous case, for every γ < κ , we have sup( f “ V γ ) < α .Since cf( α ) > κ , we have sup( f “ V κ ) = sup γ<κ sup( f “ V γ ) < α .Finally suppose k x k < κ = cf( α ). Then κ is a regular weakly LS cardinal.If there is a cofinal map from x into α , we can find a cofinal map f from x into κ . Take an elementary submodel X ≺ V α + ω such that V k x k ⊆ X , X contains all relevant objects, and the transitive collapse of X is in V κ .We know f “ x ⊆ X , hence sup( X ∩ κ ) = κ . Then ot( X ∩ κ ) = κ because κ is regular, so the transitive collapse of X cannot be in V κ . This is acontradiction. (cid:3) Lemma 4.3.
Let κ be a singular LS cardinal and ν ≤ cf( κ ) a weakly LScardinal. Then for every α > κ and x ∈ V α , there is β > α and X ≺ V β such that V κ ⊆ X , x ∈ X , X ∩ κ + ∈ κ + , V γ ( X ∩ V α ) ⊆ X for every γ < ν ,and k X k = κ .Proof. Take a large β and Y ≺ V β such that α, x, . . . ∈ Y , V ν ( Y ∩ V α + ω + ω ) ⊆ Y , cf( κ ) ⊆ Y , and the transitive collapse of Y is in V κ . Let F = { Z ≺ V α + ω | x ∈ Z , the transitive collapse of Z is in V κ } . Let X = S ( F ∩ Y ). As in the roof of Lemma 3.1, we have X ≺ V α + ω , x ∈ X , V κ ⊆ X , X ∩ κ + ∈ κ + , and k X k = κ .We have to see that V γ ( X ∩ V α ) ⊆ X for every γ < ν . Take f : V γ → X ∩ V α . Since ν is weakly LS, we can find Y ′ ≺ V β such that Y contains allrelevant objects, V γ ⊆ Y ′ , and the transitive collapse of Y ′ is in V ν . We have f ⊆ Y ′ . Let F ′ = F ∩ Y ∩ Y ′ ⊆ Y ∩ V α + ω + ω . Since V ν ( Y ∩ V α + ω + ω ) ⊆ Y and kF ′ k ≤ k Y k < ν , we have F ′ ∈ Y . Note that k Z k < κ for every Z ∈ F ′ .Since kF ′ k < ν ≤ cf( κ ), there is no cofinal map from F ′ into κ by Lemma4.2. Thus we have δ = sup {k Z k | Z ∈ F ′ } < κ . Then, as in the proof ofLemma 3.1, we can define a surjection from V δ onto S F ′ . In Y , we can find Z ′ ∈ F ∩ Y such that P ( V α ∩ S F ′ ) ⊆ Z ′ . Since f ∈ Y ′ , for each a ∈ V γ there is Z ∈ ( F ∩ Y ) ∩ Y ′ with f ( a ) ∈ Z . Hence f “ V γ ⊆ V α ∩ S F ′ , and f “ V γ ∈ Z ′ ⊆ X . (cid:3) Note 4.4.
In Lemma 4.5, if ν < cf( κ ) is a singular weakly LS cardinal,then we can require that V ν ( X ∩ V α ) ⊆ X .If κ is regular LS cardinal, we have the following parallel result: Lemma 4.5.
Let κ be an LS cardinal and ν < κ a weakly LS cardinal.Then for every α > κ , x ∈ V α , and γ < κ , there is β ≥ α and X ≺ V β suchthat x ∈ X , γ < X ∩ κ ∈ κ , V X ∩ κ ⊆ X , V δ ( X ∩ V α ) ⊆ X for every δ < ν ,and the transitive collapse of X is in V κ . For an ordinal γ , let DC γ be the assertion that for every non-empty set S and G : <γ S → P ( S ) \ {∅} , there is f : γ → S such that f ( α ) ∈ G ( f ↾ α )for every α < γ . Let DC <γ be the assertion that DC δ holds for every δ < γ . DC ω is the Dependent Choice, and it is known that AC is equivalent tothat DC γ for every γ . Note also that the following:(1) DC γ ⇒ DC γ +1 .(2) If γ is a singular ordinal and DC <γ holds, then DC γ holds as well.(3) DC γ ⇒ DC <γ + .Woodin [4] proved that collapsing a supercompact cardinal yields DC κ forsome large κ . Where κ is supercompact if for every α > κ , there is β ≥ α , atransitive set N with V α N ⊆ N , and an elementary embedding j : V β → N with critical point κ and α < j ( κ ). It is known that a supercompact cardinalis a limit of LS cardinals ([3]).We prove a similar result using LS cardinals. Definition 4.6.
Let κ be a regular cardinal and S a non-empty set. LetCol( κ, S ) be the poset of all partial functions p from κ to S with | p | < κ (so p is assumed to be well-orderable). The ordering of Col( κ, S ) is the reverseinclusion. f G is ( V, Col( κ, S ))-generic, then S G is a surjection from κ onto S . Proposition 4.7.
Let κ be a regular uncountable cardinal, and suppose DC <κ holds. Let λ > κ be a singular LS cardinal such that cf( λ ) > κ andthere is a weakly LS cardinal ν with κ ≤ ν ≤ cf( λ ) . Then Col( κ, V λ ) doesnot add new < κ -sequences, and forces ( λ + ) V = ( κ + ) V Col( κ,Vγ ) and DC κ .Proof. By DC <κ , we know that Col( κ, V λ ) is κ -closed and does not add new < κ -sequences. We can check that Col( κ, V λ ) preserves DC <κ as well.To see Col( κ, V λ ) forces ( λ + ) V = ( κ + ) V Col( κ,Vλ ) , it is clear that ( λ + ) V ≤ ( κ + ) V Col( κ,Vλ ) . For the converse, take p ∈ Col( κ, V λ ) and a name ˙ f for afunction from λ to ( λ + ) V . Define F : λ × Col( κ, V λ ) → λ + by F ( δ, q ) = η ⇐⇒ q ≤ p and q (cid:13) “ ˙ f ( δ ) = η ”. F is not cofinal by Theorem 1.2,hence the image of F has an upper bound in λ + , say η . It is clear that p (cid:13) “ ˙ f “ λ ⊆ η ”, so (cid:13) “( λ + ) V is regular”.For DC κ , take p ∈ Col( κ, V λ ) and names ˙ S and ˙ F with p (cid:13) “ ˙ F : <κ ˙ S →P ( ˙ S ) \ {∅} ”. Fix a sufficiently large α > λ + . By Lemma 4.5, we can find β > α and X ≺ V β such that X contains all relevant objects, V λ ⊆ X , X ∩ λ + ∈ λ + , V γ ( X ∩ V α + ω ) ⊆ X for every γ < ν , and k X k = λ . Take a( V, Col( κ, V λ ))-generic G and work in V [ G ]. We may assume X [ G ] ≺ V [ G ] β .We see that X [ G ] ∩ V [ G ] α is closed under < κ -sequences in V [ G ]. Take η < κ and f : η → X [ G ] ∩ V [ G ] α . By DC <κ in V [ G ], we can find a sequence ofCol( κ, V λ )-names h ˙ x i | i < η i such that ˙ x i ∈ X and f ( i ) = ( ˙ x i ) G for every i < η . We may assume ˙ x i ∈ V α + ω for i < η . Since Col( κ, V λ ) does not addnew < κ -sequences, we have h ˙ x i | i < η i ∈ V , hence h ˙ x i | i < η i ∈ X and { ( ˙ x i ) G | i < η } ∈ X [ G ].Since there is a sujrction from κ onto V λ , from V λ onto X , and from X onto X [ G ], we can obtain a bijection from κ onto X [ G ]. Let S = ˙ S G and F = ˙ F G . Since X [ G ] ≺ V [ G ] β , X [ G ] ∩ V [ G ] α is closed under < κ -sequences, and X [ G ] is well-orderable, we can take f : κ → S ∩ X [ G ] suchthat f ( i ) ∈ F ( f ↾ i ) for every i < κ . (cid:3) If there are proper class many LS cardinals, then for every cardinal κ ,there is a singular LS cardinal λ > κ such that there is an LS cardinal ν with κ < ν < cf( λ ); Take a singular LS cardinal ν > κ , and let λ be the ν + -th LS cardinal above ν . The cofinality of λ is ν + . In addition, in [3] weproved that if κ is a limit of LS cardinals, then every poset with rank < κ forces that κ is LS. Using these facts, we can obtain the following corollary: Corollary 4.8.
Suppose there are proper class many LS cardinals. Thenthere is a definable class forcing which forces
ZFC . . LS cardinals in
HOD( V λ )For a set X , let HOD( X ) be the class of all hereditarily definable sets withparameters from ON ∪ trcl( X ∪ { X } ). HOD( X ) is a transitive model of ZF with X ∈ HOD( X ). By the definition of HOD( X ), for every x ∈ HOD( X )there is an ordinal θ and a surjection σ : θ × <ω (trcl( X ∪ { X } ) → x with σ ∈ HOD( X ). If X = V α and α is limit, then V α is transitive, ordinaldefinable, and <ω V α ⊆ V α . Hence the domain of σ can be the set θ × V α inthis case. Lemma 5.1.
Let δ be a limit ordinal. Then for every limit ordinals α >β ≥ δ and x ∈ V α ∩ HOD( V δ ) , there is X ≺ V α ∩ HOD( V δ ) such that x ∈ X ∈ HOD( V δ ) , V β ∩ HOD( V δ ) ⊆ X , and k X k HOD( V δ ) = β .Proof. For given α > β ≥ δ , we can take a large ordinal θ and a sur-jection σ : θ × V δ → V α ∩ HOD( V δ ) with σ ∈ HOD( V δ ). By induc-tion on n < ω , we define X n and f n as follows. First, let X = ( V β ∩ HOD( V δ )) ∪ { x } , and take a surjection f : V β ∩ HOD( V δ ) → X with f ∈ HOD( V δ ). Suppose X n and f n are defined and f n is a surjection from V β ∩ HOD( V δ ) onto X n . For a formula ϕ ( v , . . . , v k , w ) and x , . . . , x k ∈ X n ,if ∃ wϕ ( x , . . . , x k , w ) holds in V α ∩ HOD( V δ ) then we can find the least α ϕ,x ,...,x k ∈ θ such that ∃ w ∈ σ “( { α ϕ,x ,...,x k } × V δ ) ϕ ( x , . . . , x k , w ) holdsin V α ∩ HOD( V δ ). Set X n +1 = X n ∪ S { σ “( { α ϕ,x ,...,x k } × V δ ) : ϕ is a for-mula, x , . . . , x k ∈ X n } . There is a canonical surjection from V β ∩ HOD( V δ )onto S { σ “( { α ϕ,x ,...,x k } × V δ ) : ϕ is a formula, x , . . . , x k ∈ X n } , namely h⌈ ϕ ⌉ , y , . . . , y k , z i 7→ σ ( α ϕ,f n ( y ) ,...,f n ( y k ) , z ) (where ⌈ ϕ ⌉ ∈ ω is the G¨odelnumber of ϕ ). Hence we can define f n +1 : V β ∩ HOD( V δ ) → X n +1 canon-ically using this surjection and f n . We can carry out this construction inHOD( V δ ), hence h X n , f n | n < ω i ∈ HOD( V δ ). Thus we have X = S n X n ∈ HOD( V δ ). We know V β ∩ HOD( V δ ) ⊆ X and x ∈ X . By Tarski-Vaughtcriterion, we have X ≺ V α ∩ HOD( V δ ). Moreover, there is a surjection from ω × ( V β ∩ HOD( V δ )) onto X in HOD( V δ ) constructed from h f n | n < ω i , sowe can construct a surjection f : V β ∩ HOD( V δ ) → X in HOD( V δ ), hence k X k HOD( V δ ) = β . (cid:3) The following is immediate from the previous lemma:
Corollary 5.2.
Let δ be a limit ordinal. Then for every cardinal κ > δ with k κ k = κ , κ is weakly LS in HOD( V δ ) .Proof. Take γ < κ , α > κ and x ∈ V α ∩ HOD( V δ ). We may assume α and γ are limit ordinals and γ ≥ δ . By the previous lemma, there is X ≺ V α ∩ HOD( V δ ) such that x ∈ X ∈ HOD( V δ ), V γ ∩ HOD( V δ ) ⊆ X , nd k X k HOD( V δ ) = γ . Then k X k ≤ k X k HOD( V δ ) = γ < κ , so the transitivecollapse of X must be in V κ . (cid:3) An uncountable cardinal κ is said to be inaccessible if for every γ < κ ,there is no cofinal map from V γ into κ . Every inaccessible cardinal is regular,and in ZFC , this definition is equivalent to the standard one. One can checkthat every regular weakly LS cardinal is inaccessible.The following corollary is already proved in Schlutzenberg [2].
Corollary 5.3.
Let κ be an uncountable cardinal, and suppose κ is inac-cessible in HOD( V κ ) . Then κ is a weakly LS cardinal in HOD( V κ ) . Inparticular, if κ is inaccessible then κ is regular weakly LS in HOD( V κ ) .Proof. Take γ < κ , α > κ and x ∈ V α ∩ HOD( V κ ). By Lemma 5.1, wecan find X ≺ V α ∩ HOD( V κ ) such that x ∈ X ∈ HOD( V κ ), V κ ⊆ X , and k X k HOD( V κ ) = κ . Take a surjection σ : V κ → X in HOD( V κ ). Since κ isinaccessible in HOD( V κ ), the set { η < κ | σ “ V η ≺ V α ∩ HOD( V κ ) } contains aclub in κ , hence we can find η < κ such that V γ ⊆ σ “ V η ≺ V α ∩ HOD( V κ ). (cid:3) Note 5.4.
For an uncountable cardinal κ , if κ is not regular in HOD( V κ )then cf( κ ) HOD( V κ ) = cf( κ ). Corollary 5.5.
Let δ be a limit ordinal and ν ≤ cf( δ ) a weakly LS cardinal.Then ν is weakly LS in HOD( V δ ) .Proof. For γ < ν , α > δ and x ∈ V ν ∩ HOD( V δ ), we can take X ≺ V α ∩ HOD( V δ ) such that x ∈ X ∈ HOD( V δ ), V δ ⊆ X , and there is a surjection σ : V δ → X in HOD( V δ ). Define R , R ⊆ V δ by a R b ⇐⇒ σ ( a ) ∈ σ ( b ),and a R b ⇐⇒ σ ( a ) = σ ( b ). We know R , R ∈ HOD( V δ ). Since ν isweakly LS, we can find h Y ; ∈ , R ∩ Y, R ∩ Y i ≺ h V δ ; ∈ , R , R i such that V γ ⊆ Y , k Y k < ν , and there is a ∈ Y with σ ( a ) = x . Since k Y k < ν ≤ cf( δ ),there is no cofinal map from Y to δ . In particular sup { rank( y ) | y ∈ Y } < δ ,so Y ∈ V δ and Y ∈ HOD( V δ ). Then one can check that V γ ⊆ σ “ Y ≺ V α ∩ HOD( V δ ), as required. (cid:3) Note 5.6.
HOD( V δ ) and HOD( V κ ) in this section can be replaced by L ( V δ )and L ( V κ ). 6. On elemtary embeddings
Woodin [4] proved that if κ is a singular limit of supercompact cardinalsand there is a set A of ordinals with κ + = ( κ + ) L [ A ] , then there is no non-trivial elementary embedding j : V κ +2 → V κ +2 . We can prove the sameresult using weakly LS cardinals. Note that whenever A is a set of ordinals,HOD( A ) is a transitive model of ZFC and A ∈ HOD( A ). roposition 6.1. Suppose κ is a singular weakly LS cardinal. If thereis a set A of ordinals with κ + = ( κ + ) HOD( A ) , then there is no non-trivialelementary embedding j : V κ +2 → V κ +2 .Proof. Let F be the club filter over κ + restricted to the set { α < κ + | cf( α ) = ω } , it is κ + -complete. F is ordinal definable, hence F ∩ HOD( A ) ∈ HOD( A ), and F ∩ HOD( A ) is a κ + -complete filter in HOD( A ). SinceHOD( A ) computes κ + correctly and is a model of ZFC , we have that F ∩ HOD( A ) is not κ + -saturated in HOD( A ). Take pairwise disjoint F -positive sets { E α | α < κ + } ∈ HOD( A ). Each E α is a stationary set in κ + ,and we may assume that E α ⊆ { η < κ + | cf( η ) = ω } . Hence we obtain κ -many pairwise disjoint stationary subsets of { η < κ + | cf( η ) = ω } . Thispartition can be coded in V κ +2 , and using this coded partition we can provethat there is no non-trivial elementary embedding j : V κ +2 → V κ +2 (e.g.,see Kanamori [1]). (cid:3) The assumption in the previous proposition can be weakened as follows.
Proposition 6.2.
Suppose κ is a singular weakly LS cardinal. If there is aset A of ordinals such that κ + is not measurable in HOD( A ) , then there isno non-trivial elementary embedding j : V κ +2 → V κ +2 .Proof. First note that κ is strong limit in HOD( A ). Let F be the filterdefined as in Proposition 6.1. By Tarski’s theorem, if F ∩ HOD( A ) is λ -saturated for some λ < κ in HOD( A ), then κ + is measurable in HOD( A ),this contradicts to the assumption. Hence F cannot be λ -saturated for every λ < κ , and we can take a large stationary partition of { η < κ + | cf( η ) = ω } .The rest is the same to before. (cid:3) Corollary 6.3.
Suppose there are proper class many weakly LS cardinals.If there is a non-trivial elementary embedding j : V → V , then { κ | κ issingular and κ + is measurable in HOD } forms a proper class. On ultrapowers
A cardinal κ is said to be critical if there is a transitive class M and anelementary embedding j : V → M with critical point κ (Schlutzenberg [2]).While the definition of critical cardinal is a second-order statement, in ZFC it is equivalent to a first order statement: There is a κ -complete ultrafil-ter over κ . Recently Schlutzenberg [2] showed that under the existence ofproper class many weakly LS cardinals, the definition of critical cardinal isequivalent to a certain first order sentence. Actually, under the assump-tion, he showed that the ultrapower by certain extender sequence satisfies Los’ theorem, so the critical cardinals can be formalized by the existence of uitable extenders. Moreover his argument can formalize strong cardinal aswell. See [2] for the proof of the following proposition. Proposition 7.1.
Suppose there are proper class many weakly LS cardinals.Then for every cardinal κ the following are equivalent:(1) For every α > κ , there is a weakly LS cardinal λ > α of uncountablecofinality, a transitive set N , and an elementary embedding j : V λ → N such that the critical point of j is κ , α < j ( κ ) , and V α ⊆ N .(2) For every α , there is a transitive class M and an elementary embed-ding j : V → M such that the critical point of j is κ , α < j ( κ ) , and V α ⊆ M .(3) For every α , there is a definable transitive class M and a definableelementary embedding j : V → M such that the critical point of j is κ , α < j ( κ ) , and V α ⊆ M . Other large cardinals defined by extenders, such as Woodin cardinal,superstrong cardinal, and I -embedding, can be formalized by a similarway. Acknowledgments.
This research was supported by JSPS KAKENHIGrant Nos. 18K03403 and 18K03404.
References [1] A. Kanamori,
The Higher Infinite: Large Cardinals in Set Theory from Their Be-ginnings . Springer-Verlag, 1994.[2] F. Schlutzenberg,
Reinhardt cardinals and non-definability.
Preprint. Available at https://arxiv.org/abs/2002.01215 [3] T. Usuba,
Choiceless L¨owenheim-Skolem property and uniform definability ofgrounds . To appear in the Proceedings of the Symposium on Advances in Mathe-matical Logic 2018. Available at https://arxiv.org/abs/1904.00895 [4] W. H. Woodin,
Suitable extender models I.
J. Math. Log. 10 (2010), no. 1-2, 101–339.(T. Usuba)
Faculty of Science and Engineering, Waseda University, Okubo3-4-1, Shinjyuku, Tokyo, 169-8555 Japan
E-mail address : [email protected]@waseda.jp