aa r X i v : . [ m a t h . L O ] F e b HOW STRONG IS A REINHARDT SET OVER EXTENSIONS OF CZF?
HANUL JEON
Abstract.
We investigate the lower bound of the consistency strength of
CZF with Full Separation
Sep anda Reinhardt set, a constructive analogue of Reinhardt cardinals. We show that
CZF + Sep with a Reinhardtset interprets ZF − with a cofinal elementary embedding j : V ≺ V . We also see that CZF + Sep with aReinhardt set interprets ZF − with a model of ZF + WA , the Wholeness axiom for bounded formulas. Introduction
Large cardinals are one of the important topics of set theory: the linear hierarchy of large cardinalsprovides a scale to fathom the consistency strength of a given set-theoretic statement. Reinhardt introduceda quite strong notion of large cardinal, now known as Reinhardt cardinal. Unfortunately, Kunen [15] showedthat Reinhardt cardinals do not exist in
ZFC . However, Kunen’s proof relied on the Axiom of Choice, so itremains the hope that Reinhardt cardinals are consistent if we dispose of the Axiom of Choice. Recently,there are attempts to study Reinhardt cardinals over ZF and find intrinsic evidence for choiceless largecardinals , cardinals that are incompatible with the Axiom of Choice.We may ask choiceless large cardinals are actually consistent with subtheories of ZFC , and there are somepositive answers for this question. For example, Schultzenberg [21] showed that ZF with an elementaryembedding j : V λ +2 ≺ V λ +2 is consistent if ZFC with I is. Matthews [17] proved that Reinhardt cardinal iscompatible with ZFC − if we assume the consistency of ZFC with I .We may also ask the lower bound of the consistency strength of large cardinals over subtheories of ZFC ,which is quite non-trivial if we remove axioms other than choice. For example, Schultzenberg proved that ZF with j : V λ +2 ≺ V λ +2 is equiconsistent with ZFC with I . We do not have actual bound for ZFC − with j : V ≺ V , but we can obtain the lower bound if we add the assumption V crit j ∈ V that Matthew’s modelsatisfies: in that case, we can see that V λ exists and it is a model of ZFC with the Wholeness axiom WA .In this paper, we will take another ‘subtheory’ of ZFC that lacks the law of excluded middle, namely
CZF . CZF is a weak theory in that its consistency strength is the same as that of Kripke-Platek set theory KP with Infinity. However, adding the law of excluded middle to CZF results in the full ZF . The aim of thispaper is to measure the ‘lower bound’ of the consistency strength of CZF with Full Separation
Sep and aReinhardt set. Hence we have how hard to establish the consistency of
CZF + Sep with a Reinhardt set. Themain result of this paper is as follows, which is a consequence of Theorem 5.9 and Theorem 6.10.
Proposition 1.1.
The theory
CZF + Sep + ‘there is a Reinhardt set’ can interpret the following theory: ZF − with the cofinal embedding j : V ≺ V with a transitive set K such that j ( x ) = x for all x ∈ K , K ∈ j ( K ) and Λ := S n ∈ ω j n ( K ) thinks it is a model of ZF with WA , the Wholeness axiom for ∆ -formulas. The structure of this paper.
In Section 2 and 3, we will cover relevant preliminaries. We willreview constructive set theory in Section 2 and large set axioms in Section 3, so the readers who are alreadyfamiliar with these topics may skip them. In Section 4, we review and discuss Gambino’s Heyting-valuedinterpretation [9] over the double negation formal topology Ω, which is the main tool of the paper. It forces ∆ - LEM , the law of excluded middle for bounded formulas. By assuming the Full Separation, we may turn ∆ - LEM to the full law of excluded middle. However, forcing over Ω does not preserve every axiom of
CZF :it does not preserve the Axiom of Subset Collection, a
CZF -analogue of Power Set axiom. Thus the resultingtheory of forcing over Ω is ZF − if we start from CZF + Sep . Unlike ‘small’ formal topologies, Ω is not absolutebetween transitive models, and it causes issues about absoluteness of the Heyting-valued interpretation. Wewill also discuss it in this section. Section 5 is devoted to prove elementary embeddings are preserved underΩ. However, Ω does not prove inaccessibility of a critical point of the elementary embedding, and it restricts
Mathematics Subject Classification.
Primary 03E70; Secondary 03E55. the analysis on the consistency strength of
CZF + Sep with an elementary embedding. We will deal withthis issue in Section 6 by showing that the Heyting-valued interpretation under Ω still proves the criticalpoint enjoys a strong reflection principle, which makes the critical point a transitive model of ZF with largecardinal axioms. We briefly discuss why achieving the upper bound of the consistency strength of our objecttheories in Section 7, with some concluding questions.2. Constructive set theory
In this section, we will briefly review
ZFC − , ZFC without power set and the constructive set theory. Thereare various formulations of constructive set theories, but we will focus on
CZF .2.1.
ZFC without Power Set.
We will frequently mention
ZFC without Power Set, called
ZFC − . However, ZFC − is not obtained by just dropping Power Set from ZFC : Definition 2.1.
ZFC − is the theory obtained from ZFC as follows: it drops Power Set, uses Collectioninstead of Replacement, and the well-ordering principle instead of the usual statement of Choice. ZF − is asystem obtained from ZFC − by dropping the well-ordering principle.Note that using Collection instead of Replacement is necessary to avoid pathologies. See [10] for thedetails. It is also known by [8] that ZFC − does not prove the reflection principle.2.2. Axioms of
CZF . Constructive Zermelo-Fraenkel set theory
CZF is introduced by Aczel [2] with histype-theoretic interpretation of
CZF . We will introduce subtheories called
Basic Constructive Set Theory
BCST and
CZF − before defining the full CZF . Definition 2.2.
BCST is the theory which consists of Extensionality, Pairnig, Union, Emptyset, Replace-ment, and ∆ -Separation. CZF − is obtained by adding the following axioms to BCST : Infinity, ∈ -induction,and Strong Collection which states the following: if φ ( x, y ) is a formula such that ∀ x ∈ a ∃ yφ ( x, y ) for given a , then we can find b such that ∀ x ∈ a ∃ ∈ byφ ( x, y ) ∧ ∀ y ∈ b ∃ x ∈ aφ ( x, y ) . We will also define some synonyms for frequently-mentioned axioms:
Definition 2.3.
We will use
Sep , ∆ - Sep , ∆ - LEM for denoting Full Separation (i.e., Separation for allformulas), ∆ -Separation and the Law of Excluded Middle for ∆ -formulas.Full separation proves Strong Collection from Collection, but ∆ -Separation is too weak to do it. It isalso known that ∆ -Separation is equivalent to the existence of the intersection of two sets. See Section 9.5of [4] for its proof. Proposition 2.4.
Working over
BCST without ∆ -Separation, ∆ -Separation is equivalent to the Axiom ofBinary Intersection , which asserts that a ∩ b exists if a and b are sets. (cid:3) It is convenient to introduce the notion of multi-valued function to describe Strong Collection and SubsetCollection that appears later. Let A and B be classes. A relation R ⊆ A × B is a multi-valued functionfrom A to B if dom R = A . In this case, we write R : A ⇒ B . We use the notation R : A ⇔⇒ B if both R : A ⇒ B and R : B ⇒ A hold. Then we can rephrase Strong Collection as follows: for every set a and aclass R : a ⇒ V , there is a set b such that R : a ⇒ b .Now we can state the Axiom of Subset Collection: Definition 2.5.
The Axiom of Subset Collection states the following: let R u be a class with a parameter u ∈ V . For each a, b ∈ V , we can find a set c ∈ V such that R u : a ⇒ b = ⇒ ∃ d ∈ c ( R u : a ⇒ d ) . CZF is the theory by adding Subset Collection to
CZF − .There is a simpler version of Subset Collection known as Fullness , which is a bit easier to understand.
Definition 2.6.
The Axiom of Fullness states the following: Let mv( a, b ) the class of all multi-valuedfunction from a and b . Then there is a subset c ⊆ mv( a, b ) such that if r ∈ mv( a, b ), then there is s ∈ c suchthat s ⊆ r . We call c to be full in mv( a, b ). OW STRONG IS A REINHARDT SET OVER EXTENSIONS OF CZF? 3
Then the following holds:
Proposition 2.7. (1) (
CZF − ) Subset Collection is equivalent to Fullness.(2) ( CZF − ) Power Set implies Subset Collection.(3) ( CZF − ) Subset Collection proves the function set a b exists for all a and b .(4) ( CZF − ) If ∆ - LEM holds, then Subset Collection implies Power Set.We do not provide the proof for it, and the readers may consult with [5] or [4] for its proof. Note thatSubset Collection does not increase the proof-theoretic strength of
CZF − while the Axiom of Power Set does.The following lemma is useful to establish (1) of Proposition 2.7, and is also useful to treat multi-valuedfunctions: Lemma 2.8.
Let R : A ⇒ B be a multi-valued function. Define A ( R ) : A ⇒ A × B by A ( R ) = {h a, h a, b ii | h a, b i ∈ R } , then the following holds: (1) A ( R ) : A ⇒ S ⇐⇒ R ∩ S : A ⇒ B , (2) A ( R ) : A ⇔ S ⇐⇒ S ⊆ R .Proof. For the first statement, observe that A ( R ) : A ⇒ S is equivalent to ∀ a ∈ A ∃ s ∈ S : h a, s i ∈ A ( R ) . By the definition of A , this is equivalent to ∀ a ∈ A ∃ s ∈ S [ ∃ b ∈ B : s = h a, b i ∧ h a, b i ∈ R ] . We can see that the above statement is equivalent to ∀ a ∈ A ∃ b ∈ B : h a, b i ∈ R ∩ S , which is the definitionof R ∩ S : A ⇒ B . For the second claim, observe that A ( R ) : A ⇔ S is equivalent to ∀ s ∈ S ∃ a ∈ A : h a, s i ∈ A ( R ) . By rewriting A to its definition, we have ∀ s ∈ S ∃ a ∈ A : [ ∃ b ∈ B : s = h a, b i ∈ R ] . We can see that it is equivalent to S ⊆ R . (cid:3) The following lemma provides useful applications of Strong Collection:
Lemma 2.9. If a ∈ A and R : a ⇒ A , then there is a set b ∈ A such that b ⊆ R and b : a ⇒ A .Proof. Consider A ( R ) : a ⇒ a × A . By the second-order Strong Collection over A , there is b ∈ A such that A ( R ) : A ⇔⇒ b . Hence by Lemma 2.8, we have b ⊆ R and b : a ⇒ A . (cid:3) Inductive definition.
Various recursive construction on
CZF is given by inductive definition. Thereaders might refer [5] or [4] to see general information about inductive definition, but we will review someof it for the readers who are not familiar with it.
Definition 2.10. An inductive definition Φ is a class of pairs h X, a i . For an inductive definition Φ, associateΓ Φ ( C ) = { a | ∃ X ⊆ C h X, a i ∈ Φ } . A class C is Φ-closed if Γ Φ ( C ) ⊆ C .We may think Φ as a generalization of a deductive system, and Γ Φ ( C ) a class of theorems derivable fromthe class of axioms C . Some authors use the notation X ⊢ Φ a or X/a ∈ Φ instead of h X, a i ∈
Φ.Each Inductive definitions arise the least class fixed point:
Theorem 2.11 (Class Inductive Definition Theorem).
Let Φ be an inductive definition. Then thereis a smallest Φ -closed class I (Φ) . The following lemma is the essential tool for the proof of Class Inductive Definition Theorem. See Lemma12.1.2 of [4] for its proof:
Lemma 2.12.
Every inductive definition Φ has a corresponding iteration class J , which satisfies J a =Γ (cid:0)S x ∈ a J x (cid:1) for all a , where J a = { x | h a, x i ∈ J } . HANUL JEON
CZF versus
IZF . There are two possible constructive formulations of ZF , namely IZF and
CZF , althoughwe will focus on the later.
Definition 2.13.
IZF is the theory that comprises the following axioms: Extensionality, Pairing, Union,Infinity, ∈ -induction, Separation, Collection, and Power Set.It is known that every theorem of CZF is that of
IZF . Moreover,
IZF is quite strong in the sense thatits proof-theoretic strength is the same as that of ZF . On the other hand, it is known that the proof-theoretic strength of CZF is equal to that of Kripke-Platek set theory KP with Infinity. IZF is deemed to be impredicative due to the presence of Full Separation and Power Set. On the other hand,
CZF is viewed aspredicative since it allows the type-theoretic interpretation given by Aczel [2]. However, adding the full lawof excluded middle into
IZF or CZF results in the same ZF .3. Large set axioms
In this section, we will discuss large set axioms, which is an analogue of large cardinal axioms over
CZF .Since ordinals over
CZF could be badly behaved (for example, they need not be well-ordered), we focus onthe structural properties of given sets to obtain higher infinities over
CZF . We also compare the relationbetween large cardinal axioms over well-known theories like ZF and large set axioms.3.1. Tiny and Small Large set axioms.
The first large set notions over
CZF would be regular sets .Regular sets appear first in Aczel’s paper [3] about inductive definitions over
CZF . As we will see later,regular sets can ‘internalize’ most inductive constructions, which turns out to be useful in many practicalcases.
Definition 3.1.
A transitive set A is regular if it satisfies second-order Strong Collection: ∀ a ∈ A ∀ R [ R : a ⇒ A → ∃ b ∈ A ( R : a ⇔⇒ b )] . A regular set A is S -regular if S a ∈ A for all a ∈ A . A regular set A is inaccessible if ( A, ∈ ) is a model of CZF , and furthermore, it also satisfies the second-order Subset Collection: ∀ a, b ∈ A ∃ c ∈ A ∀ u ∈ A ∀ R ( R u : a ⇒ b ) → ∃ d ∈ c ( R u : a ⇔⇒ d ) . The
Regular Extension Axiom
REA asserts that every set is contained in some regular set. The
InaccessibleExtension Axiom
IEA asserts that every set is contained in an inaccessible set.There is no ‘pair-closed regular sets’ since every regular set is closed under pairings if it contains 2:
Lemma 3.2. If A is regular and ∈ A , then h a, b i ∈ A for all a, b ∈ A . (cid:3) REA has various consequences: For example,
CZF − + REA proves Subset Collection. Moreover, it alsoproves that every bounded inductive definition Φ has a set-sized fixed point I (Φ).The notion of regular sets is quite restrictive, as it does not have Separation axioms, at least for ∆ -formulas, so we have no way to do any internal construction over a regular set. The following notion is astrengthening of regular set, which resolves the issue of internal construction: Definition 3.3.
A regular set A is BCST-regular if A | = BCST . Equivalently, A satisfies Union, Pairing,Empty set and Binary Intersection.We do not know that CZF proves every regular set is BCST-regular, although Lubarsky and Rathjen[20] proved that the set of all hereditarily countable sets in the Feferman-Levy model is functionally regularbut not S -regular. It is not even sure that the existence of a regular set implies that of BCST-regular set.However, every inaccessible set is BCST-regular, and every BCST-regular sets we work with in this paper isinaccessible.What are regular sets and inaccessible sets in classical context? The following result illustrates how thesesets look like under the well-known classical context: Proposition 3.4. (1) ( ZF − ) Every S -regular set containing 2 is a transitive model of second-order ZF − , ZF − . There is no consensus on the definition on the predicativity. The usual informal description of the predicativity is rejectingself-referencing definitions.
OW STRONG IS A REINHARDT SET OVER EXTENSIONS OF CZF? 5 (2) (
ZFC − ) Every S -regular set containing 2 is of the form H κ for some regular cardinal κ .(3) ( ZF − ) Every inaccessible set is of the form V κ for some inaccessible κ .Note that we follow the Hayut and Karagila’s definition [14] of inaccessiblity in choiceless context; thatis, κ is inaccessible if V κ | = ZF . Proof. (1) Let A be a regular set containing 2. We know that A satisfies Extensionality, ∈ -induction,Union and the second-order Collection. Hence it remains to show that the second-order Separationholds.Let X ⊆ A and a ∈ A . Fix c ∈ X ∩ a . Now consider the function f : a → A defined by f ( x ) = ( x if x ∈ X, and c otherwise . By the second-order Strong Collection over A , we have b ∈ A such that f : a ⇔⇒ b , and thus b = a ∩ X .(2) Let A be a regular set. Let κ be the least ordinal that is not a member of A . Then κ must be a regularcardinal: if not, there is α < κ and a cofinal map f : α → κ . By transitivity of A and the definitionof κ , we have α ∈ A , so κ ∈ A by the second-order Replacement and Union, a contradiction.We can see that ZFC − proves H κ is a class model of ZFC − , and A satisfies the Well-orderingPrinciple. We can also show that A ⊆ H κ = { x : | TC x | < κ } holds: We know that A ∩ Ord = H κ ∩ Ord = κ . By the second-order Separation over A , P (Ord) ∩ A = P (Ord) ∩ H κ . Hence A = H κ :for each x ∈ H κ , we can find θ < κ , R ⊆ θ × θ and X ⊆ θ such that (trcl x, ∈ , x ) ∼ = ( θ, R, X ). (Herewe treat x as a unary relation.) Then ( θ, R, X ) ∈ A , so x ∈ A by Mostowski Collapsing Lemma.(3) If A is inaccessible, then A is closed under the true power set of its elements, since the second-orderSubset Collection implies if a, b ∈ A then a b ∈ A . Hence A must be of the form V κ for some κ .Moreover, κ is inaccessible because V κ = A | = ZF − . (cid:3) Question 3.5.
Is there a characterization of regular sets over
ZFC ? How about S -regular sets over ZF − ?3.2. Large Large set axioms.
There is no reason to stop defining large set notions up to weaker ones.Hence we define stronger large set axioms. The main tool to access strong large cardinals (up to measurablecardinals) is to use elementary embedding, so we follow the same strategy:
Definition 3.6 ( CZF − ) . Working over the extended language ∈ , a unary functional symbol j and a unarypredicate symbol M . We will extend CZF − as follows: we allow j and M in ∆ -Separation and StrongCollection (also for Subset Collection if we start from CZF ), and add the following schemes:(1) M is transitive, ∀ xM ( j ( x )), and(2) ∀ ~x [ φ ( ~x ) ↔ φ M ( j ( ~x ))] for every φ which does not contain any j or M , where φ M is the relativizationof φ over M .If a transitive set K satisfies ∀ x ∈ Kj ( x ) = x and K ∈ j ( K ), then we call K a critical point and we call K a critical set if K is also inaccessible. If M = V and K is transitive and inaccessible, then we call K a Reinhardt set .We will use the term critical point and critical set simultaneously, so the readers should distinguish thedifference of these two terms. (For example, a critical point need not be a critical set unless it is inaccessible.)Note that the definition of critical sets is apparently stronger than that is suggested by Hayut and Karagila[14]. Finding the
CZF -definition of a critical set which is classically equivalent to a critical set in the style ofHayut and Karagila would be a good future work. Also, Ziegler [24] uses the term ‘measurable sets’ to denotecritical sets, but we will avoid this term for the following reasons: it does not reflect that the definition isgiven by an elementary embedding, and it could be confusing with measurable sets in measure theory.We do not know every elementary embedding j : V ≺ M over CZF enjoys cofinal properties. Surprisingly,the following lemma shows that j become a cofinal map if M = V , even under CZF without any assumptions.Note that the following lemma uses Subset Collection heavily. See Theorem 9.37 of [24] for its proof.
Lemma 3.7 (Ziegler [24],
CZF ). Let j : V ≺ V be a non-trivial elementary embedding. Then j is cofinal ,that is, we can find y such that x ∈ j ( y ) for each x . Note that Ziegler [24] uses the term set cofinality to denote our notion of cofinality. However, we will usethe term cofinality to harmonize the terminology with that of Matthews [17].
HANUL JEON Heyting-valued interpretation over the double negation formal topology
General Heyting-valued interpretation.
We will follow Gambino’s definition [9] of Heyting-valuedinterpretation. We start this section by reviewing relevant facts on the Heyting-valued interpretation.Forcing is a powerful tool to construct a model of set theory. Gambino’s definition of Heyting-valuedmodel (or alternatively, forcing) opens up a way to produce forcing models of
CZF − . His Heyting-valuedmodel starts from formal topology , which formalizes a poset of open sets with a covering relation: Definition 4.1.
A structure S = ( S, ≤ , ⊳ ) is formal topology is a poset ( S, ≤ ) endowed with ⊳ ⊆ S × P ( S )such that(1) if a ∈ p , then a ⊳ p ,(2) if a ≤ b and b ⊳ p , then a ⊳ p ,(3) if a ⊳ p and ∀ x ∈ p ( x ⊳ q ), then a ⊳ q , and(4) if a ⊳ p, q , then a ⊳ ( ↓ p ) ∩ ( ↓ q ), where ↓ p = { r ∈ S | r ≤ p } .For each formal topology S , we have a notion of nucleus p given by p = { x ∈ S | x ⊳ p } . Then the class Low( S ) of all lower subsets that are stable under (i.e., p = p ) form a set-generated frame : Definition 4.2.
A structure A = ( A, ≤ , W , ∧ , ⊤ , g ) is a set-generated frame if ( A, ≤ , W , ∧ , ⊤ ) is a completedistributive lattice with the generating set g ⊆ A , such that the class g a = { x ∈ g | x ≤ a } is a set, and a = W g a for any a ∈ A .Note that every set-generated frame has every operation of Heyting algebra: for example, we can define a → b by a → b = W { x ∈ g | x ∧ a ≤ b } , ⊥ by ∅ , and V p by W { x ∈ g | ∀ y ∈ p ( x ≤ y ) } . Proposition 4.3.
For every formal topology S , the class Low( S ) has a set-generated frame structure definedas follows: p ∧ q = p ∩ q , p ∨ q = ( p ∪ q ) , p → q = { x ∈ S | x ∈ p → x ∈ q } , W p = ( S p ) , V p = T p , ⊤ = S , ≤ as the inclusion relation, and g = {{ x } | x ∈ S } . We extend the nucleus to general classes by taking JP := S { p | p ⊆ P } , and define operations onclasses by P ∧ Q = P ∩ Q , P ∨ Q = J ( P ∪ Q ) and P → Q = { x ∈ S | x ∈ P → x ∈ Q } . For a set-indexedcollection of classes { P x | x ∈ I } , take V x ∈ I P x = T x ∈ I P x and W x ∈ I P x = J (cid:0)S x ∈ I P x (cid:1) . Note that JP = P if P is a set and we have the Full Separation.The Heyting universe V S over S is defined inductively as follows: a ∈ V S if and only if a is a functionfrom dom a ⊆ V S to Low( S ) . For each set x , we have the canonical representation ˇ x of x defined bydom ˇ x = { ˇ y | y ∈ x } and ˇ x (ˇ y ) = ⊤ . Then define the Heyting interpretation [[ φ ]] with parameters of V S asfollows: • [[ a = b ]] = (cid:16)V x ∈ dom a a ( x ) → W y ∈ dom b b ( y ) ∧ [[ x = y ]] (cid:17) ∧ (cid:16)V y ∈ dom b b ( y ) → W x ∈ dom a a ( x ) ∧ [[ x = y ]] (cid:17) , • [[ a ∈ b ]] = W y ∈ dom b b ( y ) ∧ [[ a = y ]], • [[ ⊥ ]] = ⊥ , [[ φ ∧ ψ ]] = [[ φ ]] ∧ [[ ψ ]], [[ φ ∨ ψ ]] = [[ φ ]] ∨ [[ ψ ]], and [[ φ → ψ ]] = [[ φ ]] → [[ ψ ]], • [[ ∀ x ∈ aφ ( x )]] = V x ∈ dom a a ( x ) → [[ φ ( x )]] and [[ ∃ x ∈ aφ ( x )]] = W x ∈ dom a a ( x ) ∧ [[ φ ( x )]], • [[ ∀ xφ ( x )]] = V x ∈ V S [[ φ ( x )]] and [[ ∃ xφ ( x )]] = W x ∈ V S [[ φ ( x )]].Then the interpretation validates every axiom of CZF − and more: Theorem 4.4.
Working over
CZF − , the Heyting-valued model V S also satisfies CZF − . If S is set-presentedand Subset Collection holds, then V S | = CZF . If our background theory satisfies Full Separation, then sodoes V S .Proof. The first part of the theorem is shown by [9], so we will concentrate on the preservation of FullSeparation. For Full Separation, it suffices to see that the proof for bounded separation over V S also worksfor Full Separation, since Full Separation ensures [[ θ ]] is a set for every formula θ . (cid:3) Let us finish this subsection with some constructors, which we need in a later proof.
Definition 4.5.
For S -names a and b , up ( a, b ) is defined by dom( up ( a, b )) = { a, b } and ( up ( a, b ))( x ) = ⊤ . op ( a, b ) is the name defined by op ( a, b ) = up ( up ( a, a ) , up ( a, b )) A subset p ⊆ S is a lower set if ↓ p = p . OW STRONG IS A REINHARDT SET OVER EXTENSIONS OF CZF? 7 up ( a, b ) represents the unordeded pair { a, b } over V S . Hence the name op ( a, b ) represents the orderedpair given by a and b over V S .4.2. Double negation formal topology.
Our main tool in this paper is the Heyting-valued interpretationwith the double negation formal topology . Unlike set-sized realizability or set-represented formal topology,the double negation topology and the resulting Heyting-valued interpretation need not be absolute betweenBCST-regular sets or transitive models of
CZF − . Hence we need a careful analysis of the double negationformal topology, which is the aim of this subsection. Definition 4.6.
The double negation formal topology
Ω is the formal topology (1 , = , ⊳ ), where x ⊳ p if andonly if ¬¬ ( x ∈ p ).We can see that the class of lower sets Low(Ω) is just the power set of 1, and the nucleus of S is given bythe double complement p ¬¬ = { | ¬¬ (0 ∈ p ) } . Hence the elements of Low(Ω) is the collection of all stable subsets of 1, that is, a set p ⊆ p = p ¬¬ .We will frequently mention the relativized Heyting-valued interpretation, the definition of relativized oneis not different from the usual V Ω and [[ · ]], thus we do not introduce its definition. Notwithstanding that, itis still worth to mention the notational convention for the relativization: Definition 4.7.
Let A be a transitive model of CZF − . Then A Ω := ( V Ω ) A is the relativized Ω-valueduniverse to A . If A is a set, then ˜ A denotes the Ω-name defined by dom ˜ A := A Ω and ˜ A ( x ) = ⊤ for all x ∈ dom ˜ A .Note that the definition of ˜ A makes sense due to Lemma 4.8. We often confuse ˜ A and A Ω if context isclear. It is also worth to mention that if j is an elementary embedding, then j ( ˜ K ) = ] j ( K ), so we may write j n ( ˜ K ) instead of ^ j n ( K ).We cannot expect that Low(Ω) is absolute between transitive models of CZF − , and as a result, we do notknow whether its Heyting-valued universe V Ω and Heyting-valued interpretation [[ · ]] is absolute. Fortunately,the formula p ∈ Low(Ω) is ∆ , so it is absolute between transitive models of CZF − . As a result, we havethe following absoluteness result on the Heyting-valued universe: Lemma 4.8.
Let A be a transitive model of CZF − without Infinity. Then we have A Ω = V Ω ∩ A . Moreover,if A is a set, then A Ω is also a set.Proof. We will follow the proof of Lemma 6.1 of [19]. Let Φ be the inductive definition given by h X, a i ∈ Φ ⇐⇒ a is a function such that dom a ⊆ X and a ( x ) ⊆ a ( x ) ¬¬ = a ( x ) for all x ∈ dom a. We can see that the Φ defines the class V Ω . Furthermore, Φ is ∆ , so it is absolute between transitive modelsof CZF − . By Lemma 2.12, we have a class J such that V Ω = S a ∈ V J a , and for each s ∈ V , J s = Γ Φ ( S t ∈ s J t ).Now consider the operation Υ given byΥ( X ) := { a ∈ A | ∃ Y ∈ A ( Y ⊆ X ∧ h Y, a i ∈ Φ) } . By Lemma 2.12 again, there is a class Y such that Y s = Υ( S t ∈ s Y t ) for all s ∈ V . Furthermore, we can seethat Y s ⊆ V Ω by induction on a .Let Y = S s ∈ A Y s . We claim by induction on s that J s ∩ A ⊆ Y . Assume that J t ∩ A ⊆ Y holds for all t ∈ s . If a ∈ J s ∩ A , then the domain of a is a subset of A ∩ (cid:0)S t ∈ s J s (cid:1) , which is a subclass of Y by theinductive assumption and the transitivity of A . Moreover, for each x ∈ dom a there is u such that x ∈ Y u .By Strong Collection over A , there is v ∈ A such that for each x ∈ dom a there is u ∈ v such that x ∈ Y u .Hence dom a ⊆ S u ∈ v Y u , which implies a ∈ Y v ⊆ Y .Hence V Ω ∩ A ⊆ Y , and we have Y = V Ω ∩ A . We can see that the construction of Y is the relativizedconstruction of V Ω to A , so Y = A Ω . Hence A Ω = V Ω ∩ A . If A is a set, then Υ( X ) is a set for each set X ,so we can see by induction on a that Y a is also a set for each a ∈ A . Hence A Ω = Y = S a ∈ A Y a is also aset. (cid:3) HANUL JEON
We extended nucleus to J for subclasses of Low S , and use it to define the validity of formulas of theforcing language. We are working with the specific formal topology S = Ω, and in that case, JP for a class P ⊆ JP = S { q ¬¬ | q ⊆ P } . It is easy to see that P ⊆ JP ⊆ P ¬¬ . We also define thefollowing relativized notion for any transitive class A such that 1 ∈ A : J A P = [ { q ¬¬ | q ⊆ P and q ∈ A } . If P ∈ A , then J A P = P ¬¬ , and in general, we have P ⊆ J A P ⊆ JP ⊆ P ¬¬ . Moreover, we have Lemma 4.9.
Let A and B be transitive classes such that ∈ A, B and P ⊆ be a class. (1) A ⊆ B implies J A P ⊆ J B P . (2) If P (1) ∩ A = P (1) ∩ B , then J A P = J B P . However, the following proposition shows that we cannot prove they are the same from
CZF − : Proposition 4.10. (1) If A ∩ P (1) = 2 (it holds when A = 2 or A = V and ∆ - LEM holds), then J A P = P . (2) If P ¬¬ ⊆ JP for every class P , then ∆ - LEM implies the law of excluded middle for arbitraryformulas. (cid:3) J A has a crucial role in defining Heyting-valued interpretation, but it could differ from transitive set totransitive set. It causes absoluteness problems, which is apparently impossible to emend in general. Thefollowing lemma states facts on relativized Heyting interpretations: Lemma 4.11.
Let A ⊆ B be transitive models of CZF − . Assume that φ is a formula with parameters in A Ω . (1) If φ is bounded, then [[ φ ]] A = [[ φ ]] B . (2) If φ only contains bounded quantifications, logical connectives between bounded formulas, unbounded ∀ , and ∧ , then [[ φ ]] A = [[ φ ˜ A ]] B . (3) If every conditional of → appearing in φ is bounded, then [[ φ ]] A ⊆ [[ φ ˜ A ]] B . (4) If P (1) ∩ A = P (1) ∩ B , then [[ φ ]] A = [[ φ ˜ A ]] B .Proof. If φ is bounded, then [[ φ ]] is defined in terms of double complement, Heyting connectives betweensubsets of 1, and set-sized union and intersection. These notions are absolute between transitive sets, so wecan prove [[ φ ]] is also absolute by induction on φ . (In the case of atomic formulas, we apply the inductionon A Ω -names.)The remaining clauses follow from the induction on φ : For the unbounded ∀ , we have[[ ∀ xφ ( x )]] A = ^ x ∈ A Ω [[ φ ( x )]] A ⊆ ^ x ∈ A Ω [[ φ ˜ A ( x )]] B = [[ ∀ x ∈ ˜ Aφ ˜ A ( x )]] B . under conditions in the remaining clauses. The case for ∧ and → are similar. In the case of the second orfourth clause, we can see that the above argument raises equality.For the unbounded ∃ , we have[[ ∃ xφ ( x )]] A = J A (cid:16)[ { [[ φ ( x )]] A | x ∈ A Ω } (cid:17) ⊆ J B (cid:16)[ { [[ φ ˜ A ( x )]] B | x ∈ A Ω } (cid:17) = [[ ∃ x ∈ ˜ Aφ ˜ A ( x )]] B . Note that if A ∈ B , then S x ∈ A Ω [[ φ ˜ A ( x )]] B ∈ P (1) ∩ B , thus J B (cid:16)S x ∈ A Ω [[ φ ˜ A ( x )]] B (cid:17) = (cid:16)S x ∈ A Ω [[ φ ˜ A ( x )]] B (cid:17) ¬¬ in this case.The case for ∨ is similar to that of the unbounded ∃ . For the last clause, we need J A = J B that followsfrom P (1) ∩ A = P (1) ∩ B . (cid:3) As a final remark, note that we may understand the forcing over Ω as a double negation translation `a laforcing. See 2.3 of [11] or [6] for details.
OW STRONG IS A REINHARDT SET OVER EXTENSIONS OF CZF? 9 Double negation translation of an elementary embedding
The following section is devoted to the following theorem:
Theorem 5.1. ( CZF + Sep ) Let j : V ≺ V be an elementary embedding and let K be an inaccessible criticalpoint of j ; that is, K is inaccessible, K ∈ j ( K ) and j ( x ) = x for all x ∈ K . Then there is a Heyting-valuedmodel V Ω such that V Ω | = ZF − and it thinks j : V ≺ V is cofinal and has a critical point. We will mostly follow the proof of Ziegler [24], but we need to check his proof works on our settingsince his applicative topology does not cover Heyting algebra generated by formal topologies that are notset-presentable. We will focus on Reinhardt sets, so considering the target universe M of j might beunnecessary. Nevertheless, we will consider M and we will not assume M = V unless it is necessary.We need to redefine Heyting-valued interpretations to handle critical sets and Reinhardt sets, so we definethe interpretation of M and j in the forcing language. Since j preserves names, we can interpret j as j itself.We will interpret M as the M Ω . Thus, for example, [[ ∀ x ∈ M φ ( x )]] = V x ∈ M Ω [[ φ ( x )]]. We discussed thatthe Heyting interpretation [[ φ ]] need not be absolute between transitive sets. A cacophony of absolutenessissues causes technical trouble in an actual proof, so we want to assure [[ φ ]] = [[ φ ]] M , which follows from P (1) = P (1) ∩ M . The following lemma is due to Ziegler, and see Remark 9.51 to 9.52 of [24] for its proof: Lemma 5.2.
Let p ∈ P (1) and j : V ≺ M be an elementary embedding. Then j ( p ) = p . Especially, we have P (1) = P (1) ∩ M . Hence by Lemma 4.11, [[ φ ]] M = [[ φ M Ω ]] for any formula φ with parameters in M Ω . Thus we do not needto worry about the absoluteness issue on the Heyting interpretation.We are ready to extend our forcing language to {∈ , j, M } . For each formula φ , define[[ ∀ x ∈ M φ ( x )]] := ^ x ∈ M Ω [[ φ ( x )]] and [[ ∃ x ∈ M φ ( x )]] := _ x ∈ M Ω [[ φ ( x )]]and define the remaining as given by [9]. (We may define x ∈ M by using that this is equivalent to ∃ y ∈ M ( x = y ).) From this definition, we have an analogue of Lemma 4.26 of [24], which is useful to checkthat j is still elementary over V Ω : Lemma 5.3.
For any bounded formula φ ( ~x ) with all free variables displayed in the language ∈ (that is,without j and M ), we have [[ φ ( ~a )]] = [[ φ M Ω ( j ( ~a ))]] = [[ φ ( j ( ~a ))]] for every ~a ∈ V Ω .Proof. For the first equality, we have(1) [[ φ ( ~a )]] = j ([[ φ ( ~a )]]) = [[ φ M Ω ( j ( ~a ))]] . by Lemma 5.2. Note that the last equality of (1) follows from the induction on φ . The second equality alsofollows from the induction on φ . (cid:3) Moreover, we can check the following equalities easily:
Proposition 5.4. (1) [[ ∀ x, y ( x = y → j ( x ) = j ( y ))]] = ⊤ , (2) [[ ∀ xj ( x ) ∈ M ]] = ⊤ , (3) [[ ∀ x ( x ∈ M → ∀ y ∈ x ( y ∈ M ))]] = ⊤ .Proof. The first equality follows from [[ x = y ]] = [[ j ( x ) = j ( y )]], and the remaining two follow from the directcalculation. (cid:3) Lemma 5.5.
For every ~a ∈ V Ω and a formula φ that does not contain j or M , we have [[ φ ( ~a ) ↔ φ M ( j ( ~a ))]] = ⊤ .Proof. Lemma 5.3 shows that this lemma holds for bounded formulas φ . We will use full induction on φ toprove [[ φ ( ~a )]] = [[ φ M ( j ( ~a ))]] for all ~a ∈ V Ω . If φ is ∀ xψ ( x, ~a ), we have [[ ∀ xφ ( x, ~a )]] = V x ∈ V Ω [[ φ ( x, ~a )]]. Since0 ∈ ^ x ∈ V Ω [[ φ ( x, ~a )]] ⇐⇒ ∀ x ∈ V Ω (0 ∈ [[ φ ( x, ~a )]]) ⇐⇒ ∀ x ∈ ( V Ω ) M (0 ∈ [[ φ ( x, j ( ~a ))]]) , where the last equivalence follows from applying j to the above formula. Since the last formula is equivalentto 0 ∈ V x ∈ M Ω [[ φ ( x )]], we have V x ∈ V Ω [[ φ ( x )]] = V x ∈ M Ω [[ φ ( x )]] = [[ ∀ x ∈ M φ M ( x )]].If φ is ∃ xψ ( x, a ), we have [[ ∃ xφ ( x, a )]] = W x ∈ V Ω [[ φ ( x, ~a )]]. Moreover,0 ∈ _ x ∈ V Ω [[ φ ( x, ~a )]] ⇐⇒ ∃ p ⊆ " p ⊆ [ x ∈ V Ω [[ φ ( x, ~a )]] and 0 ∈ p ¬¬ ⇐⇒ ∃ p ⊆ p ⊆ [ x ∈ ( V Ω ) M [[ φ M ( x, j ( ~a ))]] and 0 ∈ p ¬¬ Hence 0 ∈ W x ∈ V Ω [[ φ ( x, ~a )]] if and only if 0 ∈ W x ∈ M Ω [[ φ ( x, j ( ~a ))]] (cid:3) We extended the language of set theory to treat elementary embedding j and the target universe M . Wealso expand the Full Separation Sep and Strong Collection to the extended language. Thus we need to checkthat Full Separation and Strong Collection under the extended language are also persistent under the doublenegation interpretation. We can see that the proof given by [9] and Theorem 4.4 carries over, so we havethe following:
Proposition 5.6. (1)
If we assume Full Separation for the extended language, then V Ω also thinks FullSeparation for the extended language holds. (2) V Ω thinks Strong Collection for the extended language holds. The essential property of a critical set is that it is inaccessible. Unfortunately, there is no hope to preservethe inaccessibility of a critical set. The main reason is that an inaccessible set must satisfy second-orderSet Collection, which is not preserved by forcing under Ω. Fortunately, being a critical point is preservedprovided if it is regular:
Lemma 5.7.
Let K be a regular set such that K ∈ j ( K ) and j ( x ) = x for all x ∈ K . Then [[ ˇ K ∈ j ( ˜ K ) ∧∀ x ∈ ˜ K ( j ( x ) = x )]] = ⊤ .Proof. Since ( j ( K ) is inaccessible) M , j ( K ) Ω = ( j ( K ) Ω ) M = j ( K ) ∩ V Ω is a set by Lemma 4.8. Also, K ∈ j ( K ) implies ˜ K ∈ j ( K ). Since the domain of j ( ˜ K ) = ] j ( K ) is j ( K ) ∩ V Ω , we have ˜ K ∈ dom j ( ˜ K ), whichimplies [[ ˜ K ∈ j ( ˜ K )]] = ⊤ . For the second assertion, observe that if x ∈ dom ˜ K then j ( x ) = x , so we have thedesired conclusion. (cid:3) Note that the canonical name ˇ K , is also a critical point in V Ω . However, we stick to use ˜ K , whose reasonbecomes apparent in Section 6.We do not make use of Full Separation or Subset Collection until now. However, the following proofrequires Full Separation (or at least, Separation for Σ-formulas) or REA . Lemma 5.8. If j : V ≺ V is a cofinal elementary embedding, then V Ω thinks j is cofinal.Proof. Let a ∈ V Ω . Then there is a set X such that a ∈ j ( X ). If we assume Full Separation, then X ∩ V Ω is a set and j ( X ∩ V Ω ) = j ( X ) ∩ V Ω . Let b be a name such that dom b = X ∩ V Ω and b ( y ) = 1 for all y ∈ dom b . Then [[ a ∈ j ( b )]] = ⊤ .We need some work if we assume REA instead: Take a set X such that a ∈ j ( X ). By REA , we can find aregular Y such that X ∈ Y . By Lemma 4.8, Y Ω = Y ∩ V Ω is a set. The remaining argument is identical tothe previous one. (cid:3) Combining the above lemmas shows our main theorem and more:
Theorem 5.9.
The consistency of the former implies the consistency of the latter. Here the former alwaysassume that the critical point of j is regular (but not for the latter), M can be equal to V , and we alwaysassume that j is non-trivial: (1) CZF − + j : V ≺ M and CZF − + ∆ - LEM + j : V ≺ M , (2) CZF + REA + j : V ≺ V and CZF − + ∆ - LEM + j : V ≺ V is cofinal, (3) CZF − + Sep + j : V ≺ V (is cofinal) and ZF − + j : V ≺ V (is cofinal), (4) CZF + Sep + j : V ≺ V and ZF − + j : V ≺ V is cofinal, (5) IZF + j : V ≺ M and ZF + j : V ≺ M , OW STRONG IS A REINHARDT SET OVER EXTENSIONS OF CZF? 11
Proof.
It follows from our previous lemmas in this section, Lemma 3.7, and that Heyting-valued interpreta-tion preserves
Sep and
Pow . (cid:3) Unfortunately, the above result answers little about the consistency strength of
CZF + Sep with a Rein-hardt set, since we know little about the consistency strength of ZF − with a cofinal embedding j : V ≺ V .The author thought that its consistency strength is very high from the following argument: consider theprinciple known as the Relation Reflection Scheme ( RRS ) defined by Aczel [1]. It is known that
RRS ispersistent under Heyting-valued interpretation, and
ZFC − proves the equivalence between RRS and the re-flection principle. Moreover, in an earlier version of [17], Matthews claimed that
ZFC − with the reflectionprinciple and the existence of a cofinal elementary embedding is sufficient to derive the contradiction. HenceReinhardt cardinals are incompatible with choice over ZF − with the reflection principle. However, Matthewspointed out to the author that the claim as stated above is incorrect: the problem is that the reflectionprinciple over ZFC − does not ensure that V crit j is a set, which seems necessary to derive the contradiction.In spite of that, we can still see that assuming DC µ -scheme for every cardinal µ proves there is no cofinalelementary embedding over ZFC − .Can we see the revised result as a form of Kunen’s inconsistency phenomenon, so that it is evidenceof the consistency strength of ZF − with a cofinal elementary embedding? We can still prove that ZFC − and ZFC − with DC µ -scheme for all µ are equiconsistent since L satisfies global choice. We know that L iscompatible with small large cardinals over ZFC , so we may guess the same holds for
ZFC − . However, wedo not aware well about large cardinals over ZFC − to conclude that the consistency strength of ZF − witha cofinal elementary is high. Even worse, L is not compatible with large large cardinals. Thus we cannotextend the same argument further. Analyzing the notion of large cardinals over ZFC − and their consistencywith DC µ -scheme and reflection principles would be an interesting topic, but beyond the scope of this paper. Question 5.10.
What is the exact consistency strength of
CZF + Sep with a Reinhardt set?Despite that, we will see in the next section that
CZF + Sep with a Reinhardt set is still quite strong.6.
An analysis on the critical point
In the previous section, we saw that non-trivial elementary embeddings over
CZF + Sep have a strongconsistency strength. However, we observed nothing about how much large cardinal properties of the criticalpoint of an j are preserved by the double negation translation. In this section, we will extract the largecardinal properties of the critical point of an elementary embedding. Especially, we will focus on the criticalpoint of a Reinhardt embedding.The main result of this section is as follows: Theorem 6.1.
Let j : V ≺ M be a elementary embedding with an inaccessible critical point K . Then V Ω thinks ∀ n, m ∈ ω ( n < m → j n ( ˜ K ) is BCST-regular) j m ( ˜ K ) .Furthermore, if j : V ≺ V , then V Ω also thinks j ↾ j n ( ˜ K ) ∈ j n +1 ( ˜ K ) . The main strategy of this theorem is to internalize the proof of V Ω | = CZF − into a BCST-regular set.We mostly follow the proof of [9], but for the sake of verification, we will provide most of the detail ofrelevant lemmas and their proof. Throughout this section, A ∈ B sets of the same power set of 1 (i.e., P (1) ∩ A = P (1) ∩ B ) such that A is BCST-regular and B is a transitive model of CZF − , and R ∈ B is amulti-valued function unless specified. Lemma 6.2.
Let a ∈ A , R : a ⇒ A , Q ⊆ a × A , and Q ∈ B . Moreover, assume that (1) h x, y i ∈ R → y ⊆ Q x = { y | h x, y i ∈ Q } , and (2) (Monotone Closeness) h x, y i ∈ R and y ⊆ z ⊆ Q x implies h x, z i ∈ R ,then there is f ∈ A ∩ a A such that h x, f ( x ) i ∈ R for all x ∈ a .Proof. By Lemma 2.9, there is b ∈ A such that b ⊆ R and b : a ⇒ A . Take f as x S b x = S { y | h x, y i ∈ b } .Then f ∈ A , since A satisfies Union and the second-order Replacement. Moreover, by monotone-closednessof R , we have h x, f ( x ) i ∈ R for all x ∈ a . (cid:3) Lemma 6.3.
Let P ⊆ A , P ∈ B , a ∈ A and R : a ⇒ A ∩ P ( P ) . Furthermore, assume that R is monotoneclosed , that is, y, z ∈ A , y ⊆ z ⊆ P , and h x, y i ∈ R implies h x, z i ∈ R . Then there is b ∈ A such that b ⊆ P and h x, b i ∈ R for all x ∈ a . Proof.
Applying Lemma 6.2 to R provides a function f ∈ A ∩ a A such that h x, f ( x ) i ∈ R for all x ∈ A . Nowtake b = S { f ( x ) | x ∈ a } ∈ A , then we have h x, b i ∈ R by monotone closeness of R . (cid:3) The following lemma has a critical role in the proof of our theorem. Moreover, this lemma requires P (1) ∩ A = P (1) ∩ B : Lemma 6.4.
Let a ∈ A Ω , R : a ⇒ A , R ∈ B and p ∈ A be such that p ⊆ and p = p ¬¬ . Define P = {h x, y, t i ∈ dom a × A Ω × | t ∈ ( p ∧ a ( x ) ∧ [[ op ( x, y ) ∈ R ]] B ) } . Furthermore, assume that we have p ⊆ [[ R : a ⇒ ˜ A ]] B . Then there is r ∈ A such that r ⊆ P and p ∧ a ( x ) ⊆ { t | ∃ y ∈ A Ω h x, y, t i ∈ r } ¬¬ .Proof. Let Q = {h x, t i | ∃ y ∈ A Ω ( h x, y, t i ∈ P ) } and Q x = { t | h x, t i ∈ Q } ⊆
1. Then Q x ∈ P (1) ∩ B = P (1) ∩ A . From p ⊆ [[ R : a ⇒ ˜ A ]] B , we can deduce p ∧ a ( x ) ⊆ Q ¬¬ x . By applying Lemma 6.2 to the relation {h x, v i ∈ dom a × ( P (1) ∩ A ) | p ∧ a ( x ) ⊆ v ¬¬ and v ⊆ Q x } we have a function f ∈ dom a A ∩ A such that p ∧ a ( x ) ⊆ f ( x ) ¬¬ and f ( x ) ⊆ Q x for all x ∈ dom a . (Thecondition P (1) ∩ B = P (1) ∩ A is necessary to ensure the above relation is a multi-valued function of domaindom a .) Now let q = {h x, t i | x ∈ dom a and t ∈ f ( x ) } . Then ∀h x, t i ∈ q ∃ y ∈ A Ω ( h x, y, t i ∈ P ) holds. That is, P : q ⇒ A Ω . By Lemma 2.9, there is r ∈ A such that r ⊆ P and r : q ⇒ A Ω . It is easy to see that r satisfies the desired property. (cid:3) There is some technical note for the proof: there is no need that P , Q , and Q x are definable over A ingeneral. The reason is that we do not know either R or [[ · ]] B is accessible from A . However, we do not needto care about it since we are relying on the second-order Strong Collection over A . Theorem 6.5.
Let A be a BCST-regular set. Then B Ω thinks ˜ A is BCST-regular.Proof. It is easy to see that B Ω thinks ˜ A is transitive, and closed under Pairing, Union and Binary Intersec-tion. Hence it remains to show that B Ω thinks ˜ A satisfies second-order Strong Collection, that is,[[ ∀ a ∈ ˜ A ∀ R [ R : a ⇒ ˜ A → ∃ b ∈ ˜ A ( R : a ⇔⇒ b )]]] B = ⊤ . Take a ∈ dom ˜ A , R ∈ B Ω and p ∈ A such that p ⊆ p = p ¬¬ . We claim that if p ⊆ [[ R : a ⇒ ˜ A ]] B ,then there is b ∈ dom ˜ A such that p ⊆ [[ R : a ⇔⇒ b ]] B . By Lemma 6.4, we have r ∈ A such that r ⊆ P and p ∧ a ( x ) ⊆ { t | ∃ y h x, y, z i ∈ r } ¬¬ . Define b such that dom b = { y | ∃ x, t ( h x, y, t i ∈ r ) } and b ( y ) = { | ∃ x h x, y, i ∈ r } ¬¬ . for y ∈ dom b . (Note that b ( y ) ∈ A .) Then we have p ⊆ [[ R : a ⇔⇒ b ]] B . (cid:3) Hence we have
Corollary 6.6. [[ j n ( ˜ K ) is BCST-regular]] j m ( K ) = ⊤ . Furthermore, [[( j n ( ˜ K ) is BCST-regular) j m ( ˇ K ) ]] = ⊤ .Proof. The first statement follows from Theorem 6.5 by taking A = K and B = j m ( K ) for n = 1, andapplying j n − (cid:3) However, it does not directly result in our desired theorem, since we do not know Ω forces j n ( ˜ K ) is uniformly BCST-regular. We need some work to see this.There are two possible meanings of j n ( K ): the first is applying j n times to K . Here n must be a naturalnumber over the metatheory, and this description lacks a way to describe the sequence h j n ( K ) | n ∈ ω i . Thesecond way to see j n ( K ) is to understand it as it is given by the following recursion: j ( K ) = K and j n +1 ( K ) = j ( j n ( K )) . Thus the formal statement of φ ( j n ( K )) is ∃ f [dom f = ω ∧ f (0) = K ∧ ∀ m ∈ ω ( f ( m + 1) = j ( f ( m )))] ∧ φ ( f ( n )) . OW STRONG IS A REINHARDT SET OVER EXTENSIONS OF CZF? 13
Theorem 6.7. V Ω thinks the following statement is valid: (2) ∃ f [ f is a function ∧ dom f = ω ∧ f (0) = K ∧ ∀ m ∈ ω ( f ( m + 1) = j ( f ( m )))] ∧ ∀ n, m ∈ ω [ n < m → ( f ( n ) is BCST-regular) f ( m ) ] . Proof.
Let dom f = { op (ˇ n, j n ( ˜ K )) | n ∈ ω } . We claim that f witnesses our theorem. The first threeconditions are easy to prove. To see the last condition, let n, m ∈ ω . We can show the following facts byinduction on m :(1) If 0 ∈ [[ˇ n < ˇ m ]] then n < m .(2) [[ f ( m ) = j m ( ˜ K )]] = ⊤ .By combining these facts with Corollary 6.6, we have [[ˇ n < ˇ m ]] ⊆ [[( f ( n ) is BCST-regular) f ( m ) ]]. Hence theresult follows. (cid:3) Hence ˜ K in V Ω has the following reflection property: For every n < m , j m ( ˜ K ) | = ( j n ( ˜ K ) | = CZF − ).How much is this reflection principle strong? To see this, assume that we started from CZF + Sep with anelementary embedding j : V ≺ V and produced the Heyting interpretation under Ω. Then the Heytinginterpretation interprets ZF − with a cofinal elementary embedding j : V ≺ V . By Theorem 6.1, a criticalpoint ˜ K of j satisfies j ( ˜ K ) | = ( ˜ K | = CZF − ). Due to the help of the classical logic, we have j ( ˜ K ) | =( ˜ K | = ZF − ). For each x ∈ ˜ K , we have j ( ˜ K ) | = ∃ X ( x ∈ X ∧ X | = ZF − ). By the property of j , we have˜ K | = ∀ x ∃ X ( x ∈ X ∧ X | = ZF − ). That is, ˜ K satisfies REA . Since
REA implies the Axiom of Subset Collection,which is equivalent to Power Set in the classical context, we have ˜ K | = ZF and j ( ˜ K ) | = ( ˜ K | = ZF )! Wemay extend it further to stronger notions of large cardinal properties, like inaccessibility, Mahloness, orindescribability. However, the following example describes there is a limit of large cardinal property we canachieve from the reflection property of ˜ K : Example 6.8.
Work over ZF with a Reinhardt cardinal. Let j : V ≺ V be an elementary embedding and κ = crit j . Take K = L κ . Since κ is a critical cardinal, it is strongly inaccessible by Proposition 3.3 of [14].Hence L j ( κ ) also thinks κ is inaccessible. Especially, L j ( κ ) thinks V κ = L κ is a model of ZFC .Since L is incompatible with large cardinals stronger than the existence of 0 ♯ , the above example showsthe previous argument with K does not yield large cardinal properties stronger than the existence of 0 ♯ .However, it does not mean there is no room for stronger properties of K if j has a stroger property. Thefollowing result shows we can extract more large cardinal properties from K if j : V ≺ V , by bringing theelementary embedding j over CZF to an elementary embedding of j ω ( ˜ K ) in the Heyting interpretation: Lemma 6.9 (
CZF − ). Let j : V ≺ V . Then V Ω thinks j ↾ j n ( ˜ K ) ∈ j n +1 ( ˜ K ) for all n ∈ ω .Proof. Observe that j n +1 ( K ) is regular and j ↾ j n ( K ) : j n ( K ) → j n +1 ( K ) is a multi-valued function. ByLemma 2.9, there is b ∈ j n ( K ) such that b ⊆ j ↾ j n ( K ) and b : j n ( K ) ⇒ j n +1 ( K ). Since j ↾ j n ( K ) is afunction, b is also a function of domain j n ( K ). Hence j ↾ j n ( K ) = b ∈ j n +1 ( K ).From the previous argument, we also have j ↾ j n ( ˜ K ) = ( j ↾ j n ( K )) ↾ j n ( ˜ K ) ∈ j n +1 ( K ). Now let c be aname such that dom c = { op ( x, y ) | h x, y i ∈ j ↾ j n ( ˜ K ) } andand c ( x ) = ⊤ for all x ∈ dom c . By definition of c , c ∈ j n +1 ( ˜ K ). Moreover, it is easy to see that V Ω thinks c is a function of the domain j n ( ˜ K ), and direct calculation shows [[ ∀ x ∀ y op ( x, y ) ∈ c → j ( x ) = y ]] = ⊤ . Hence[[ c = j ↾ j n ( ˜ K )]] = ⊤ . (cid:3) Note that this lemma does not work for general j : V ≺ M , since we do not know j n ( K ) is regular in V . Theorem 6.10 (
CZF + Sep ). V Ω thinks j ω ( ˜ K ) := S n ∈ ω j n ( ˜ K ) satisfies ZF with a cofinal elementary em-bedding j from itself to itself. Moreover, j ω ( ˜ K ) satisfies ∆ -Separation with j be allowed to appear.Proof. Let ˜ K = j ( ˜ K ) and Λ := j ω ( ˜ K ). We will rely on a completely internal argument to V Ω , which is amodel of ZF − .Assume that Λ | = φ ( ~a ), where φ ( ~x ) is a ∆ -formula with all free variables displayed in the language ∈ (i.e., without j .) Since j (Λ) = Λ, we have Λ | = φ ( j ( ~a )). Hence j : V ≺ V . It remains to show that Λ satisfies ∆ -separation for formulas with j be allowed to appear. Let a ∈ Λ,then there is n such that a ∈ j n ( ˜ K ). For a bounded formula φ with parameters in j n ( ˜ K ), let φ ′ be theformula obtained by every occurrence of j to j ↾ j n ( ˜ K ). Then { x ∈ a | φ ( x ) } = { x ∈ a | φ ′ ( x ) } ∈ j n +1 ( ˜ K ).Thus Λ satisfies ∆ -separation for formulas with j . (cid:3) How much is the resulting theory strong? We can see that Λ is a model of ZF with the Wholeness axiomfor ∆ -formulas WA proposed by Hamkins [13]. The author does not know the exact consistency strengthof ZF + WA in ZFC -context. However, we can still find a lower bound of it: we can see that Λ also thinks κ = rank K is a critial point of j , and the critical sequence defined by κ = κ and κ n +1 = j ( κ n ) is cofinalover Ord. (Note that cofinal sequence is still definable, although it may not a set. See Proposition 3.2 of [7]for details.) From this, we have Lemma 6.11 ( ZF + WA ). If the critical sequence is cofinal, then κ is extendible.Proof. Let η be an ordinal. Take n such that η < j n ( κ ), then j n : V κ + η ≺ V j n ( κ + η ) and crit j n = κ . Hence κ satisfies η -extendibility. (cid:3) By an easy reflection argument, we can see also see that ZF + WA with the cofinal critical sequence provesnot only there is an extendible cardinal, but also the consistency of ZF with a proper class of extendible car-dinals, an extendible limit of extendible cardinals, and many more. Since extendible cardinals are preservedby Woodin’s forcing (Theorem 226 of [23]), we have a lower bound of the consistency strength of ZF + WA ,e.g., ZFC with there is a proper class of extendible cardinals.7.
Concluding Questions
We may wonder how to find the upper bound of the consistency strength of
CZF + Sep with a non-trivialelementary embedding, in terms of extensions of
ZFC or ZFC − . Most construction of interpretations of CZF from classical theories rely on realizability, and employ type-theoretic interpretations (like [18] or [12]) orset-as-tree interpretation (like [16] or functional realizability model over
ZFC − given by Swan [22]. ) Theseinterpretations usually satisfy the Axiom of Subcountability , which states every set is an image of a subsetof ω . However, Ziegler [24] proved that the existence of non-trivial elementary embedding j : V ≺ M contradicts with the Axiom of Subcountability. The author thinks that the Axiom of Subcountability comesfrom that we are using countable pca to construct interpretations, but delimiting the size of pca does notguarantee that we can reach the upper bound of CZF with a non-trivial elementary embedding. We donot know about the type-theoretic analogue of
CZF with an elementary embedding, so we do not know wecan employ type-theoretic interpretations. Functional realizability or set-as-tree interpretations also have aproblem. We need a pca of size greater than not only the critical point, but also its successive application tothe elementary embedding to ensure the critical set exists under the interpretation, since the size of the pcadelimits the size of every set under the interpretation. However, nothing much is known about realizabilityunder large pcas. Especially, these large pcas are not fixed under the elementary embedding, which makesthem hard to handle. One possible way to control the large pcas under the elementary embedding j is toadd j into the pca. However, finding the realizer of φ M ( j ( x )) → φ ( x ) seems not easy. Question 7.1.
What is the upper bound of the consistency strength of
CZF + Sep with a critical set or aReinhardt set? Can we find the bound in terms of large cardinals compatible with the Axiom of Choice?Our method is also restricted to the analysis on
CZF + Sep , mainly because the forcing over the doublenegation topology only provides ∆ - LEM . Full Separation is necessary to turn it to the full excluded middle.We may ask we can analyze the strength of
CZF with large large set axioms without any help of fullSeparation.
Question 7.2.
Is there any non-trivial result for the consistency strength of
CZF with a critical set or aReinhardt set?Improving the lower bound of the consistency strength of
CZF + Sep with a Reinhardt set is also an issue.We have not made use of the full strength of the resulting theory stated in Proposition 1.1 to get the lowerbound of the consistency strength. For example, we never used the cofinality of j . Even worse, we did not Swan worked over
ZFC , but his proof for soundness of functional realizability still works over
ZFC − . Also, note that hisfunctional realizability is a part of his two-stage Kripke model. EFERENCES 15 extract the full consistency strength of ZF + WA : extendibility (or a proper class of extendibles) is far below WA . One may try to force the Axiom of Choice over ZF + WA , by using Woodin’s forcing (see Theorem226 of [23]) and a method given by Hamkins [13]. However, the quotient of the limit stages of Woodin’sforcing is not sufficiently γ -closed, so Hamkins’ argument is not applied well. Question 7.3.
Can we obtain a better lower bound of the consistency strength of ZF − with a cofinalembedding j : V ≺ V , with a critical point K | = ZF such that j ω ( K ) | = ( K | = ZF − )? Especially, are ZF + WA and ZFC + WA equiconsistent? Acknowledgements
The author want to thank Richard Matthews for pointing out an error and providing comments on earlierversions of this manuscript.
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