Choiceless large cardinals and set-theoretic potentialism
aa r X i v : . [ m a t h . L O ] J u l Choiceless large cardinals andset-theoretic potentialism
Raffaella Cutolo and Joel David HamkinsJuly 6, 2020
Abstract
We define a potentialist system of ZF-structures, that is, a collectionof possible worlds in the language of ZF connected by a binary acces-sibility relation , achieving a so called “potentialist account” of the fullbackground set-theoretic universe V . The definition involves Berkeleycardinals, the strongest known large cardinal axioms, inconsistent withthe Axiom of Choice. In fact, as background theory we assume just ZF.It turns out that the propositional modal assertions which are valid atevery world of our system are exactly those in the modal theory S4.2.Moreover, we characterize the worlds satisfying the potentialist maxi-mality principle, and thus the modal theory S5, both for assertions inthe language of ZF and for assertions in the full potentialist language. In the current scenario of set theory, we are faced with a conflict between largecardinal axioms and the Axiom of Choice. In fact, there is a whole new hierar-chy of ZF large cardinals - the Berkeley hierarchy - which contradict AC andlie beyond the Kunen inconsistency of Reinhardt cardinals. Such “choiceless”large cardinals have been recently introduced in [1] and the investigation oftheir consistency is very involved in the present main foundational questionsconcerning the universe of set theory. But let us point out something else thatis of interest here, namely: if we drop AC then the set-theoretic universe V grows upward. This observation raises a potentialist perspective, that is, onein which the universe of set theory reveals gradually, and never completely, aswe progressively take under consideration new fragments of it; indeed, we canactually think of we access higher and higher parts of the set-theoretic universeby considering stronger and stronger large cardinals.Recent works of the second author focus on the idea of set-theoretic poten-tialism and the analysis of the modal principles validated by specific poten-tialist systems. The general definition is stated below. Definition 1.1. A potentialist system is a collection W of structures ina common language L called “worlds”, equipped with a binary accessibilityrelation R , such that: 1 R is reflexive and transitive; • whenever M R N , then M is - or embeds to - a substructure of N .So, a potentialist system is a Kripke model of L -structures for some language L . In order to study how truth of an assertion ϕ propagates through theworlds of W , one adds to the basic language L the modal operators ♦ and (cid:3) ,expressing, respectively, the notions of possibility and necessity : • ♦ ϕ holds at a world M (that is, “ ϕ is possible over M ”) if ϕ holds atsome world N such that M R N ; • (cid:3) ϕ holds at a world M (that is, “ ϕ is necessary over M ”) if ϕ holds atall worlds N such that M R N .Now one can ask which propositional modal assertions are valid in the wholesystem W (that is, hold in every world of W ); the point is that determiningthe modal validities of a potentialist system gives a precise account of how itsworlds interact with respect to their respective truths.Let us turn to our particular case, whose hallmark is to combine choicelesslarge cardinals with the potentialist ideas. Indeed, we consider the concept ofset-theoretic potentialism that arises from elementary embeddings of a tran-sitive set into itself, where we view M as accessing N ⊇ M whenever therestriction to M of any elementary embedding j : N → N yields an elemen-tary embedding j ′ : M → M . Such a definition of the accessibility relationresults in an interesting case as in the context of Berkeley cardinals, one canarrange non-trivial elementary embeddings fixing any desired set. The keypoint is that every given set is definable in some big transitive set and if thereis a Berkeley cardinal δ then, by definition, any transitive set M containing δ as a member admits non-trivial elementary embeddings j : M → M , whosecritical points are in fact cofinal in δ . Definition 1.2.
A cardinal δ is a Berkeley cardinal if for every transitiveset M such that δ ∈ M , and for every ordinal η < δ , there exists a non-trivialelementary embedding j : M → M with η < crit ( j ) < δ . It turns out that the set-theoretic universe V equals the union of the worldsof our potentialist system, and given any world M and any set a , there is aworld N accessed by M such that a ∈ N . Thus, truth in V is approximated bytruth in our worlds: we can assert any property concerning any set of V fromany of the worlds of our system by using the diamond operator, and we canprogressively move from any world to the wider perspective of another worldwhich is “closer” to V in that it contains additional sets and is capable to satisfy In the choiceless context, being non-trivial means j is not the identity on the ordinals. V We start with a preliminary lemma motivating the definition of the accessibilityrelation we shall consider.
Lemma 2.1.
For every transitive set M , for every set a , there exists a transi-tive set N ⊇ M with a ∈ N such that every elementary embedding j : N → N lifts some elementary embedding j ′ : M → M . Proof.
Let M be a transitive set and let a be any set. As shown in [1], thereexists a transitive set N such that M, a ∈ N and M is definable (withoutparameters) in N . Thus, M ⊆ N and every j : N → N fixes M , whichimplies j “ M ⊆ M . Therefore j ↾ M : M → M , and so actually j lifts j ′ = j ↾ M : M → M . Definition 2.2.
Let δ be a Berkeley cardinal. • Let M δ = { M : M is transitive ∧ δ ∈ M } . • Let R be the binary relation on M δ defined as follows: for M, N ∈ M δ , M R N iff M ⊆ N and every elementary embedding j : N → N liftssome elementary embedding j ′ : M → M ; that is, M R N iff M ⊆ N and for every elementary embedding j : N → N , j ↾ M : M → M .Note that the assumption that there is a Berkeley cardinal δ and the choiceof M δ as collection of worlds ensure that every world M admits non-trivialelementary embeddings j : M → M , so R is not merely reduced to the subsetrelation. It is trivial that R is reflexive; also, R is transitive: in fact, if j : H → H lifts j ′ : N → N , which in turn lifts j ′′ : M → M , then j lifts j ′′ , as j ↾ M = ( j ↾ N ) ↾ M = j ′ ↾ M = j ′′ . Therefore, hM δ , Ri isa potentialist system of ZF-structures. Since V α ∈ M δ for any α > δ , wehave that V = S M ∈M δ M . Moreover, we show that M δ provides a potentialist ccount of the set-theoretic universe V , meaning that every world in M δ isa substructure of V and for every M ∈ M δ and every set a there is a world N ∈ M δ accessed by M such that a ∈ N . Lemma 2.3. M δ provides a potentialist account of the universe V . Proof.
First, for every M ∈ M δ , h M, ∈i is a substructure of h V, ∈i (i.e., M ⊆ V and ∈ M = ∈ V ↾ M ). Further, if M ∈ M δ and a ∈ V , then by Lemma 2.1 thereexists a transitive set N such that { M, a } ∈ N and for every elementaryembedding j : N → N , j ↾ M : M → M . Since δ ∈ M ⊆ N , we have that N ∈ M δ ; since a ∈ N and N is accessed by M , we are done. Remark 2.4.
Notice that:1. By Lemma 2.1, for every M ∈ M δ there exist cofinally many N ∈ M δ which are accessed by M , meaning that such N can accommodate anygiven set.2. In particular, for every M ∈ M δ , for every set a , there exists a rank initialsegment V α ⊇ M with a ∈ V α such that every elementary embedding j : V α → V α lifts some elementary embedding j ′ : M → M , that is, suchthat V α is accessed by M .For any assertion ϕ in the language of ZF, the potentialist translation ϕ ⋄ isthe assertion in the potentialist language ZF ⋄ (which augments the languageof ZF with the modal operators ♦ and (cid:3) ) achieved by replacing every instanceof ∃ x with ♦ ∃ x and every instance of ∀ x with (cid:3) ∀ x . As an immediate corollaryof Lemma 2.3, we get that truth in V is equivalent to potentialist truth at theworlds of M δ . Corollary 2.5.
For any ZF-formula ϕ and for any a , . . . , a n ∈ V , we have: V | = ϕ ( a , . . . , a n ) iff M | = M δ ϕ ⋄ ( a , . . . , a n ),for any M ∈ M δ in which a , . . . , a n exist.Let us now state what it precisely means for a modal assertion to be valid withrespect to our potentialist system. Definition 2.6.
A modal assertion ϕ ( p , . . . , p n ) in the language of proposi-tional modal logic is valid at a world M in M δ for a certain class of assertions,if all the resulting substitution instances ϕ ( ψ , . . . , ψ n ), where assertion ψ i fromthe allowed class is substituted for the propositional variable p i , are true at M . ϕ is valid in M δ if it is valid at every world of M δ .Of course, the main question arising here is the following: Question.
What is the modal logic of M δ ? That is, which are the modalprinciples valid in M δ ? 4 The modal logic of M δ In this section we provide lower and upper bounds on the modal validities of M δ , and finally prove that the modal logic of M δ is exactly S4.2. Definition 3.1.
The modal theory S4 is obtained from the following axiomsby closing under modus ponens and necessitation. • (K) (cid:3) ( ϕ → ψ ) → ( (cid:3) ϕ → (cid:3) ψ ) • (Dual) ¬ ♦ ϕ ↔ (cid:3) ¬ ϕ • (S) (cid:3) ϕ → ϕ • (4) (cid:3) ϕ → (cid:3)(cid:3) ϕ Theorem 3.2.
The modal theory S4 is valid at every world of M δ . Proof.
Let M ∈ M δ . • (K). Suppose (cid:3) ( ϕ → ψ ) and (cid:3) ϕ hold in M . Then, ϕ → ψ and ϕ holdin any world N accessed by M . Therefore, by modus ponens, ψ holds inany such N , that is, (cid:3) ψ holds in M . • (Dual). Immediate. • (S). Follows immediately from the fact that every world accesses itself. • (4). If ϕ holds in any world N accessed by M , then so does (cid:3) ϕ , as anyworld H accessed by N is also accessed by M . Definition 3.3.
The modal theory S4.2 is obtained from S4 by adding theaxiom (.2) ♦(cid:3) ϕ → (cid:3)♦ ϕ . Theorem 3.4.
The modal theory S4.2 is valid at every world of M δ . Proof. (.2). Let M ∈ M δ . Assume ♦(cid:3) ϕ holds in M , that is, there exists N ∈ M δ such that M R N and (cid:3) ϕ holds in N . Let H ∈ M δ be such that M R H . We need to show that ♦ ϕ holds in H . Note that there exists atransitive set K such that h N, H i is definable in K . Since K is transitive and N, H ⊆ K , we have that δ ∈ K and so K ∈ M δ . Now, take any non-trivialelementary embedding j : K → K . Then, j ( N ) = N and j ( H ) = H . So, j ↾ N : N → N and j ↾ H : H → H . Therefore, N and H both access K .Since K is accessed by N , K satisfies ϕ ; but then, since K is accessed by H and ϕ holds in K , ♦ ϕ holds in H . In fact, every potentialist system validates S4. Unless otherwise specified, the validities hold for all assertions in ZF ⋄ , with parameters. - and therefore validates (.2), establishes a firstsignificant lower-bound result. In order to provide upper bounds on the validi-ties of M δ , and then determine the exact set of modal principles valid throughthe whole system, we recall the definitions of switches , buttons and dials , spe-cific kinds of control statements first introduced in [5]; in particular, we willbe interested in finding assertions satisfying such definitions which also havethe property of being independent . Definition 3.5.
An assertion s is a switch if both ♦ s and ♦ ¬ s are true atevery world, that is, both (cid:3)♦ s and (cid:3)♦ ¬ s hold. s is a switch at a particularworld M if ♦ s and ♦ ¬ s are true at all the worlds accessed by M . Definition 3.6. A button is a statement b such that ♦(cid:3) b is true at everyworld, that is, (cid:3)♦(cid:3) b holds. The button is pushed at a world if (cid:3) b holds atthat world, and otherwise unpushed. Definition 3.7.
A (possibly infinite) list of statements d , d , d , . . . is a dial if every world satisfies exactly one of the statements d i and every world canaccess another world with any prescribed dial value. If a world satisfies d i ,then we say that the dial value is i in that world. Definition 3.8.
A family of switches is independent if one can always flipthe truth values of any finitely many of the switches so as to realize any desiredfinite pattern of truth.
Definition 3.9.
A family of buttons and switches is independent if thereis a world at which the buttons are unpushed, and every world M accesses aworld N in which any additional button may be pushed without pushing anyother as-yet unpushed button from the family, while also setting any finitelymany of the switches so as to have any desired pattern in N ; and similarlywith dials. Definition 3.10.
The modal theory S5 is obtained from S4 by adding theaxiom (5) ♦(cid:3) ϕ → ϕ , which we call potentialist maximality principle (MP).The following theorem summarizes some key results - first proved in [5], anddeveloped further in [3] - we shall use. Theorem 3.11.
The following hold.1. If W is a Kripke model and a world M ∈ W admits arbitrarily largefinite collections of independent switches, then the propositional modalassertions valid at M are contained in the modal theory S5. In particular, In fact, it shows that whenever M R N and M R H then there exists K such that N R K and H R K , and so, that M δ has amalgamation .
6f the switches work throughout W , then the validities of every world of W , and so the validities of W , are contained within S5.2. A Kripke model W admits arbitrarily large finite families of independentswitches if and only if it admits arbitrarily large finite dials.3. If W is a Kripke model that admits arbitrarily large finite families ofindependent buttons and switches, or independent buttons independentof a dial, then the propositional modal validities of W are contained inS4.2. The validities of any particular world in which the buttons are notyet pushed are contained in S4.2, and in any case, are contained in S5. Theorem 3.12.
The propositional modal validities of M δ are contained inthe modal theory S5. Proof.
It suffices to show that M δ admits arbitrarily large finite dials. Weshall show that it admits in fact an infinite dial (notice that from an infinitedial, we can construct finite dials of any given size by keeping any desired finitenumber of dial statements and adding the statement that none of them holds).For i < ω , let d i be the assertion that the height of the ordinals is λ + i , where λ is a limit ordinal or zero. These statements are expressible in the languageof ZF (without parameters or modal vocabulary), correctly interpreted insideany transitive set, and so inside any world M ∈ M δ . Let us show that theyform a dial. First, since any ordinal is uniquely expressed as λ + i for some limitordinal λ or zero and some finite i < ω , every world in M δ satisfies exactly oneof the statements d i . It remains to prove that every world can access anotherworld with any desired dial value. Let M ∈ M δ and fix i < ω . Let V θ be suchthat M ⊆ V θ . Let N = V θ ∪ tr cl( {h θ, M i} ) ∪ ( θ + i ). Then N ∈ M δ and M isdefinable in N , which implies M R N , and the dial value in N is i . Theorem 3.13. M δ satisfies exactly S4.2, that is, the modal logic of M δ isS4.2. Proof.
We show that M δ admits arbitrarily large finite families of independentbuttons independent of a dial, which implies that the modal validities of M δ are contained in, and hence by Theorem 3.4 equal to, S4.2. For i < ω , let d i bethe assertion that the height of the ordinals is λ + i , where λ is a limit ordinalor zero; we already showed that these statements form a dial. For m < ω , let b m be the assertion that the set m · N = { m · k : k < ω } exists; let us showthat these statements are independent buttons independent of the above dial.Since for every m < ω , if the assertion b m is true in some transitive set then itwill continue to be true in any larger transitive set, each b m is a button. Thesebuttons are independent because every world M accesses a world N in whichany additional button b m may be pushed without pushing any other as-yet7npushed button from the family. Finally, the above buttons and dial valuescan be controlled independently of each other.By Theorem 3.12, S5 is a definite upper bound on the validities of M δ . Aninteresting point would therefore be to determine which worlds of M δ realizethe maximum set of validities. In other words: Question.
Which worlds of M δ satisfy the potentialist maximality principle? We now give a characterization of the worlds of M δ satisfying S5. Dependingon the language we consider, we get different criteria. The following conceptsare involved. Definition 4.1.
An ordinal θ is Σ n - correct if V θ ≺ Σ n V , meaning that V θ and V agree on the truth of Σ n formulas with parameters from V θ . Definition 4.2.
A cardinal θ is correct if it is Σ n -correct for every n , thatis, if it realizes the scheme V θ ≺ V . Theorem 4.3.
The following are equivalent.1. The potentialist maximality principle holds in a world M ∈ M δ forassertions in the language of ZF with parameters from M .2. M = V θ for some Σ -correct cardinal θ > δ . Proof.
Let M ∈ M δ . • (1 ⇒ M satisfies (5) ♦(cid:3) ϕ → ϕ for assertions in the languageof ZF with parameters from M . First, note that M must be a V θ with θ limit. In fact, for all a ∈ M , the existence of the power set of a ispossibly necessary, and it follows that M thinks P ( a ) exists, and that M computes the power sets correctly; moreover, M computes V α correctlyfor any ordinal α ∈ M , since the existence of V α is possibly necessary.Also, for any set a , it is possibly necessary that a ∈ V α for some ordinal α ,and so this is already true in M . Now, suppose ϕ ( ~a ) is a Σ statementtrue in V , with ~a ∈ M . This is witnessed by the existence of someordinal α for which V α satisfies ψ ( ~a ) for some assertion ψ . So, it ispossibly necessary the statement that there is an ordinal α for which V α exists and satisfies ψ ( ~a ). Thus, by (5), this statement must be true in M , and so ϕ ( ~a ) is true in M . Therefore, θ is Σ -correct. Note: this concept is not expressible as a single assertion in the language of set theory,although it can be expressed as a scheme of statements. (2 ⇒ M = V θ for θ > δ a Σ -correct cardinal. Suppose M satisfies ♦(cid:3) ϕ ( ~a ), ~a ∈ M . Then, there exists N ∈ M δ accessed by M that satisfies (cid:3) ϕ ( ~a ). Since ♦(cid:3) ϕ ( ~a ) is a Σ statement in V , it must betrue inside M , being M Σ -correct. So, there exists m ∈ M such that m satisfies (cid:3) ϕ ( ~a ). Without loss of generality, there exists m = V α likethis (inside M ), and so the smallest one is definable and so it accesses M . Since the statement that V α satisfies (cid:3) ϕ ( ~a ) is a Π statement, it isabsolute between M and V by the Σ -correctness of M , and so, it holdsin V . Thus, ϕ ( ~a ) holds in M , which therefore satisfies (5). Theorem 4.4.
The following are equivalent.1. The potentialist maximality principle holds in a world M ∈ M δ forassertions in the potentialist language ZF ⋄ with parameters from M .2. M = V θ for some correct cardinal θ > δ . Proof.
Let M ∈ M δ . • (1 ⇒ M satisfies (5) ♦(cid:3) ϕ → ϕ for assertions in the poten-tialist language ZF ⋄ with parameters from M . Then by Theorem 4.3, M = V θ for θ Σ -correct. By the potentialist translation, truth in V isexpressible as ϕ ⋄ -truth in M . Thus, M can express the statement thatthere exists a Σ n -correct cardinal (as this is definable in V ). For each n ,this statement is a button in ZF ⋄ . So M is a limit of Σ n -correct cardinals,and therefore M is fully correct. • (2 ⇒ M = V θ for θ > δ a correct cardinal. Suppose M satisfies ♦(cid:3) ϕ ( ~a ), where ~a ∈ M and ϕ is a ZF ⋄ assertion. Then, thereexists N ∈ M δ accessed by M that satisfies (cid:3) ϕ ( ~a ). Since the existenceof such a set N and the potentialist semantics are expressible in thelanguage of set theory, it follows from M ≺ V that there is such a setinside M . So, there exists m ∈ M such that m satisfies (cid:3) ϕ ( ~a ). Withoutloss of generality, this m has form V α , and so the smallest one is definableand therefore accesses M . Thus, ϕ ( ~a ) holds in M . Remark 4.5.
Note that one can view Theorem 4.4 as a ZF theorem schemeasserting the equivalence of two schemes; or, one could view it as a theorem ofG¨odel-Bernays set theory augmented with the assumption that there is a pred-icate for first-order set-theoretic truth (that theory is provable, for example,in Kelley-Morse set theory). 9
Consistency of MP and conclusive remarks
Theorem 4.3 and Theorem 4.4 characterize the worlds of M δ satisfying MP,respectively, for assertions in the language of set theory and for assertions inthe full potentialist language, with parameters. Let us remark that there existindeed such worlds in M δ . In fact, for the first case, note that by the L´evy-Montague reflection theorem (which is a ZF result), the class of all Σ -correctcardinals is closed and unbounded in the ordinals. For the second case, observethat although the existence of a correct cardinal is not provable in ZF, it isrelatively consistent with ZF (see [2]); so, it is relatively consistent with ZFthat there exist worlds in M δ satisfying MP for assertions in the potentialistlanguage ZF ⋄ .Recall that the definition of our potentialist system leverages on the as-sumption that there is a Berkeley cardinal δ ; one may ask if there is any worldin M δ which satisfies that there exists a Berkeley cardinal, and the answeris yes: in fact, the property of being a Berkeley cardinal is Π , so for anyΣ -correct cardinal θ , V θ correctly recognizes the Berkeley cardinals below θ .In other words, inside all the worlds of M δ satisfying MP, δ itself is still aBerkeley cardinal; but these are not the only worlds in M δ recognizing δ isBerkeley: in fact, as noted in [1], if δ is a Berkeley cardinal then for all limitordinals λ > δ , V λ thinks that δ is Berkeley.Finally, since every world M in M δ can access a V λ with λ limit, theassertion ϕ BC that there exists a Berkeley cardinal is possible over any M ,that is, ♦ ϕ BC holds at every world. References [1] Bagaria J., Koellner P., Woodin W. H.
Large cardinals beyond Choice . Bulletin of Symbolic Logic , vol. 25 (2019), pp. 283-318.[2] Hamkins J. D.
A simple maximality principle . The Journal of Sym-bolic Logic , vol. 68 (2003), pp. 527-550.[3] Hamkins J. D., Leibman G., L¨owe B.
Structural connections between aforcing class and its modal logic . Israel Journal of Mathematics ,vol. 207 (2015), pp. 617-651.[4] Hamkins J. D., Linnebo Ø.
The modal logic of set-theoretic potentialismand the potentialist maximality principles . To appear in
The Review ofSymbolic Logic .[5] Hamkins J. D., L¨owe B.