End Extending Models of Set Theory Via Power Admissible Covers

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In this paper we explore the variant power admissible cover of the notion of admissible cover in order to obtain new results in the model theory of set theory. The main inspiration for our results on end extensions arose from our joint work with Kaufmann [EKM] on automorphisms of models of set theory (see Theorem 5.10). The canonical extension KPP of Kripke-Platek set theory KP plays a key role in our work. KPP is intimately related to Friedman’s so-called power admissble system PAdms, whose well-founded models are the the so-called power admissible sets [Fri].3 These two systems can accommodate constructions by Σ1-recursions relative to the power set operation. The system KPP has been closely studied by Mathias [Mat01] and Rathjen P [Rat14], [Rat20]. In the latter paper Rathjen proves that Σ1 -Foundation is provable in KPP + AC (where AC is the axiom of choice). The highlights of the paper are as follows. In Corollary 3.3 we refine Rathjen’s P P aforementioned result by showing that Σ1 -Foundation is provable outright in KP . The rudiments of power admissible covers are developed in Section 4. In Section 5 the machinery of power admissible covers is put together with results of earlier sections to establish new results about powerset-preserving end extensions and rank extensions of models of set theory. For example in Theorem 5.7 we show that every countable model of M |= KPP has a topless rank extension, i.e., M has a proper rank extension N |= KPP such that OrdN \ OrdM has no least element. This result generalises a classical theorem of Friedman that shows that every countable well-founded model of KPP has a topless rank extension.

2 Background

We use L throughout the paper to denote the language {∈, =} of set theory. We will make reference to generalisations of the Lévy hierarchy of formulae in languages extending L that possibly contain constant and function symbols. ′ ′ Let L be a language extending L. We use ∆0(L ) to denote the smallest class of L′-formulae that is closed under the connectives of propositional logic and quantification in the form ∃x ∈ t and ∀x ∈ t, where t is a term of L′ and x is a variable that does ′ ′ ′ not appear in t. The classes Σ1(L ), Π1(L ), Σ2(L ), . . . are defined inductively from ′ ∆0(L ) in the usual way. We will write ∆0,Σ1, Π1, . . . instead of ∆0(L), Σ1(L), Π1(L), ′ ′ ..., and we will use Π∞ and Π∞(L ) to denote the class of all L-formulae and L - ′ ′ formulae respectively. An L -formula is ∆n(L ), for n > 0, if it is equivalent to both a ′ ′ Σn(L )-formula and a Πn(L )-formula. P The class ∆0 is the smallest class of L-formulae that is closed under the connectives of propositional logic and quantification in the form Qx ⊆ y and Qx ∈ y where Q is P P P ∃ or ∀, and x and y are distinct variables. The Takahashi classes ∆1 ,Σ1 , Πn , ...are P defined from ∆0 in the same way as the classes ∆1,Σ1, Π1, ...are defined from ∆0. If Γ is a collection of L′-formulae and T is an L′-theory, then we write ΓT for the class of L′-formulae that are provably in T equivalent to a formula in Γ. We will use capital calligraphic font letters (M, N , . . . ) to denote L-structures. If M is an L-structure, then, unless we explicitly state otherwise, M will be used to denote the underlying set of M and EM will be used to denote the interpretation of ∈ in M.

3The precise relationship between Friedman’s system and KPP is worked out in Section 6.19 of [Mat01].

2 Let L′ be a language extending L and let M be an L′-structure with underlying set M. If a ∈ M, then a∗ is deﬁned as follows:

a∗ := {x ∈ M | M |= (x ∈ a)}, as long as the structure M is clear from the context. Let Γ be a class of formulae. We say that A ⊆ M is Γ-deﬁnable over M if there exists a Γ-formula φ(x, ~z) and ~a ∈ M such that A = {x ∈ M | M |= φ(x,~a)}. Let M and N be L′-structures. We will partake in the common abuse of notation and write M⊆N if M is a substructure of N .

• We say that N is an end extension of M, and write M⊆e N , if M⊆N and for all x,y ∈ N, if y ∈ M and N |= (x ∈ y), then x ∈ M. • P We say that N is a powerset-preserving end extension of M, and write M⊆e N , if M⊆e N and for all x,y ∈ N, if y ∈ M and N |= (x ⊆ y), then x ∈ M. • We say that N is a topless powerset-preserving end extension of M, and write P P ∗ M⊆topless N , if M⊆e N , M 6= N and for all c ∈ N, if c ⊆ M, then c ∈ M. • We say that N is a blunt powerset-preserving end extension of M, and write P P M⊆blunt N , if M⊆e N , M 6= N and N is not a topless powerset-preserving end extension of M.

Let Γ be a class of L-formulae. The following deﬁne the restriction of the ZF-provable schemes Separation, Collection, and Foundation to formulae in the class Γ:

(Γ-Separation) For all φ(x, ~z) ∈ Γ,

∀~z∀w∃y∀x(x ∈ y ⇐⇒ (x ∈ w) ∧ φ(x, ~z)).

(Γ-Collection) For all φ(x, y, ~z) ∈ Γ,

∀~z∀w((∀x ∈ w)∃yφ(x, y, ~z) ⇒∃C(∀x ∈ w)(∃y ∈ C)φ(x, y, ~z)).

(Γ-Foundation) For all φ(x, ~z) ∈ Γ,

∀~z(∃xφ(x, ~z) ⇒∃y(φ(y, ~z) ∧ (∀x ∈ y)¬φ(x, ~z))).

If Γ= {x ∈ z} then we will refer to Γ-Foundation as Set-foundation.

We will also make reference to the following fragments of Separation and Foundation for formulae that are ∆n with parameters:

(∆n-Separation) For all Σn-formulae, φ(x, ~z), and for all Πn-formulae, ψ(x, ~z),

∀~z(∀~x(φ(x, ~z) ⇐⇒ ψ(x, ~z)) ⇒∀w∃y∀x(x ∈ y ⇐⇒ (x ∈ w) ∧ φ(x, ~z))).

(∆n-Foundation) For all Σn-formulae, φ(x, ~z), and for all Πn-formulae, ψ(x, ~z),

∀~z(∀x(φ(x, ~z) ⇐⇒ ψ(x, ~z)) ⇒ (∃xφ(x, ~z) ⇒∃y(φ(y, ~z) ∧ (∀x ∈ y)¬φ(x, ~z))).

3 P P Similar deﬁnitions can also be used to express ∆n -Separation and ∆n -Foundation. We use TCo to denote the axiom that asserts that every set is contained in a transitive set. We will consider extensions of the following subsystems of ZFC:

• S1 is the L-theory with axioms: Extensionality, Emptyset, Pair, Union, Set diﬀerence, and Powerset.

• M is obtained from S1 by adding TCo, Inﬁnity, ∆0-Separation, and Set-foundation. • Mac is obtained from M by adding AC (the axiom of choice).

• M− is obtained from M by removing Powerset.

• KP is the L-theory with axioms: Extensionality, Pair, Union, ∆0-Separation, ∆0- Collection and Π1-Foundation. − • KP is obtained from KP by removing Π1-Foundation. • KPI is obtained KP by adding Inﬁnity. • P P P KP is obtained from M by adding ∆0 -Collection and Π1 -Foundation.

• MOST is obtained from M by adding Σ1-Separation and AC. In subsystems of ZFC that include Inﬁnity we can also consider the following restriction of Γ-Foundation:

(Γ-Foundation on ω) For all φ(x, ~z) ∈ Γ,

∀~z((∃x ∈ ω)φ(x, ~z) ⇒ (∃y ∈ ω)(φ(y, ~z) ∧ (∀x ∈ y)¬φ(x, ~z))).

The second family of theories that we will be concerned with are extensions of the variant of Kripke-Platek Set Theory with urelements that is introduced in [Bar75, Ap- pendix]. Let L∗ be obtained from L by adding a second binary relation E, a unary predicate U, and a unary function symbol F. The intended interpretation of U is to distinguish urelements from sets. The binary relation E is intended to be a membership relation that holds between urelements, and ∈ is intended to be a membership relation that can hold between sets or urelements and sets. ∗ ∗ ∗ Let LP be obtained from L by adding a new unary function symbol P. An LP- A A A A structure is a structure AM = hM; A, ∈ , F , P i, where M = hM, E i, M is the extension of U, A is the extension of ¬U, ∈A is the interpretation of ∈, EA is the interpretation of E, FA is the interpretation of F, and PA is the interpretation of P. L∗-structures will be presented in the same format, but without an interpretation ∗ ∗ A of P. The L - and LP-theories presented below will ensure that E ⊆ M × M, and ∈A ⊆ (M ∪ A) × A. ∗ ∗ Following [Bar75], we simplify the presentation of L - and LP-formulae by treating these languages as two-sorted rather than one-sorted. ∗ ∗ When writing L - and LP-formulae we will use the convention below of Barwise [Bar75].

4 • The variables p,q,p1,... range over elements of the domain that satisfy U (urelements).

• the variables a, b, c, d, f, . . . range over elements of the domain that satisfy ¬U (sets); and

• the variables x,y,z,w... range over all elements of the domain.

Therefore, ∀p(· · · ) is an abbreviation of ∀x(U(x) ⇒ ··· ), ∃a(· · · ) is an abbreviation of ∃x(¬U(x) ∧··· ), etc. ∗ ∗ In section 4, we will see that certain L-structures can interpret L - and LP -structures in which the urelements are isomorphic to the original L-structure. It is this interaction that motivates our unorthodox convention of using EM, EN , . . . to denote the interpretation of ∈ in the L-structures M, N , . . . It should be noted that this convention diﬀers from Barwise [Bar75] where E is consistently used to denote the interpretation of ∈ in L-structures. The following are analogues of axioms, fragments of axiom schemes and fragments ∗ ∗ of theorem schemes of ZFC in the languages L and LP: (Extensionality for sets) ∀a∀b(a = b ⇐⇒ ∀x(x ∈ a ⇐⇒ x ∈ b)).

(Pair) ∀x∀y∃a∀z(z ∈ a ⇐⇒ z = x ∨ z = y).

(Union) ∀a∃b(∀y ∈ a)(∀x ∈ y)(x ∈ b).

∗ Let Γ be a class of LP-formulae. (Γ-Separation) For all φ(x, ~z) ∈ Γ,

∀~z∀a∃b∀x(x ∈ b ⇐⇒ (x ∈ a) ∧ φ(x, ~z)).

(Γ-Collection) For all φ(x, y, ~z) ∈ Γ,

∀~z∀a((∀x ∈ a)∃yφ(x, y, ~z) ⇒∃b(∀x ∈ a)(∃y ∈ b)φ(x, y, ~z)).

(Γ-Foundation) For all φ(x, ~z) ∈ Γ,

∀~z(∃xφ(x, ~z) ⇒∃y(φ(y, ~z) ∧ (∀w ∈ y)¬φ(w, ~z))).

The following axiom in the language L∗ describes the desired behaviour of the function symbol F: (†) ∀p∀x(xEp ⇐⇒ x ∈ F(p)) ∧∀a(F(a)= ∅).

∗ The next axiom, in the language LP, says that the function symbol P is the usual powerset function: (Powerset) ∀a∀b(b ∈ P(a) ⇐⇒ b ⊆ a). We will have cause to consider the following theories:

∗ • KPUCov is the L -theory with axioms: ∃a(a = a), ∀p∀x(x∈ / p), Extensionality ∗ ∗ ∗ for sets, Pair, Union,∆0(L )-Separation, ∆0(L )-Collection, Π1(L )-Foundation and (†).

5 • P ∗ ∗ KPUCov is the LP-theory obtained from KPUCov by adding Powerset, ∆0(LP)- ∗ ∗ Separation, ∆0(LP)-Collection and Π1(LP)-Foundation.

Deﬁnition 2.1 Let M = hM, EMi be an L-structure. An admissible set covering M is an L∗-structure

A A AM = hM; A, ∈ , F i |= KPUCov. such that ∈A is well-founded. ∗ A power admissible set covering M is a structure LP-structure A A A A P M = hM; A, ∈ , F , P i |= KPUCov such that ∈A is well-founded. We use CovM = hM; AM, ∈, FMi to denote the smallest admissible set covering M whose membership relation ∈ coincides with the membership relation of the metatheory. C P We use ovM = hM; AM, ∈, FM, PMi to denote the smallest power admissible set covering M whose membership relation coincides with the membership relation of the metatheory. A A Note that if AM = hM; A, ∈ , F ,...i is an admissible set covering M, then AM is isomorphic to a structure whose membership relation ∈ is the membership relation of the metatheory.

Deﬁnition 2.2 Let M = hM, EMi be an L-structure, and let

A A A A P M = hM; A, ∈ , F , P i |= KPUCov.

A We use WF(A) to denote the largest B ⊆e A such that hB, ∈ i is well-founded. A ∗ The well-founded part of M is the LP-structure A A A WF(AM)= hM; WF(A), ∈ , F , P i.

A ∗ Note that WF( M) is always isomorphic to an LP-structure whose membership relation ∈ coincides with the membership relation of the metatheory.

− − As usual, in the theories M , KP and KPUCov the ordered pair hx,yi is coded by the set {{x}, {x,y}}. This deﬁnition ensures that there is a ∆0-formula OP(x) that says that x is an ordered pair, and functions

fst(hx,yi)= x and snd(hx,yi)= y, whose graphs are deﬁned by ∆0-formulae. In KPUCov the rank function, ρ, and support function, sp, are deﬁned by recursion:

ρ(p) = 0 for all urelements p, and ρ(a) = sup{ρ(x) + 1 | x ∈ a} for all sets a;

sp(p)= {p} for all urelements p, and sp(a)= sp(x) for all sets a. x∈a [ ∗ The theory KPUCov proves that both of these are total and their graphs are ∆1(L ). In the theory KP, in which everything is a set, the rank function, ρ, is ∆1 and remains

6 provably total. We say that x is a pure set if sp(x)= ∅. We say that x is an ordinal if x is a hereditarily transitive pure set; where:

Transitive(x) ⇐⇒ ¬U(x) ∧ (∀y ∈ x)(∀z ∈ y)(z ∈ x), and

Ord(x) ⇐⇒ (Transitive(x) ∧ (∀y ∈ x)(Transitive(y)). ∗ Therefore, both ‘x is transitive’ and ‘x is an ordinal’ can be expressed using ∆0(L )- formulae. In the theories M− and KP, we can omit the reference to the predicate U in the deﬁnition of ‘x is transitive’, thus making both the property of being transitive and the property of being an ordinal into ∆0 properties. The rank function allows us to strengthen the notion of powerset-preserving end extensions for models of KP. Let L′ be a language extending L. Let M and N be L′-structures that satisfy KP. • rk P We say that N is a rank extension of M, and write M ⊆e N , if M ⊆e N and for all x,y ∈ N, if y ∈ M and N |= (ρ(x) ≤ ρ(y)), then x ∈ M. • rk We say that N is a topless rank extension of M, and write M ⊆topless N , if rk ∗ M⊆e N , M 6= N and for all c ∈ N, if c ⊆ M, then c ∈ M. • rk rk We say that N is a blunt rank extension of M, and write M⊆blunt N , if M⊆e N , M 6= N and N is not a topless rank extension of M.

− − Note that KP is a subtheory of M + ∆0-Collection. We will make use of the following results: • P A consequence of [Mat01, Theorem Scheme 6.9(i)] is that M proves ∆0 -Separation. • The availability of the collection scheme for the relevant class of formulae means that the class of formulae that are equivalent to a Σ1-formula and the class of formulae that are equivalent to a Π1-formula are closed under bounded quantifica- − P tion in the theory KP ; the class of formulae equivalent to a Σ1 -formula and the P class of formulae that are equivalent to a Π1 -formula are closed under bounded − P quantification in the theory M +∆0 -Collection; the class of formulae equivalent ∗ ∗ to a Σ1(L )-formula and the class of formulae that are equivalent to a Π1(L )- formula are closed under bounded quantification in the theory KPUCov; and the ∗ class of formulae equivalent to a Σ1(LP)-formula and the class of formulae that ∗ are equivalent to a Π1(LP)-formula are closed under bounded quantification in the P theory KPUCov. • The proof of [Bar75, I.4.4] shows:

− 1. KP ⊢ Σ1-Collection; − P P 2. M +∆0 -Collection ⊢ Σ1 -Collection; ∗ 3. KPUCov ⊢ Σ1(L )-Collection; and P ∗ 4. KPUCov ⊢ Σ1(LP)-Collection. • The argument used in [Bar75, I.4.5] shows:

− 1. KP ⊢ ∆1-Separation;

7 − P P 2. M +∆0 -Collection ⊢ ∆1 -Separation; ∗ 3. KPUCov ⊢ ∆1(L )-Separation; and P ∗ 4. KPUCov ⊢ ∆1(LP)-Separation. The following is Mathias’s calibration [Mat01, Proposition Scheme 6.12] of [Tak, Theorem 6].

Theorem 2.3 The following inclusions hold between the indicated classes of formulae (n ≥ 1):

P MOST P S1 1. Σ1 ⊆ (∆1 ) and ∆0 ⊆ ∆2 . P MOST 2. Σn+1 ⊆ (Σn ) . P MOST 3. Πn+1 ⊆ (Πn ) . P MOST 4. ∆n+1 ⊆ (∆n ) .

P S1 5. Σn ⊆ Σn+1.

P S1 6. Πn ⊆ Πn+1.

P S1 7. ∆n ⊆ ∆n+1.

• As noted by Mathias in [Mat01, Corollary 6.15], MOST+Π1-Collection and MOST+ P ∆0 -Collection axiomatise the same theory. This fact follows from part 1 of Theo- rem 2.3 and the results mentioned above and will be repeatedly used throughout this paper.

P P The Σ1 -Recursion Theorem [Mat01, Theorem 6.26] shows that the theory KP is capa- ble of constructing the levels of the cumulative hierarchy:

V0 = ∅ and for all ordinals α,

Vα+1 = P(Vα) and, if α is a limit ordinal, Vα = Vβ. β∈α [ More precisely, let RK(α, f) be the L-formula:

(f is a function) ∧ (α is an ordinal) ∧ dom(f)= α ∧ (β is a limit ordinal) ⇒ f(β)= f(γ) ∧ (∀β ∈ α) γ∈β . (∃γ ∈ β)(β =γ + 1) ⇒ ((∀x ⊆ f(γ))(x ∈ f(β))S∧ (∀x ∈ f(β))(x ⊆ f(γ))) ! P Note that RK(f, α)isa∆0 -formula.

Lemma 2.4 The theory KPP proves

(I) for all ordinals α, there exists f such that RK(α, f);

(II) for all ordinals α and for all f, if RK(f, α + 1), then

f(α)= {x | ρ(x) < α}.

8 P Therefore, the theory KP proves that the function α 7→ Vα is total and that the graph P of this function is ∆1 -deﬁnable. In contrast, the theory MOST does not prove that the function α 7→ Vα is total (see Example 2.7 below). Note that the availability of AC in MOST allows us to identify P 4 cardinals with initial ordinals. Consider the ∆0 -formula BFEXT(R, X) deﬁned by: (R is an extensional relation on X with a top element) ∧ (∀S ⊆ X)(S 6= ∅⇒ (∃x ∈ S)(∀y ∈ S)(hy,xi ∈/ R)).

The following lemma captures two important features of the theory MOST that follow from [Mat01, Theorem 3.18].

Lemma 2.5 The theory MOST proves the following statements:

(I) for all hX,Ri with BFEXT(X,R), there exists a transitive set T such that hX,Ri =∼ hT, ∈i;

(II) there exist arbitrarily large initial ordinals;

(III) for all cardinals κ, the set H≤κ = {x | |TC(x)|≤ κ} exists.

P In the theory MOST, the formula “X = H≤κ” is ∆1 with parameters X and κ: (κ is a cardinal)∧ (∀R ⊆ κ × κ)(BFEXT(R, κ) ⇒ (∃x,f,T ∈ X)(T = TC({x}) ∧ f : R =∼∈↾ T ))∧ . (∀x ∈ X)(∃T,f ∈ X)(T = TC({x}) ∧ (f : T −→ κ is injective))

The next result is a special case of [Gor, Corollary 6.11]:

P P rk ✷ Lemma 2.6 Let M and N be models of KP . If M⊆e N , then M⊆e N .

The following examples show that neither of assumptions that M in Lemma 2.6 P P satisfies ∆0 -Collection and Π1 -Foundation can be removed. The structure M defined in P P Example 2.7 satisfies all of the axioms of KP except ∆0 -Collection. The structure M P P defined in Example 2.8 satisfies all of the axioms of KP except Π1 -Foundation.

Example 2.7 Let N = hN, EN i |= ZF + V = L, and

N ∗ N M = h(Hℵω ) , E i. P Then M |= MOST + Π∞-Separation, and M⊆blunt N , but N is not a rank extension of M.

Example 2.8 Let N = hN, EN i be an ω-nonstandard model of ZF + V = L. Let M = hM, EN i, where N ∗ M = (Hℵn ) . n∈ω [ P Then M |= MOST + Π1-Collection, and M⊆topless N , but N is not a rank extension of M. 4The abbreviation BFEXT has long been used by NF-theorists for well-founded extensional relations with a top, it is an abbreviation of Bien Fond´ee Extensionnelle, extensively employed by the French- speaking NF-ists in Belgium.

9 The following recursive definition can be carried out within KPUCov thanks to the ∗ ability of KPUCov to carry out Σ1(L )-recursions. The recursion defines an operation ee ee p·q for coding the infinitary formulae of L∞ω, where L be the language obtained from L by adding new constant symbolsa ¯ for each urelement a and a new constant symbol c.

• for all ordinals α, pvαq = h0, αi, • for all urelements a, pa¯q = h1, ai,

• pcq = h2, 0i, • ee if φ is an L∞ω-formula and x is a free variable of φ, then p∃xφq = h3, pxq, pφqi,

• ee if φ is an L∞ω-formula and x is a free variable of φ, then p∀xφq = h4, pxq, pφqi,

• ee if Φ is a set of L∞ω-formulae such that only ﬁnitely many variables appear as a free variable of some formula in Φ, then

p φq = h5, Φ∗i, where Φ∗ = {pφq | φ ∈ Φ}, φ∈ _Φ • ee if Φ is a set of L∞ω-formulae such that only ﬁnitely many variables appear as a free variable of some formula in Φ, then

p φq = h6, Φ∗i, where Φ∗ = {pφq | φ ∈ Φ}, ∈ φ^Φ • ee if φ is an L∞ω-formula, then p¬φq = h7, pφqi, • ee if s and t are terms of L∞ω, then ps = tq = h8, psq, ptqi, • ee if s and t are terms of L∞ω, then ps ∈ tq = h9, psq, ptqi. M A A Let M = hM, E i be an L-structure and let AM = hM; A, ∈, F , P i be a power admissible set covering M. We use Lee to denote the fragment of Lee that is coded AM ∞ω in A . The Lee -formulae in the form s = t or s ∈ t, where s and t are Lee -terms, M AM AM are the atomic formulae of Lee . The formula that identifies the codes of the atomic AM formulae of Lee is ∆ (L∗)-definable over A . Similarly, other important properties AM 1 M of codes of Lee constituents, such as being a variable, constant, well-formed formula, AM sentence, ..., are all ∆ (L∗)-definable over A . We will often equate an Lee -theory 1 M AM T with that subset of A of codes of Lee -sentences in T . M AM The following is the Barwise Compactness Theorem ([Bar75, III.5.6]) tailormade for ∗ countable admissible LP-structures. A A Theorem 2.9 (Barwise Compactness Theorem) Let AM = hM; A, ∈, F , P i be a power admissible set covering M. Let T be an Lee -theory that is Σ (L∗ )-definable over A AM 1 P M and such that for all T0 ⊆ T , if T0 ∈ A, then T0 has a model. Then T has a model.

10 P 3 The scheme of Σ1 -Foundation Motivated by the apparent reliance of the constructions presented in the next section P on Σ1 -Foundation, this section investigates the status of this scheme in the theories P P P MOST + Π1-Collection and KP . We begin by showing that KP proves Σ1 -Foundation. P In contrast, Σ1 -Foundation is not provable in MOST + Π1-Collection but does hold in every ω-standard model of this theory. P P 5 In [Rat20, Lemma 4.4] it is shown that KP + AC proves Σ1 -Foundation . Here we use a modiﬁcation of a choiceless scheme of dependant choices introduced in [Rat92] to P P show that Σ1 -Foundation can be proved in KP . The following is [Rat92, Deﬁnition 3.1]:

Deﬁnition 3.1 Let φ(x, y, ~z) be an L-formula. Deﬁne δφ(a, b, f, ~z) to be the formula:

f is a function ∧ dom(f)= a + 1 ∧ f(0) = {b}∧ a is an ordinal ⇒ (∀x ∈ f(u))(∃y ∈ f(u + 1))φ(x, y, ~z)∧ .  (∀u ∈ a)  (∀y ∈ f(u + 1))(∀x ∈ f(u))φ(x, y, ~z)   In the next deﬁnition we introduce a formula that, given b, f and ~z, says that f is a function with domain ω and for all n ∈ ω, δφ(n + 1, b, f ↾ (n + 1), ~z).

φ Deﬁnition 3.2 Let φ(x, y, ~z) be an L-formula. Deﬁne δω(b, f, ~z) to be the formula:

f is a function ∧ dom(f)= ω ∧ f(0) = {b}∧ (∀x ∈ f(u))(∃y ∈ f(u + 1))φ(x, y, ~z)∧ . (∀u ∈ ω) (∀y ∈ f(u + 1))(∀x ∈ f(u))φ(x, y, ~z) P φ Note that if φ(x, y, ~z) is a ∆0 -formula (∆0-formula) then both δ (a, b, f, ~z) and φ P δω(b, f, ~z) are both ∆0 -formulae (respectively ∆0-formulae). The following is a modiﬁ- cation of Rathjen’s ∆0-weak dependant choices scheme (∆0-WDC) from [Rat92]: P P (∆0 -WDCω) For all ∆0 -formulae, φ(x, y, ~z), φ ∀~z(∀x∃yφ(x, y, ~z) ⇒∀w∃fδω(w, f, ~z)).

The next result is based on the proof of [Rat92, Proposition 3.2]:

P P Theorem 3.3 The theory M +∆0 -WDCω proves Σ1 -Foundation.

P Proof Work in the theory M +∆0 -WDCω. Suppose, for a contradiction, that there is P P an instance of Σ1 -Foundation that fails. Let φ(x, y, ~z)bea∆0 -formula and let ~a be a ﬁnite sequence of sets such that the class C = {x |∃yφ(x, y,~a)} is nonempty and has no ∈-least element. Let b and d be such that φ(b, d,~a) holds. Now, since C has no ∈-least element, ∀x∀u∃y∃v(φ(x, u,~a) ⇒ (y ∈ x) ∧ φ(y, v,~a)). Therefore, we have have ∀x∃yθ(x, y,~a) where θ(x, y,~a) is

x = hx0,x1i∧ y = hy0,y1i∧ (φ(x0,x1,~a) ⇒ (y0 ∈ x0) ∧ φ(y0,y1,~a)).

5 P Rathjen proves the scheme that asserts that set induction holds for all Π1 -formulae, which, in the P P theory KP , is equivalent to Σ1 -Foundation.

11 P P Note that θ(x, y,~a)isa∆0 -formula. Therefore, using ∆0 -WDCω, let f be such that θ P P δω(hb, di, f,~a). Now, ∆0 -Separation facilitates induction for ∆0 -formulae and proves that for all n ∈ ω,

f(n) 6= ∅∧ (∀x ∈ f(n))(x = hx0,x1i∧ φ(x0,x1,~a))∧ (∀x ∈ f(n))(∃y ∈ f(n + 1))(x = hx0,x1i∧ y = hy0,y1i∧ y0 ∈ x0)∧ . (∀y ∈ f(n + 1))(∃x ∈ f(n))(y = hy0,y1i∧ x = hx0,x1i∧ y0 ∈ x0)

Let B = TC({b}). Induction for ∆0-formulae suﬃces to prove that for all n ∈ ω,

(∀x ∈ f(n))(x = hx0,x1i∧ x0 ∈ B). Consider

A = x ∈ B | (∃n ∈ ω)(∃z ∈ f(n)) ∃y ∈ z (z = hx,yi) , n [ o which is a set by ∆0-Separation. Now, let x ∈ A. Let y and n ∈ ω be such that hx,yi ∈ f(n). Therefore, there exists w ∈ f(n + 1) such that w = hu, vi and u ∈ x. So u ∈ A and u ∈ x, which shows that A has no ∈-least element. This contradicts Set-Foundation in M and proves the theorem. ✷

P P P The fact that KP proves Σ1 -Foundation follows from the fact that KP proves P ∆0 -WDCω. The proof of Theorem 3.5 is inspired by the argument used in the proof of [FLW, Theorem 4.15]. The stratification of the universe into ranks allows us to select P sets of paths through a relation defined by a ∆0 -formula φ(x, y, ~z) with parameters ~z. Definition 3.4 Let φ(x, y, ~z) be an L-formula. Define ηφ(a, b, f, ~z) by

δφ(a, b, f, ~z)∧ (α is an ordinal) ∧ (X = Vα) ∧ (∀x ∈ f(u + 1))(x ∈ X) ∧   . (∀u ∈ a)∃α∃X (∀y ∈ X)(∀x ∈ f(u))(φ(x, y, ~z) ⇒ y ∈ f(u + 1)) ∧  Y = V ⇒   (∀β ∈ α)(∀Y ∈ X) β   (∃x ∈ f(u))(∀y ∈ Y )¬φ(x, y, ~z)      The formula ηφ(a, b, f, ~z) asserts that f is a function with domain a + 1 such that f(0) = {b} and for all u ∈ a, f(u + 1) is the set of y of rank α such that there exists x ∈ f(u) with φ(x, y, ~z) and α is the minimal ordinal such that for all x ∈ f(u), there exists y of rank α such that φ(x, y, ~z). Recall that, in the theory KPP , the formula P P ‘X = Vα’is∆1 with parameters X and α. Therefore, if φ(x, y, ~z)isa∆0 -formula, then φ P P η (a, b, f, ~z) is equivalent to a Σ1 -formula in the theory KP .

P P Theorem 3.5 The theory KP proves ∆0 -WDCω.

P P Proof Work in the theory KP . Let φ(x, y, ~z)bea∆0 -formula. Let ~a be sets such that ∀x∃yφ(x, y,~a) holds. Let b be a set. We begin by claiming that for all n ∈ φ P ω, ∃fη (n, b, f,~a). Suppose, for a contradiction, that this does not hold. Using Π1 - Foundation, there exists a least m ∈ ω such that ¬∃fηφ(m, b, f,~a). It is straightforward to see that m 6= 0. Therefore, there exists a function g with dom(g) = m such that ηφ(m − 1, b, g,~a) holds. Consider the class

A = {α ∈ Ord |∀X(X = Vα ⇒ (∀x ∈ g(m − 1))(∃y ∈ X)φ(x, y,~a))}.

12 P Applying ∆0 -Collection to the formula φ(x, y,~a) shows that A is nonempty. Therefore, P by Π1 -Foundation, there exists a least element β ∈ A. Let

C = {y ∈ Vβ | (∃x ∈ g(m − 1))φ(x, y,~a)},

P φ which is a set by ∆0 -Separation. Now, let f = g ∪ {hm,Ci}. So, η (m, b, f,~a), which is a contradiction. Therefore, for all n ∈ ω, ∃fηφ(n, b, f,~a). Note that for all n ∈ ω and φ φ P for all f and g, if η (n, b, f,~a) and η (n, b, g,~a), then f = g. Now, using Σ1 -Collection, we can ﬁnd a set D such that (∀n ∈ ω)(∃f ∈ D)ηφ(n, b, f,~a). Let

f = {hn, Xi∈ ω × TC(D) | (∃g ∈ D)(ηφ(n, b, g,~a) ∧ g(n)= X)}

{hn, Xi∈ ω × TC(D) | (∀g ∈ D)(ηφ(n, b, g,~a) ⇒ g(n)= X)}. P P ✷ Now, f is a set by ∆1 -Separation and f is the function required by ∆0 -WDCω.

Combining Theorems 3.3 and 3.5 yields:

P P ✷ Corollary 3.6 KP ⊢ Σ1 -Foundation.

P We now turn to investigating Σ1 -Foundation in the theory MOST + Π1-Collection. The following is an instance of [PK, Proposition 2] in the context of set theory:

P P Theorem 3.7 Let Σ denote Σ1 -Foundation on ω, and Π denote Π1 -Foundation on ω. Then we have:

− P − M +∆0 -Collection + Π ⊢ Σ, and M +Σ ⊢ Π.

− P Proof To see that Π implies Σ, work in the theory M +∆0 -Collection. We prove P the contrapositive. Let φ(x, ~z)beaΠ1 -formula and let ~a be sets such that the class {x ∈ ω | φ(x,~a)} is nonempty and has no least element. Let p ∈ ω be such that φ(p,~a). Let C = {x ∈ ω |∃x(x + w = p ∧ (∀y ∈ w)¬φ(y,~a))}. P P Note that ∆0 -Collection implies that C isaΣ1 -deﬁnable subclass of ω. Moreover, p ∈ C and 0 ∈/ C. Identical reasoning to that used above shows that C has no least element. P Therefore Σ1 -Foundation on ω fails. To see that Σ implies Π, work in the theory M−. Again, we prove the contrapositive. P Let φ(x, ~z)beaΣ1 -formula and let ~a be the sequence of set parameters such that the class {x ∈ ω | φ(x,~a)} is nonempty and has no least element. Let p ∈ ω be such that φ(p,~a). Let C = {x ∈ ω |∀w(x + w = p ⇒ (∀y ∈ w)¬φ(y,~a)}. P Note that C is a Π1 -deﬁnable subclass of ω, p ∈ C and 0 ∈/ C. Suppose that q ∈ C is a least element of C. Let u ∈ ω be such that q + u = p. Now, φ(u,~a), since q is the least of C, and (∀y ∈ u)¬φ(y,~a). But then u is a least element of {x ∈ ω | φ(x,~a)}, which is P ✷ a contradiction. Therefore Π1 -Foundation on ω fails.

An examination of the proof of [Mat01, Proposition 9.22] yields:

P ✷ Theorem 3.8 The consistency of Mac is provable in M + Π1 -Foundation on ω.

13 The results of [Mat01] and [M] (see [M, Corollary 3.5]) show that Mac and MOST + Π1-Collection have the same consistency strength. Therefore, Theorem 3.7 yields:

Theorem 3.9 The consistency of MOST+Π1-Collection is provable in MOST+Π1-Collection+ P ✷ Σ1 -Foundation.

P Therefore, Σ1 -Foundation is not provable in MOST+Π1-Collection. However, we can P show that Σ1 -Foundation does hold in every model of MOST + Π1-Collection in which the natural numbers are standard. In the context of the theory MOST + Π1-Collection, we can use the stratiﬁcation of the universe into the sets H≤κ in the same way that we used the stratiﬁcation of the universe into ranks in the proof of Theorem 3.5.

Deﬁnition 3.10 Let φ(x, y, ~z) be an L-formula. Deﬁne χφ(a, b, f, ~z) by

δφ(a, b, f, ~z)∧ (X = H≤κ) ∧ (∀x ∈ f(u + 1))(x ∈ X)∧ (∀y ∈ X)(∀x ∈ f(u))(φ(x, y, ~z) ⇒ y ∈ f(u + 1))∧   . (∀u ∈ a)∃κ∃X BFEXT(R,λ) ⇒    (∀λ ∈ κ)(∃x ∈ f(u))(∀R ⊆ λ × λ)  T = TC({y})∧    ∃T,f,y   f : R =∼∈↾ T ∧ ¬φ(x, y, ~z)      The formula χφ(a, b, f, ~z) asserts that f is a function with domain a + 1 such that f(0) = {b} and for all u ∈ a, f(u + 1) is the set of all y in H≤κ such that there exists an x ∈ f(u) with φ(x, y, ~z) and κ is the minimal cardinal such that for all x ∈ f(u), there P exists y ∈ H≤κ with φ(x, y, ~z). Recall that the formula expressing “X = H≤κ” is ∆1 P with parameters X and κ in the theory MOST. Therefore, if φ(x, y, ~z)isa∆0 -formula, φ P then χ (a, b, f, ~z) is equivalent to a Σ1 -formula in the theory MOST + Π1-Collection. In the proof of the next theorem the formula χφ(a, b, f, ~z) plays the role of ηφ(a, b, f, ~z) in the proof of Theorem 3.5.

M Theorem 3.11 Let M = hM, E i be an ω-standard model of MOST + Π1-Collection. P Then M |=∆0 -WDCω.

P Proof Let φ(x, y, ~z)bea ∆0 -formula. Let ~a be sets such that M |= ∀x∃yφ(x, y,~a). Let b a set. We begin by showing that

M |= (∀n ∈ ω)∃fχφ(n, b, f,~a).

Work inside M. Suppose, for a contradiction, that there exists n ∈ ω such that ¬∃fχφ(n, b, f,~a) holds. Therefore, since M is ω-standard, there is a least m ∈ ω such that ¬∃fχφ(m, b, f,~a). It is straightforward to see that m 6= 0. Therefore, there exists a function g with dom(g)= m such that χφ(m − 1, b, g,~a). Consider

M A = {κ ∈ Ord | M |= ∀X(X = H≤κ ⇒ (∀x ∈ g(m − 1))(∃y ∈ X)φ(x, y,~a))}

M = {κ ∈ Ord | M |= ∃X(X = H≤κ ∧ (∀x ∈ g(m − 1))(∃y ∈ X)φ(x, y,~a))}. P P Applying ∆0 -Collection to φ(x, y,~a) shows that A is nonempty. Therefore, ∆1 -Separation ensures that A has an ∈-least element λ. Let

C = {y ∈ H≤λ | (∃x ∈ g(m − 1))φ(x, y,~a)},

14 P φ which is a set by ∆0 -Separation. Now, let f = g ∪ {hm,Ci}. So, χ (m, b, f,~a) which is a contradiction. This shows that

M |= (∀n ∈ ω)∃fχφ(n, b, f,~a).

Work inside M. Note that for all n ∈ ω and for all f and g, if χφ(n, b, f,~a) and φ P χ (n, b, g,~a), then f = g. Now, using Σ1 -Collection, there exists D such that

(∀n ∈ ω)(∃f ∈ D)χφ(n, b, f,~a).

Let f = {hn, Xi∈ ω × TC(D) | (∃g ∈ D)(χφ(n, b, g,~a) ∧ g(n)= X)} = {hn, Xi∈ ω × TC(D) | (∀g ∈ D)(χφ(n, b, g,~a) ⇒ g(n)= X)}, P which is a set by ∆1 -Separation. Therefore δω(b, f,~a) holds, which completes the proof P ✷ that ∆0 -WDCω holds in M.

Combining Theorems 3.3 and 3.11:

Corollary 3.12 If M is an ω-standard model of MOST + Π1-Collection, then M |= P ✷ Σ1 -Foundation.

4 Obtaining CovM from M

[Bar75, Appendix] shows how the admissible cover, CovM, can be built from an L- structure M that satisfies KP+Σ1-Foundation. The construction proceeds in two stages. The first stage interprets a model of KPUCov inside M. The second stage takes the well- founded part of this interpreted model of KPUCov to obtain an admissible set covering M that [Bar75, Appendix] shows is minimal. It should be noted that [Bar75, Appendix] starts with a structure M that satisfies full Π∞-Foundation. It is noted in [Res, Chapter 2] that all of the elements of Barwise’s construction of CovM can be carried out when M satisfies Π1 ∪ Σ1-Foundation. The aim of this section is to review the construction of CovM from M and investigate the influence of the theory of M on CovM. In particular, we will show that if M is a model of P P C KP + powerset +∆0 -Collection +Σ1 -Foundation, then P can be interpreted in ovM to make it a power admissible set. Throughout this section we will work with a fixed L-structure M = hM, EMi that P P satisfies KP + Powerset +∆0 -Collection +Σ1 -Foundation. We begin by expanding the interpretation of the theory KPUCov inside M presented in [Bar75, Appendix Section ∗ 3] to obtain LP-structure that satisfies powerset. Working inside M, define the unary relations N and Set, the binary relations E and E′, and unary function symbols F¯ and P¯ by: N(x) iff ∃y(x = h0,yi); xE′y iff ∃w∃z(x = h0, wi∧ y = h0, zi∧ w ∈ z); Set(x) iff ∃y(x = h1,yi∧ (∀z ∈ y)(N(z) ∨ Set(z))); xEy iff ∃z(y = h1, zi∧ x ∈ y); F¯(x)= h1, Xi where X = {h0,yi|∃w(x = h0, wi∧ y ∈ w)};

15 P¯(x)= h1, Xi where X = {h1,yi|∃w(x = h1, wi∧ y ⊆ w)}. ′ [Bar75, Appendix Section 3] notes that N, E , E and F¯ are deﬁned by ∆0-formulae in M, and, using the Second Recursion Theorem ([Bar75, V.2.3.]), Set can be expressed using a Σ1-formula. [Res, Chapter 2] notes that the Second Recursion Theorem can be ¯ P proved in KP +Σ1-Foundation. The function y = P(x) is deﬁned by a ∆0 -formula:

y = P¯(x) iﬀ

OP(x) ∧ OP(y) ∧ fst(x) = 1 ∧ fst(y) = 1∧ (∀z ⊆ snd(x))(h1, zi ∈ snd(y)) ∧ (∀w ∈ snd(y))(snd(w) ⊆ snd(x)). ∗ These deﬁnitions yield an interpretation, I, of an LP-structure that is summarised in Table 1 that extends the table in [Bar75, p. 373]:

∗ Table 1: The interpretation I of an LP-structure in M ∗ LP Symbol L expression under I

∀x ∀x(N(x) ∨ Set(x) ⇒··· ) = = U(x) N(x) xEy xE′y x ∈ y xEy F(x) F¯(x) P(x) P¯(x)

M M M M M ′ M In other words, AN = hN ; Set , E , F¯ , P¯ i, where N = hN , (E ) i, is an ∗ ∗ I LP-structure. If φ is an LP-formula, then we write φ for the translation of φ into an L-formula of M described in Table 1. Note that the map x 7→ h0,xi is an isomorphism between M and N . The following is the reﬁnement of [Bar75, Appendix Lemma 3.2] noted by [Res, Chapter 2]:

Theorem 4.1 AN |= KPUCov. ✷

We now turn to showing that axioms and axiom schemes transfer from M to AN .

Lemma 4.2 AN |= Powerset.

Proof Let a be a set of AN . To see that P¯(a) exists, note that a = h1, a0i and P¯(a)= h1, Xi where X = {1} × P(a0). Therefore, the powerset axiom in M ensures that P¯ is total in AN . Now, let b be a set of AN . Work inside M. Now, b = h1, b0i. And,

bEP¯(a) iﬀ b0 ⊆ a0,

iﬀ for all x, if xEb, then xEa, iﬀ (b ⊆ a)I .

Therefore, AN satisﬁes Powerset. ✷

16 ∗ I P Lemma 4.3 Let φ(~x) be a ∆0(LP)-formula. Then φ (~x) is equivalent to a ∆1 -formulae in M.

Proof We prove this lemma by induction on the complexity of φI . Note that, by the ′ above observations, N(x) and xE y can be written as ∆0-formulae. Moreover, y = F¯(x) ¯ P ¯ is equivalent to a ∆0-formula, and y = P(x) is equivalent to a ∆0 -formula. Now, yEF(x) iﬀ fst(y) = 0 ∧ snd(y) ∈ snd(x), which is ∆0. Similarly, yEP¯(x) iﬀ

fst(y) = 1 ∧ snd(y) ⊆ snd(x),

∗ I I which is also ∆0. Now, suppose that t(x) is an LP-term and both y = t (x) and yEt (x) P ¯ I are ∆1 in M. Now, y = P(t (x))

iﬀ ∃w(w = tI(x) ∧ y = P¯(w)),

iff ∀w(w = tI (x) ⇒ y = P¯(w)). Similarly, yEP¯(tI (x)) iff ∃w(w = tI (x) ∧ yEP¯(w)), iff ∀w(w = tI (x) ⇒ yEP¯(w)). ¯ I ¯ I P ¯ I Therefore, both y = P(t (x)) and yEP(t (x)) are ∆1 in M. Now, y = F(t (x))

iﬀ ∃w(w = tI (x) ∧ y = F¯(w)),

iff ∀w(w = tI(x) ⇒ y = F¯(w)). And, yEF¯(tI (x)) iff ∃w(w = tI (x) ∧ yEF¯(w)), iff ∀w(w = tI (x) ⇒ yEF¯(w)). ¯ ¯ ∗ Since F and P are both unary functions, this shows that for every LP-term t(x), both I I P y = t (x) and yEt (x) are ∆1 in M. Finally, we need an induction step that allows us ∗ to deal with bounded quantification. Let ψ(x0,...,xn−1) be an LP-formula such that I P I ψ (x0,...,xn−1) is ∆1 in M. Now, (∃x0Exn)ψ (x0,...,xn−1)

I iﬀ (∃x0 ∈ snd(xn))ψ (x0,...,xn−1).

I I P Therefore, (∃x0Exn)ψ (x0,...,xn−1) = ((∃x0 ∈ xn)ψ(x0,...,xn−1)) is ∆1 in M. Let ∗ I I t(x) be an LP-term. Now, (∃x0Et (xn))ψ (x0,...,xn−1)

I I iﬀ ∃w(w = t (xn) ∧ (∃x0 ∈ snd(w))ψ (x0,...,xn−1)),

I I iﬀ ∀w(w = t (xn) ⇒ (∃x0 ∈ snd(w))ψ (x0,...,xn−1)). I I I P Therefore (∃x0Et (xn))ψ (x0,...,xn−1) = ((∃x0 ∈ t(xn))ψ(x0,...,xn−1)) is ∆1 in M. The Lemma now follows by induction. ✷

A ∗ Lemma 4.4 N |=∆0(LP)-Separation.

17 ∗ A Proof Let φ(x, ~z)bea∆0(LP)-formula, ~v be sets and/or urelements of N and a a set of AN . Work inside M. Now, a = h1, a0i. Let I b0 = {x ∈ a0 | φ (x, ~v)}, P which is a set by ∆1 -Separation. Let b = h1, b0i. Therefore, for all x such that Set(x), xEb iﬀ xEa ∧ φI (x, ~v). A ∗ ✷ Therefore, N satisﬁes ∆0(LP)-Separation. A ∗ Lemma 4.5 N |=∆0(LP)-Collection. ∗ Proof Let φ(x, y, ~z)bea∆0(LP)-formula. Let ~v be a sequence of sets and/or urelements of AN and let a be a set of AN such that

AN |= (∀x ∈ a)∃yφ(x, y,~v).

Work inside M. Since a is a set of AN , a = h1, a0i. We have (∀xEa)∃y((N(y) ∨ Set(y)) ∧ φI (x, y,~v)). And, I (∀x ∈ a0)∃y((N(y) ∨ Set(y)) ∧ φ (x, y,~v)). I P P So, since (N(y) ∨ Set(y)) ∧ φ (x, y,~v) is equivalent to a Σ1 -formula, we can apply ∆0 - Collection to obtain b such that I (b) (∀x ∈ a0)(∃y ∈ b)((N(y) ∨ Set(y)) ∧ φ (x, y,~v)) . (b) Let b0 = {y ∈ b | (N(y) ∨ Set(y)) }, which is a set by ∆0-Separation. Let b1 = h1, b0i. Therefore Set(b1) and I (∀xEa)(∃yEb1)φ (x, y,~v). So, AN |= (∀x ∈ a)(∃y ∈ b1)φ(x, y,~v). A ∗ ✷ This shows that N satisﬁes ∆0(LP)-Collection. A ∗ Lemma 4.6 N |=Σ1(LP)-Foundation. ∗ Proof Let φ(x, ~z)beaΣ1(LP)-formula. Let ~v be a sequence of sets and/or urelements be such that {x ∈ AN | AN |= φ(x, ~v)} is nonempty. Work inside M. Consider θ(α, ~z) deﬁned by (α is an ordinal) ∧∃x((Set(x) ∨ N(x)) ∧ ρ(x)= α ∧ φI (x, ~z)). P P Note that θ(α, ~z) is equivalent to a Σ1 -formula. Therefore, using Σ1 -Foundation, let β be an ∈-least element of {α ∈ M | M |= θ(α, ~v)}. Let y be such that (N(y) ∨ Set(y)), ρ(y) = β and φI (y,~v). Note that if xEy, then ρ(x) < ρ(y). Therefore y is an E-least element of

{x ∈ AN | AN |= φ(x, ~v)}. ✷

18 The following combines [Bar75, II.8.4] with the characterisation of CovM proved in [Bar75, Appendix Section 3]:

∗ − M M M Theorem 4.7 The L -reduct of WF(AN ), WF (AN )= hN ; WF(Set ), E , F¯ i is an admissible set covering N that is isomorphic to CovM. ✷

We now turn to extending this result to show that WF(AN ) is a power admissible set covering N and therefore the least power admissible set covering N .

M M M M Theorem 4.8 The structure WF(AN )= hN ; WF(Set ), E , F¯ , P¯ i is a power ad- A C P missible set covering N . Moreover, WF( N ) is isomorphic to ovM.

Proof Note that it follows immediately from Theorem 4.7 that M M M M WF(AN )= hN ; WF(Set ), E , F¯ , P¯ i satisfies all of the axioms of KPUCov plus full A P A A Foundation. The fact that WF( N ) ⊆e N implies that WF( N ) satisfies Powerset and ∗ A ∗ ∆0(LP)-Separation. To show that WF( N ) satisfies ∆0(LP)-Collection, let φ(x, y, ~z) be ∗ A a ∆0(LP)-formula. Let ~v be sets and/or urelements of WF( N ) and let a be a set of WF(AN ) such that WF(AN ) |= (∀x ∈ a)∃yφ(x, y,~v). Consider the formula θ(β, ~z) defined by

(β is an ordinal) ∧ (∀x ∈ a)(∃α ∈ β)∃y(ρ(y)= α ∧ φ(x, y, ~z). A P A A A Since WF( N ) ⊆e N , if β is a nonstandard ordinal of N , then N |= θ(β,~v). ∗ ∗ A Using ∆0(LP)-Collection, θ(β, ~z) is equivalent to a Σ1(LP)-formula in N . Therefore, by ∗ A Σ1(LP)-Foundation, {β | N |= θ(β,~v)} has a least element γ. Note that γ is an ordinal of WF(AN ). Now, consider the formula ψ(x, y, ~z, γ) deﬁned by

φ(x, y, ~z) ∧ (ρ(y) < γ).

Note that AN |= (∀x ∈ a)∃yψ(x, y,~v, γ). ∗ A A Using ∆0(LP)-Collection in N , there exists a set b of N such that

AN |= (∀x ∈ a)(∃y ∈ b)ψ(x, y,~v, γ). A ∗ Let c = {x ∈ b | ρ(x) < γ}, which is a set in N by ∆1(LP)-Separation. Now, c is a set of WF(AN ) and WF(AN ) |= (∀x ∈ a)(∃y ∈ c)φ(x, y,~v). A ∗ A Therefore, WF( N ) satisﬁes ∆0(LP)-Collection. And so, WF( N ) is a power admissi- ∗ ble set covering N . Finally, since the L -reduct of WF(AN ) is isomorphic to CovM, A C P ✷ WF( N ) is isomorphic to ovM.

The following theorem summarises the analysis undertaken in this section:

P P Theorem 4.9 If M |= KP + Powerset +∆0 -Collection +Σ1 -Foundation, then there is C C P ✷ an interpretation of P in ovM that yields the power admissible set ovM.

This yields a version of [Bar75, Corollary 2.4.] that will be useful for the compactness arguments in the next section.

19 M P P Theorem 4.10 Let M = hM, E i |= KP + Powerset +∆0 -Collection +Σ1 -Foundation. For all A ⊆ M, there exists a ∈ M such that

∗ P a = A iﬀ A ∈ CovM. ✷

5 End extension results

ee In this section we use the Barwise Compactness Theorem for LC P to show that every ovM P P countable model of KP + powerset + ∆0 -Collection + Σ1 -Foundation has a powerset- preserving end extension. The following is an immediate consequence of Theorem 4.10:

M P P Lemma 5.1 Let M = hM, E i |= KP+Powerset+∆0 -Collection+Σ1 -Foundation, and ee C P let T0 be an LC P -theory. If T0 ∈ ovM, then there exists b ∈ M such that ovM ∗ b = {a ∈ M | a¯ is mentioned in T0}. ✷

The next result expands on comments made in [Bar75, p. 637] and connects deﬁn- C P ability in M to deﬁnability in ovM.

M P P Lemma 5.2 Let M = hM, ∈ i |= KP + Powerset +∆0 -Collection +Σ1 -Foundation, P ˆ ∗ and let φ(~z) be a ∆0 -formula. Then there exists a formula φ(~z) that is ∆1(LP) in the P theory KPUCov such that for all ~z ∈ M, P ˆ M |= φ(~z) iﬀ CovM |= φ(~z).

Proof Let φ(~z)bea∆0-formula. We prove the lemma by structural induction on the complexity of φ. Without loss of generality we can assume that the only connectives of propositional logic appearing in φ are ¬ and ∨. If φ(z1, z2) is z1 ∈ z2, then let φˆ(z1, z2) ∗ be the ∆0(LP)-formula z1Ez2. Therefore, for all z1, z2 ∈ M, P ˆ M |= φ(z1, z2) iﬀ CovM |= φ(z1, z2). If φ(~z) is ¬ψ(~z) and the lemma holds for ψ(~z), then let φˆ(~z) = ¬ψˆ(~z). So, φˆ(~z) is ∗ P ∆1(LP) in the theory KPUCov and for all ~z ∈ M, P ˆ M |= φ(~z) iﬀ CovM |= φ(~z).

Suppose that φ(~z) is ψ1(~z) ∨ ψ2(~z) and the lemma holds for ψ1(~z) and ψ2(~z). Let φˆ(~z) ˆ ˆ ˆ ∗ P be ψ1(~z) ∨ ψ2(~z). Therefore, φ(~z) is ∆1(LP) in the theory KPUCov and for all ~z ∈ M, P ˆ M |= φ(~z) iff CovM |= φ(~z). Suppose φ(y, ~z) is (Qx ∈ y)ψ(x, y, ~z), where Q ∈ {∃, ∀}, and the lemma holds for ˆ ˆ ˆ ∗ ψ(x, y, ~z). Let φ(y, ~z) be (Qx ∈ F(y))ψ(x, y, ~z). So, φ(~z) is ∆1(LP) in the theory P C P KPUCov. Since ovM satisfies (†), for all y, ~z ∈ M, M |= φ(y, ~z) iff M |= (Qx ∈ y)ψ(x, y, ~z)

20 P ˆ iff CovM |= (Qx ∈ F(y))ψ(x, y, ~z) P ˆ iff CovM |= φ(y, ~z). Suppose that φ(y, ~z) is (Qx ⊆ y)ψ(x, y, ~z), where Q ∈ {∃, ∀}, and the lemma holds for ψ(x, y, ~z). Let φˆ(y, ~z) be (Qx ∈ P(F(y)))∃p(F(p) = x ∧ ψˆ(p, y, ~z)). Note that, in the P theory KPUCov, for all urelements ~z, (Qx ∈ P(F(y))) ∃p(F(p)= x ∧ ψˆ(p, y, ~z)) ⇐⇒ (Qx ∈ P(F(y))) ∀p(F(p)= x ⇒ ψˆ(p, y, ~z)). P ˆ ∗ Therefore, in the theory KPUCov, φ(y, ~z) is ∆1(LP). Moreover, by Theorem 4.10 and (†), for all ~z ∈ M, M |= φˆ(y, ~z) iff M |= (Qx ⊆ y)ψ(x, y, ~z) iff (Qx ∈ P(F(y))) ∃p(F(p)= x ∧ ψˆ(p, y, ~z)) P ˆ iff CovM |= φ(y, ~z). Therefore, the lemma follows by induction. ✷

We are now able to use the machinery we have developed to establish the following result.

Theorem 5.3 Let S be a recursively enumerable L-theory such that

P P S ⊢ KP + Powerset +∆0 -Collection +Σ1 -Foundation, and let M be a countable model of S. Then there exists an L-structure N such that P M⊆e N |= S, and for some d ∈ N, and for all x ∈ M, N |= (x ∈ d).

ee Proof Let T be the LC P -theory that contains: ovM • S;

• for all a, b ∈ M with M |= (a ∈ b),a ¯ ∈ ¯b;

• for all a ∈ M,

∀x x ∈ a¯ ⇐⇒ (x = ¯b) ; b∈a ! _ • for all a ∈ M,

∀x x ⊆ a¯ ⇐⇒ (x = ¯b) ;   ⊆ b_a   • for all a ∈ M,a ¯ ∈ c. C P ∗ C P Lemma 5.2 shows that T ⊆ ovM is Σ1(LP)-deﬁnable over ovM. Let T0 ⊆ T be such C P that T0 ∈ ovM. Using Lemma 5.1, there exists c ∈ M such that ∗ c = {a ∈ M | a¯ is mentioned in T0}.

Therefore, by interpreting eacha ¯ that is mentioned in T0 by a ∈ M and interpreting ′ c by c, we can expand M to a model M that satisﬁes T0. Therefore, by the Barwise Compactness Theorem, there exists N |= T . It is straightforward to see that the L- reduct of N has is the desired extension of M. ✷

21 We ﬁrst apply this result to show that countable models of KPP have topless rank extensions that satisfy KPP . This generalises [Fri, Theorem 2.3], which shows that every countable transitive model of KPP has a topless rank extension that satisﬁes KPP .

Theorem 5.4 (Friedman) Let S be a recursively enumerable L-theory such that S ⊢ KPP . If M is a countable transitive model of S, then there exists N |= S such that rk ✷ M⊆topless N .

It follows from [Gor, Theorem 4.8] that every countable nonstandard model of KPP + P rk Σ1 -Separation, N , is isomorphic to substructure M of N such that M⊆topless N . We will make use of this result in the following form:

P P Theorem 5.5 (Gorbow) Let M be a countable nonstandard model of KP +Σ1 -Separation. rk ✷ Then there exists N ≡ M such that M⊆topless N .

We next note that if a model of KPP has a blunt rank extension, then that model must satisfy the full scheme of separation.

M N P rk Lemma 5.6 Let M = hM, E i and N = hN, E i be models of KP . If M ⊆blunt N , then M |= Π∞-Separation.

M N rk Proof Assume that M ⊆ N, E = E ↾ M and M⊆blunt N . Let c ∈ N be such that ∗ rk c ⊆ M and c∈ / M. Working inside N , let α = ρ(c). Therefore, since M⊆blunt N ,

N ∗ x ∈ (Vα ) if and only if N |= (ρ(x) < α) if and only if N |= (∃y ∈ c)(ρ(x) ≤ ρ(y)) if and only if x ∈ M.

N ∗ So, M = (Vα ) and every instance of Π∞-Separation in M can be reduced to an instance P of ∆0-Separation in N and, since M ⊆e N , the resulting set will be in M. Therefore, M |= Π∞-Separation. ✷

Theorem 5.7 Let S be a recursively enumerable L-theory such S ⊢ KPP . If M is a rk countable model of S, then there exists N |= S such that M⊆topless N .

Proof Let M be a countable model of S. If M is well-founded, then M is isomorphic to a transitive model of KPP and we can use Theorem 5.4 to find an L-structure N |= S rk such that M ⊆topless N . Therefore, assume that M is nonstandard. By Corollary 3.6, P P M satisfies KP + Powerset +∆0 -Collection +Σ1 -Foundation. Therefore, using Theorem P 5.3, we can find an L-structure N |= S such that M 6= N and M⊆e N . So, by Lemma rk rk rk 2.6, M⊆e N . If M⊆topless N , then we are done. Alternatively, if M⊆blunt N , then, by Lemma 5.6, M |= Π∞-Separation. Therefore, since M is nonstandard, we can apply ′ rk ′ ✷ Theorem 5.5 to obtain an L-structure N ≡ M such that M⊆topless N .

We now turn to showing that every countable model of MOST + Π1-Collection + P Σ1 -Foundation has a topless powerset-preserving end extension that satisﬁes MOST + P Π1-Collection +Σ1 -Foundation. We ﬁrst prove an analogue of Lemma 5.6 for models of MOST.

22 M N P Lemma 5.8 Let M = hM, E i and N = hN, E i be models of MOST. If M⊆blunt N , then M |= Π∞-Separation.

M N rk Proof Assume that M ⊆ N, E = E ↾ M and M⊆blunt N . Let c ∈ N be such that c∗ ⊆ M and c∈ / M. Work inside N . Let κ = |TC(c)| and note that κ∈ / M. Consider

A = {λ ∈ κ | (∃y ∈ c)(λ = |TC(y)|)}, which is a set by Σ1-Separation. Let µ = sup A and note that µ is an initial ordinal. + N + M N Work in the metatheory again. If µ ∈ M, then so is (µ ) = (µ ) ∈ M. So, Hµ+ ∈ M and N |= (c ⊆ Hµ+ ). And, c ∈ M, which is a contradiction. Therefore µ∈ / M. Now,

N ∗ x ∈ (Hµ ) if and only if N |= (|TC(x)| <µ) if and only if N |= (∃y ∈ c)(|TC(x)| < |TC(y)|) if and only if x ∈ M.

N ∗ So, M = (Hµ ) and every instance of Π∞-Separation in M can be reduced to an instance P of ∆0-Separation in N and, since M ⊆e N , the resulting set will be in M. Therefore, M |= Π∞-Separation. ✷

P Theorem 5.9 Let S be an L-theory such that S ⊢ MOST+Π1-Collection+Σ1 -Foundation. P If M is a countable model of S, then there exists a model N such that M⊆topless N |= S. Proof This can be proved using an identical argument to the proof of Theorem 5.7 P after observing that every transitive model of MOST + Π1-Collection +Σ1 -Foundation P P P is a model of KP and KP +Σ1 -Separation is a subtheory of MOST + Π1-Collection + Π∞-Separation. ✷

The work [EKM] studies the class C of structures Ifix(j) where j : M −→ M is a nontrivial automorphism, M is an L-structure that satisfies MOST, j fixes every point M ∗ in (ω ) and Ifix(j) is the substructure of M that consists of elements x of M such that j fixes every point in (TCM({x}))∗. The results of [EKM, Section 3] show that every structure in C satisfies MOST + Π1-Collection. Conversely, [EKM, Section 4] shows that a sufficient condition for a countable structure M that satisfies MOST + Π1-Collection P to be in C is that there exists M⊆topless N such that N satisfies MOST + Π1-Collection. Theorem 5.9 allows us to extend [EKM, Theorem B] by showing that C contains all P countable models of MOST + Π1-Collection +Σ1 -Foundation. M Theorem 5.10 Let M = hM, E i be a countable model of MOST + Π1-Collection + P N Σ1 -Foundation. Then there exists a model N = hN, E i that satisfies MOST and a non- ∼ trivial automorphism j : N −→N such that M = Ifix(j), where Ifix(j) is the substructure of N with underlying set N ∗ Ifix(j) = {x ∈ N | (∀y ∈ (TC ({x})) )(j(y)= y)}. ✷

Combined with Corollary 3.12 and [EKM, Theorem 5.6] this shows that the class C contains every countable recursively saturated model of MOST+Π1-Collection and every countable ω-standard model of MOST+Π1-Collection, providing a partial positive answer to Question 5.1 of [EKM]. A positive answer to the following question would positively answer Question 5.1 of [EKM]:

23 Question 5.11 Does every countable ω-nonstandard model of MOST + Π1-Collection have a topless powerset-preserving end extension that satisﬁes MOST + Π1-Collection?

Note that [EKM, Theorem 5.6] shows that this question has a positive answer when the countable model is recursively saturated.

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