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Automorphisms, Mahlo Cardinals, and NFU
Ali Enayat
Abstract. In what follows, ZF (L) is the natural extension of Zermelo-Fraenkel
set theory ZF in the language L={∈,C}, GW is the axiom “Cis a global
well-ordering”, and Φ := {ϕn:n∈ω}, where ϕnasserts the existence of a
Σn-reflecting n-Mahlo cardinal. Our principal results are Theorem A, and its
“reversal” Theorem B below.
Theorem A. Suppose Tis a consistent completion of ZF C + Φ.There is
a model Mof T+Z F (L) + GW such that Mhas a proper elementary end
extension Nthat possesses an automorphism jwhose fixed point set is M.
Theorem B. There is a weak fragment Wof Zermelo-set theory plus GW
such that if some model N= (N, ∈N,CN)of Whas an automorphism whose
fixed point set Mforms a proper C-initial segment of N,then
MZF (L) + GW + Φ,
where Mis the submodel of Nwhose universe is M.
We also explain how Theorems A and B can be used to fine tune the work
of Solovay and Holmes concerning the relationship between Mahlo cardinals
and the extension N F UA of the Quine-Jensen system NF U.
Department of Mathematics and Statistics, American University, Washington,
DC, 20016-8050
1. INTRODUCTION AND PRELIMINARIES
In this paper we establish an intimate relationship between Mahlo cardinals
and automorphisms of models of set theory. The models of set theory involved
here are not well-founded since a routine argument by transfinite induction reveals
that a well-founded model satisfying the extensionality axiom only admits the triv-
ial automorphism. The source of our inspiration was Solovay’s calibration of the
consistency strength of the unorthodox system of set theory N F U A, but our re-
sults can also be motivated in an orthodox fashion by comparing the behavior of
automorphisms of models of Z F C (Zermelo-Fraenkel set theory with the axiom of
choice), with those of ZF C ’s sister theory P A (Peano Arithmetic). Our point of
departure in this comparative context is Gaifman’s seminal work [G] on the model
theory of P A. Gaifman refined the MacDowell-Specker method [MS] of building
elementary end extensions by introducing the powerful machinery of minimal end
extension types, which can be used to produce a variety of models of P A with spe-
cial properties. In particular, one can use minimal end extension types to establish
that every model M= (M , +,·)of P A has a proper elementary end extension N
1
2 ALI ENAYAT
that possesses an automorphism jwhose fixed point set is precisely M(as observed
by Kossak, this is a corollary of [G, Theorems 4.9, 4.10 and 4.11]). Therefore,
Theorem 1.1. Every consistent completion of P A has a model Nthat possesses
a nontrivial automorphism whose fixed point set is a proper initial segment of N.
A natural question is whether there is an analogue of Theorem 1.1 for ZF C .
We answer this question by showing, in Theorem A (Section 3) and Theorem B
(Section 4), that such an analogue exists precisely for consistent completions of the
theory ZF C + Φ, where
Φ := {∃κ(κis n-Mahlo and Vκis a Σn-elementary submodel of V) : n∈ω}.
(intuitively, Φ says that the class of ordinals behaves like an ω-Mahlo cardinal).
The prehistory of the scheme Φ goes back to the groundbreaking work of Schmerl
and Shelah [SS] on the Hanf-number computation of power-like models in terms of
Mahlo cardinals, but Φ itself was first explicitly introduced and studied by Kauf-
mann [Ka] and the author [E-1, 2] in the context of identifying the right “comple-
tion” of ZF C which resembles P A from the point of view of model theory. For
example, every consistent completion of ZF C + Φ has a κ-like model for any un-
countable cardinal κ([Kau, Theorem 3.10], [E-1, Theorem 3.8]). Moreover, if ℵ2
has the tree property, then every ℵ2-like model of ZF C already satisfies Φ [E-3,
Theorem 4.8]. By [E-1, Remark 1.6] there is no analogue of minimal end exten-
sion types for models of Z F C in any finite alphabet extending {∈},but our work
here shows that, in contrast, there is such an analogue for the enriched theory
ZF C (L) + Φ(L), where Lis an expansion of {∈} by infinitely many predicates.
This provides further a posteriori evidence for the maxim:
ZF C + Φ imitates P A, vis-`a-vis model theory.
In Section 5 we use the results obtained about automorphisms of models of Z F C
to gain insight into models of the set theoretical system N F UA (an extension of the
Quine-Jensen system N F U of set theory). In particular, Theorems A and B allow
us to pinpoint the first order theories of the so-called Cantorian initial segments
CZ Aof Zermelian structures ZAarising from models Aof N F U A, which in turn
lead to a new proof of the following result:
Theorem 1.2.1(Solovay, 1995) The following theories are equiconsistent:
T1:= ZF C +{“there is an n-Mahlo cardinal”:n∈ω},
T2:= N F U A.
There is also an arithmetical counterpart to our set theoretical story. Recent
work of Solovay and the author has revealed a close relationship between classical
arithmetical theories (ranging from a fragment of P A to second order arithmetic)
and natural extensions of the theory obtained by adding the negation of the axiom
of infinity to N F U. The picture that emerges from this work shows that N F U
provides a robust framework for the formulation of elegant theories whose consis-
tency strength varies from the tangible territory of the fragment I-∆0+Exp of
P A, all the way up to the heights of systems of set theory with large cardinals. In
light of these developments, the Quine-Jensen paradigm emerges as a fascinating
1This result is announced (without proof ) in Solovay’s online manuscript [So].
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 3
foundational alternative that not only offers a coherent rival account of set theory2,
but also unifies seemingly unrelated systems of classical arithmetic and set theory.
I am grateful to Robert Solovay for pointing the way in [So], enduring my
rough drafts, and providing precious advice. I am also indebted to Randall Holmes
for helpful discussions about N F U , and to the anonymous referee for suggested
improvements.
Preliminaries: Theories and Models
A. Theories
•For a relational language L, Σn(L),Πn(L), and ∆n(L) are sets of first
order formulas built up from Lthat (respectively) belong to the Σn-level,
Πn-level, and ∆n-level of the well-known L´evy hierarchy. We use Σ∞(L)
for S
n∈ω
Σn(L).
•For a class Γ of formulas, Γ-Separation consists of formulas of the form
∀a∃b∀x(x∈b←→ x∈a∧ϕ(x)),
for all formulas ϕ∈Γ in which bdoes not occur free.
•For a class Γ of formulas, Γ-Replacement consists of formulas of the form
[∀x∈a∃!yϕ(x, y)] →[∃b∀y(y∈b↔ ∃x∈a∧ϕ(x, y ))],
for all formulas ϕ∈Γ in which bdoes not occur free.
•ZF (L) is Z F augmented by Σ∞(L)-Replacement, where L ⊇ {∈}.
•ES T (L) [Elementary Set Theory] is obtained from the usual axiomati-
zation of ZF C (L) by deleting Power Set and Σ∞(L)-Replacement, and
adding ∆0(L)-Separation. More explicitly, it consists of Extensional-
ity, Foundation (every nonempty set has an ∈-minimal member), Pairs,
Union, Infinity, Choice, and ∆0(L)-Separation. The theory K P + Infin-
ity (where KP is Kripke-Platek set theory as in [Bar]) is a significantly
stronger theory than ES T {∈} since KP includes Σ∞-Foundation, and
∆0-Replacement.
•GW [Global Well-ordering] is the axiom in the language L={∈,C},
expressing “Cwell-orders the universe”. More explicitly, GW is the con-
junction of the axioms “Cis a total order” and “every nonempty set has
aC-least element”. A well-known forcing argument shows that every
countable model of ZF C expands to a model of Z F {∈,C}+GW (see,
e.g. [Fe]).
•GW ∗is the strengthening of GW obtained by adding the following two
axioms to GW :
(a) ∀x∀y(x∈y→xCy);
(b) ∀x∃y∀z(z∈y←→ zCx).
It is easy to see that every model of Z F ({∈,C}) + GW (C) can be ex-
panded to a model of Z F ({∈,J}) + GW ∗(J) since the desired ordering
Jsatisfying GW ∗is defined by:
xJy⇔[(xCyand ρ(x) = ρ(y)) or ρ(x)∈ρ(y)],
2Rosser’s classic text [Ro] offers a development of mathematics within NF . See also Forster’s
[Fo] and Holmes’s [Ho-1] for samples of recent work in this area.
4 ALI ENAYAT
where ρis the usual ordinal-valued rank function. However, over weaker
set theories GW ∗might be stronger than GW . Also, note that if αis an
admissible ordinal, or a limit of admissible ordinals, and Lαis the α-th
level of the constructible hierarchy, then by Σ1-reflection [Bar, Theorem
4.3],
(Lα,∈, <Lα)ES T ({∈,C}) + GW ∗,
where <Lαis G¨odel’s canonical well-ordering of Lα.
•For a finite relational language L, Φ(L) is the scheme {ϕn(L) : n∈ω},
where ϕn(L) is the formula expressing:
∃κ(κ is n-Mahlo and (Vκ,∈,···)≺Σn(L)(V,∈,···).
•GBC is the G¨odel-Bernays theory3of classes GB with the class form of
the axiom of choice.
B. Models
•Models of set theory are of the form M= (M, E, ···) and N= (N , F, ···),
where E=∈M, and F=∈N.
•Mis a submodel of N, written M⊆N,if M⊆Nand E=F∩M2.
•For M= (M, E , ···), and a∈M, aE={b∈M:bEa}.
•Nend extends M(equivalently: Mis an initial submodel of N), written
M⊆eN,if Mis a submodel of Nand for every a∈M, aE=aF.
•We abbreviate “elementary end extension” by “e.e.e.”. It is well-known
that if Nis an e.e.e. of a model Mof Z F , then Nis a rank extension of
M, i.e., whenever a∈Mand b∈N\M, then Nρ(a)∈ρ(b).
•Nis said to be a conservative extension of M, if M⊆Nand for all
parametrically definable subsets Xof N,X∩Mis definable in M. For
M≺eNZF, this is equivalent to saying that for all c∈N\M, cF∩M
is parametrically definable in M.
•Suppose L ⊇ {∈} is a relational language. Given structures Mand Nof
the same language L,M≺Σn(L)Nmeans that Mis a Σn(L)-elementary
submodel of N,i.e., every formula of Σn(L) is absolute in the passage
between Mand N.For finite L, Σn(L)-truth is Σn(L)-definable within
ZF (L) for n≥1 [J, Section 14]. Therefore, for every finite language
L, and every natural number nthere is a single L-formula ϕn(x) such
that for all models Mof Z F (L), and all α∈OrdM,Mϕ(α) iff
(Vα,∈,···)M≺Σn(L)M.
•Models of GB can be written in the two-sorted form (M,A), where Mis a
model of Z F , and Ais a family of subsets of M. Since coding of sequences
is available in GB, we shall use expressions such as “f∈ A”, where fis
a function, as a substitute for the precise but lengthier expression “the
canonical code of fis in A”. S∈ A is said to form a proper class if there
is no c∈Msuch that cE=S, else Sis said to form a set. It is well-known
that for A⊆P(M),and MZF , (M,A)GB iff (M, S )S∈A ZF (L),
where L={∈} ∪ {S:S∈ A}.
3We have followed Mostowski’s lead in our adoption of the appellation GB, but Jech’s text
[Jec] dubs this theory BG. To make matters more confusing, the same theory is also known in
the literature as V NB (von Neumann-Bernays) and N B G (von Neumann-Bernays-G¨odel).
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 5
2. THE STRENGTH OF “ORD IS WEAKLY COMPACT”
In this section we probe the strength of the assertion “Ord is weakly compact”
within GBC. Here “Ord is weakly compact” is the statement in class theory as-
serting that every Ord-tree has a branch4.Ord-trees are defined in analogy with
the familiar notion of κ-trees in infinite combinatorics: (τ , <τ) is an Ord-tree, if
(τ, <τ) is a well-founded tree of height Ord such that the collection of nodes of
any prescribed ordinal rank form a set. The principal results of this section are
Theorems 2.1 and 2.2. Theorem 2.1 reveals an unexpected equivalence between the
class theory GBC+ “Ord is weakly compact” and the set theory Z F C + Φ, which
can be paraphrased as:
Φ/ZF C ∼“Ord is weakly compact”/GBC.
Theorem 2.2, on the other hand, establishes a combinatorial theorem that plays a
vital role in the proof of Theorem A.
It is a classical theorem of ZF C set theory that if κis a weakly compact
cardinal, then κis κ-Mahlo. In contrast, Corollary 2.1.1 of Theorem 2.1 implies
that the set theoretical consequences of GBC + “Ord is weakly compact” are
precisely the same as those of ZF C + Φ. Consequently, even though the theory
GBC + “Ord is weakly compact” can prove the family of statements of the form
“Ord is n-Mahlo” for standard natural numbers n, it is unable to prove “Ord is
ω-Mahlo”. To see this, note that
GBC + “Ord is ω-M ahlo”`“Z F C + Φ is finitely satisfiable”.
Therefore, since Corollary 2.1.1 is a theorem of P A,
GBC + “Ord is ω-M ahlo”`Con(GBC + “Ord is weakly compact”).
So, by G¨odel’s second incompleteness theorem,
GBC + “Ord is weakly compact” 0“Ord is ω-Mahlo”.
Theorem 2.1.
1. If (M,A)GBC + “Ord is weakly compact”, then MZF C + Φ.
2. Every consistent completion of Z F C + Φ has a countable model that has an
expansion to a model of GBC + “Ord is weakly compact”.
Before proceeding with the proof of Theorem 2.1, let us derive two corollaries.
Corollary 2.1.1. The following are equivalent for a sentence ψin the language of
set theory:
1. ZF C + Φ `ϕ.
2. GBC + “Ord is weakly compact ”`ψV.
Proof .This is an immediate consequence of coupling Theorem 2.1 with the
completeness theorem for first order logic.
4The discriminating reader might prefer “Ord has the tree property” to express this property.
But all is well, since one can use the methods of this section to show that within GBC the statement
“Every Ord-tree has a branch” is equivalent to the statement “If Tis a proper class of sentences
in the infinitary logic L∞,∞(allowing less than Ord-many conjunctions and strings of quantifiers)
every subset of which has a model, then Thas a model”.
6 ALI ENAYAT
Corollary 2.1.2. Suppose GBC + “Ord is weakly compact ”is consistent. If ψis
a sentence in the language of set theory satisfying either
1. ZF `“ψholds in L”, or
2. ZF C `“for some poset P, 1Pψ”,
then there is a model of GBC + “Ord is weakly compact ” + ψV.
Proof .This follows from Corollary 2.1.1 and the observation that if Msatisfies
ZF C + Φ,then (1) Φ holds in the constructible universe LMof M, and (2) Φ holds
in every P-generic extension of M, where P∈M. (1) is easy to verify. (2) is a
consequence of the preservation of both (a) the n-Mahlo property of a cardinal κ,
and (b) the property “Vκ≺ΣnV”, in P-generic extensions satisfying P∈Vκ. (a)
is established along the lines of the proof of [Kan, Proposition 10.13]; (b) follows
from a standard forcing argument.
The proof of part (1) of Theorem 2.1 is based on the methods introduced in [E-
2, E-3]. Notice that the proof would have been routine if GBC were to be replaced
by the stronger theory KM C (Kelley-Morse theory of classes with global choice),
since the usual ZF C -proof establishing the κ-Mahlo property of a weakly compact
cardinal κcan be conveniently implemented within KM C. We need the following
refinement of the KM C-proof, tailor made for GBC .
Lemma 2.1.2. Suppose (M,A)is a model of GBC + “Ord is weakly compact ”
and S∈ A.For every natural number n,there is a Σn-e.e.e. (N, S∗)of (M, S)
such that OrdN\Mhas a minimum element, and {cF∩M:c∈N} ⊆ A, where
F=∈N.
Proof .By global choice, there is some C∈ A such that (M , C)GW. Given
any S∈ A, (M, f, S)Z F ({∈,C, S}),and therefore, as shown in [E-3, Section 3],
for every natural number nthere is a tree τnsuch that (i) through (iii) below hold:
(i) τnis definable in (M,C, S);
(ii) (M,C, S)“τnis an Ord-tree”;
(iii) If τnhas a branch B, then there is a model (N, S∗), and an embedding
j: (M, S)→(N, S ∗),
where both (N, S∗) and jare definable within (M,C, S, B),such that
(M,C, S, B)“j(M, S ) is a Σn-elementary initial segment of (N, S∗)”.
(i) through (iii) together imply that for every natural number n,
(iv) (M,A)“(M, S) has a Σn-e.e.e. (N, S ∗)”.
It remains to show that (N, S∗) can be arranged to have a least new ordinal.
This is a consequence of the following result, which can be established by adapting
the proof of [Kau-2, Lemma 3.2] or [E-2, Theorem 3.3] to the present setting.
Proposition 2.1.3. Suppose Mis a model of ZF C (L)for a finite relational
language Lsuch that for some n∈ω,there is a model Nsuch that
(1) M≺e,Σn+2(L)N,and
(2) for every parametrically N-definable subset Xof N,
(M, X ∩M)ZF (L∪{S}),
where Sis a unary predicate interpreted by X∩M.There is a submodel N0of N
such that M≺e,Σn(L)N0and OrdN0\Mhas a least element.
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 7
Proof of part 1 of Theorem 2.1. We are now in a position to verify that if
(M,A) is a model of GBC + “Ord is weakly compact”, then Msatisfies Φ. We
shall use an external induction to show that for each standard natural number n,
if Nis a Σ1-e.e.e. of Msatisfying
(z){cF∩M:c∈N}⊆A,where F=∈N,
and γ= min(OrdN\M), then γis n-Mahlo in N.Coupled with Lemma 2.1.2, this
shows that Φ holds in M. Suppose Nis a Σ1-e.e.e. of Msatisfying (z) with a
least new ordinal γ. The case n= 0 follows from the simple observation that γ
cannot be singular in Nbecause any potential singularizing map in Nis already in
Aby (z), and GB rules out such an entity. Moreover, γmust be a strong limit
cardinal because if λ < γ then (2λ)M= (2λ)Nsince “xis the power set of y” is a
Π1-predicate and, of course, (2λ)Mis a member of Mand therefore is less than γ.
Now assume the inductive hypothesis for some n∈ω, and assume on the contrary
that γis not (n+ 1)-Mahlo in N. Let Cbe a closed unbounded subset of γall of
whose members fail to be n-Mahlo. Since C∈ A by (z), we can use Lemma 2.1.2
to get hold of a Σ1-e.e.e. (M∗, C∗) of (M, C ) with a least new ordinal γ∗.Since
(M, C) satisfies the Σ0-statement
“C∗is closed and no member of C∗is an n-Mahlo cardinal”,
(M∗, C∗) satisfies the same sentence. But then we have a contradiction because
on one hand γ∗∈C∗since C∗is closed, thus implying that γ∗is not n-Mahlo
in the eyes of M∗, and on the other hand γ∗is n-Mahlo in M∗by our inductive
hypothesis. Therefore γmust be (n+ 1)-Mahlo in N.
Proof of part 2 of Theorem 2.1. We use the strategy of Schmerl and Shelah
[SS] of building models with “built-in” e.e.e.’s using Mahlo cardinals of finite order.
Let Tbe a consistent completion of ZF C + Φ.We wish to describe a theory T
extending Tin a new language L ⊇ {∈,C}with the following two key properties:
(∗)ZF (L) + GW ⊆T.
(∗∗) Every model Mof Thas a conservative e.e.e..
Note that if Mis a model of Tand Ais the family of M-definable subsets of M, then
by (∗) (M,A) is a model of GBC. Moreover, it is easy to see5that (∗∗) implies that
(M,A) satisfies “Ord is weakly compact”. Therefore, to prove part (2) of Theorem
2.1 it is sufficient to construct a consistent extension Tof Tsatisfying properties
(∗)and (∗∗)above. This will be accomplished in Lemmas 2.1.4 and 2.1.5.
The language Lis obtained by enriching {∈} with a binary relation symbol C,
and two sets {Sn:n∈ω}and {Un:n∈ω}of unary predicates.
•Let L0:= {∈,C}, and Ln+1 := L0∪ {Si:i≤n+ 1}∪{Ui:i≤n+ 1}.
The desired theory Tis Taugmented with 4 sets of axioms A1through A4in the
extended language L, as described below:
• A1:= ZF (L) + GW (C).
5See the proof of (3) ⇒(1) of Lemma 3.3 for more detail.
8 ALI ENAYAT
• A2:= {σn:n∈ω},where σnasserts that Sn+1 is a satisfaction predicate
over (V,∈,C, Si, Ui)i≤n. More specifically, σnsays that Sn+1 contains the
Σ0(Ln)-theory of (V,∈,C, Si, Ui)i≤n, and Sn+1 satisfies Tarski’s inductive
truth clauses.
• A3:= {ψn:n∈ω},where ψnasserts that Un+1 is a class of ordinal codes
of unary formulas that describe a nonprincipal Ord-complete ultrafilter
over subsets of Ord that are definable in
(V,∈,C, Si, Ui)i≤n.
Note that by GW (C) there is a definable bijection between the universe
and the class of ordinals, and therefore, every object can be canonically
coded by an ordinal. Also note that to formulate ψnwe need to use the
predicate Sn+1 to express definability within (V,∈, Si, Ui)i≤n.Finally, we
point out that the Ord-complete condition of Un+1 can be expressed in
one sentence asserting
pτ(x, c)∈κq∈Un+1 ⇒ ∃α < κ pτ(x, c) = αq∈Un+1 ,
where τis a definable term of Ln, c is a parameter in V,κis a cardinal,
and ϕ7→ pϕqis a canonical coding of formulas by ordinals.
• A4:= {Un⊆Un+1 :n∈ω}.
The following proposition is essentially due to Schmerl and Shelah [SS] and is
our main tool in the construction of a model of T . First we need a definition:
•Suppose A= (A, ···) and B= (B , · · ·) are models in common language,
and A≺B.Ais relatively saturated in B, written A≺RS B,if for every
X⊆Awith |X|<|A|, every 1-type over Xrealized in Bis already
realized in A, i.e., for every b∈B, there is some a∈Asuch that
(A, x, a)x∈X≡(B, x, b)x∈X.
Proposition .Suppose θis an (n+ 1)-Mahlo cardinal for some n∈ω, and for
some k∈ω,Pi⊆Vθfor i < k. There is an n-Mahlo cardinal γ < θ such that
(Vγ,∈, Pi∩Vγ)i<k ≺RS (Vθ,∈, Pi)i<k.
Proof .Let C1:= {α < θ : (Vα,∈, Pi∩Vα)i<k ≺(Vθ,∈, Pi)i<k }.C1is a closed
subset of θby Tarski’s elementary chain theorem, and it is unbounded in θby a
Skolem hull argument. Since (Vθ,∈)Z F ,
(∗)∀α∈C1|Vα|=iα=α.
Let C2be the subset of C1consisting of elements α∈C1satisfying:
∀γ < α ∀a∈Vθ∃b∈Vα(Vθ, x, a)x∈Vγ≡(Vθ, x, b)x∈Vγ.
Note that if γ∈C2is an inaccessible cardinal, then
(Vγ,∈, Pi∩Vγ)i<k ≺RS (Vθ,∈, Pi)i<k.
So the proof would be complete once we verify that C2is a closed unbounded subset
of θ. It is clear that C2is a closed subset of θ. To see that it is also unbounded,
suppose α < θ. Using the inaccessibility of θwe can define a sequence hαn:n∈ωi
of elements of C1such that the following two conditions are satisfied for every
n∈ω:
(1) α < αn< αn+1 < θ;
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 9
(2) Every 1-type realized in (Vθ,∈, Pi, x)i<k,x∈Vαnis realized already by an element
in Vαn+1 .
Let δ:= S
n∈ω
αn.Clearly δ > α and δ∈C2because of (1), (2) and (∗).
Lemma 2.1.4. Thas a model.
Proof .Since Φ ⊆T , by the compactness theorem there is a model Mof T
containing a nonstandard element H∈ωM, and some θ0∈OrdMsuch that
VM
θ0≺Mand M“θ0is (H+ 1)-Mahlo”.
Fix r∈Msuch that M“rwell-orders Vθ0”. Since θis an inaccessible cardinal
of M,
(Vθ0,∈, r)MZ F (L0) + GW (C).
Reasoning within M, we plan to construct a strictly decreasing sequence hθj:j < Hi
of elements of OrdM, and two sequences of elements hsj:j < Hiand huj:j < Hi
of Mto construct the sequence of models hAj:j < H isuch that ∀j < H P (j)
holds in M,where
•P(j) := “For all Lj-sentences ϕ∈T , Ajϕ”, and
•Aj:= (Vθj,∈, rj, si, ui)i≤j, where uiand sirespectively interpreteUiand
Si,and rj:= r∩Vθjinterprets C.
Note that Tis a recursive theory and therefore we can meaningfully speak about T
in M(with the proviso that TMcontains formulas of infinite length and properly
extends T). The following recursive construction of length Hshould be understood
to take place within M.
To begin with, let s0=u0=∅.Note that P(0) holds. By Proposition we can
choose an H-Mahlo cardinal θ1< θ0,such that:
(1) (Vθ1,∈, r1)≺(Vθ0,∈, r).
(note: for this step, we do not require relative saturation). Let s1⊆θ1code the
satisfaction predicate for (Vθ1,∈, r1), and let U1be the ultrafilter on the paramet-
rically (Vθ1,∈, r1)-definable subsets of θ1that is generated by α0:= θ1∈θ0,i.e., if
Y⊆θ1is definable in (Vθ1,∈, r1) by a formula ϕ(x),then
Y∈ U1iff (Vθ0,∈, r)ϕ(α0).
Let u1⊆θ1code U1. Note that P(1) holds. The next step of the construction is
more subtle. By Proposition , there is some (H−1)-Mahlo cardinal θ2< θ1such
that:
(2) (Vθ2,∈, r2, s1∩θ2, u1∩θ2)≺RS A1:= (Vθ1,∈, r1, s1, u1).
Let s2⊆θ2code the satisfaction predicate for (Vθ2,∈, r2, s1∩θ2, u1∩θ2). We wish
to extend the ultrafilter coded by u1∩θ2to an ultrafilter U2on the family of all
subsets Yof θ2that are definable in the model (Vθ2,∈, r2, s1∩θ2, u1∩θ2) such
that P(2) holds. By (2) there is some α1∈θ1such that α0and α1have the same
L1-type in A1over Vθ2.Therefore, we can use α1to define the desired U2extending
U ∩ θj+1 via
Y∈ U2iff (Vθ1,∈, r1, s1, u1)ϕ(α1),
where ϕdefines Yin (Vθ2,∈, r2, s1∩θ2, u1∩θ2). It is clear that if u2⊆θ2codes U2,
then P(2) holds. Since θ0was chosen to be (H+ 1)-Mahlo, thanks to Proposition
10 ALI ENAYAT
this process can be continued a total of Hsteps to produce sequences hθj:j < H i,
hαj:j < H i,hsj:j < Hiand huj:j < H isuch that for all j < H the following
hold in M:
(a) θj+1 is (H−j)-Mahlo and αj+1 ∈θj+1;
(b) αjand αj+1 have the same Lj+1-type in Aj+1 over Vθj+2;
(c) uj+1 ⊆θj+1 codes the ultrafilter generated by αj;
(d) sj+1 ⊆θj+1 codes the satisfaction predicate for (Vθj+1 ,∈, rj+1, si, ui)i≤j+1.
This makes it evident that M∀j < H P (j). The desired model of Tis
(VθK,∈)M, rK, si, uii<ω ,
where K < H is any nonstandard element of ωM.
Lemma 2.1.5. Every model Mof Thas a conservative e.e.e. .
Proof .Suppose M= (M, E , ···) is a model of T. Let
U:= [
n∈ω
(Un)E.
By design, Ucodes an OrdM-complete nonprincipal ultrafilter Uover the Boolean
algebra of subsets of OrdMthat are parametrically definable in M. Let Nbe the
definable ultrapower6of Mmodulo U. Usual arguments show that Nis an e.e.e.
of M. Moreover Nis a conservative extension of Msince any c∈Nis of the form
[τ]Ufor some Ln-term τ, and therefore
cF∩M={m∈M:M“{α∈Ord :m∈τ(α)} ∈ Un”}.
A Canonical Partition Relation
We close this section by discussing an important combinatorial principle that
will be used in the proof of Theorem A.
•For a natural number n≥1, let Ord →(Ord)nbe the statement in class
theory asserting that for every f: [Ord]n→ {0,1}there is a proper class
H⊆Ord such that fis constant on [H]n.
•Given f: [Ord]n→Ord, and H⊆Ord, H is f-canonical if there is some
S⊆ {1,···, n}such that for all sequences α1<·· · < αn, and β1<·· · < βn
of elements of H,
f(α1,···, αn) = f(β1,···, βn)⇔ ∀i∈S(αi=βi).
Note that if S=∅, then fis constant on [H]n, and if S={1,···, n},then
fis injective on [H]n.
•Ord → ∗(Ord)nis the statement in class theory asserting that for every
f: [Ord]n→Ord there is some proper class H⊆Ord such that His
f-canonical.
It is a theorem of ZF C [Bau, Theorem 2] that if κis a weakly compact cardinal
then κ→ ∗(κ)nholds for all n<ω. This is based on Baumgartner’s observation
that the proof of the Erd¨os-Rado canonical partition theorem [ER] for ℵ0→ ∗ (ℵ0)n
shows that κ→ ∗(κ)nfollows from κ→(κ)2n.
6See the proof of part of Lemma 3.3 for more detail on the construction of such ultrapowers.
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 11
Theorem 2.2.
1. ∀n∈ω,GBC + “Ord is weakly compact” `Ord →(Ord)n.
2. ∀n∈ω,GBC + “Ord is weakly compact” `Ord → ∗(Ord)n.
Proof .The usual ramification tree proof from Z F C establishing the partition
properties of weakly compact cardinals (as in [Kan, Theorem 7.8]) works in the
GBC context to prove (1) as well, by an external induction on n. (2) is a corollary
of (1), and the fact that the aforementioned Erd¨os-Rado ZF C proof of κ→ ∗(κ)n
from κ→(κ)2ncan be conveniently implemented in the GBC context to establish
GBC +Ord →(Ord)2n`Ord → ∗ (Ord)n.
Remark 2.2.1. If GBC is replaced by KM C (Kelley-Morse theory of classes with
global choice), then in both parts of Theorem 2.2 “∀n∈ω” can be moved to the
right hand side of the provability symbol `.
3. AUTOMORPHISMS FROM MAHLO CARDINALS
The main result of this section is Theorem A, which is the analogue of Theorem
1.1 for ZF C + Φ.In the next section we shall prove a “reversal” of Theorem A by
showing that over the weak fragment EST ({∈,C}) + GW ∗the assumption that T
contains ZF C + Φ is indeed necessary.
Theorem A. Suppose Tis a consistent completion of Z F C+ Φ.There is a model
Mof T+ZF (C) + GW such that Mhas a proper e.e.e. Nthat possesses an
automorphism whose fixed point set is M.
The proof of Theorem A is presented at the end of the present section, once
the machinery of generic ultrafilters and iterated ultrapowers are put into place.
However, we can describe the high-level strategy of the proof here: Mis the {∈
,C}-reduct of the model of GBC + “Ord is weakly compact” whose existence is
certified by part (2) of Theorem 2.1; Nis the Z-iterated ultrapower of Mmodulo a
“generic ultrafilter” (where Zis the linearly ordered set of integers); and the desired
automorphism of Nis induced by the automorphism n7→ n+ 1 of Z.
For a countable model (M,A) of GBC, let Bbe the Boolean algebra
{S∈ A :S⊆OrdM}.
Our first goal is to construct ultrafilters Uover Bwith certain combinatorial prop-
erties that yield desirable structural properties of U-based ultrapowers. We shall
employ the conceptual framework of forcing in order to efficiently present the nec-
essary bookkeeping arguments. Our notion of forcing is the poset P, where
P:= {S∈B:Sis unbounded in OrdM},
ordered under inclusion.
•A subset Dof Pis dense if for every X∈Pthere is some Y∈ D with
Y⊆X.
• U ⊆ Pis a filter if it satisfies (1) ∅/∈U; (2) Uis closed under intersections;
and (3) Uis upward closed.
•A filter U ⊆ Pis P-generic over (M,A) if U ∩ D 6=∅whenever Dis a
dense subset of Pthat is parametrically definable in (M,A).
12 ALI ENAYAT
•A filter U ⊆ Pis (M,A)-complete if for every f:OrdM→κ, where
κ∈OrdMand f∈ A,there is some H∈ U such that fis constant on U.
Note that if a filter U ⊆ Pis (M,A)-complete, then Uis a nonprincipal ultrafilter
on Bsince for each Y∈B,the characteristic function of Yis constant on some
member of U. We therefore refer to (M,A)-complete filters as ultrafilters.
•Let Γ be a canonical bijection between OrdM×OrdMand OrdM. Every
g:OrdM→ {0,1}codes a sequence DSg
α:α∈OrdMEof subsets of
OrdM, where Sg
α:= {β∈OrdM:g(Γ(α, β)) = 1}.
•A filter U ⊆ Pis (M,A)-iterable7if Uis (M,A)-complete, and for every
g∈ A and g:OrdM→ {0,1},{α∈Ord :Sg
α∈U}∈A.
•A filter U ⊆ Pis (M,A)-canonically Ramsey if for every
f: [OrdM]n→OrdM,
where nis a standard natural number, and f∈ A, f is canonical on some
H∈ U, i.e., there is some S⊆ {1,· · ·, n}such that for all sequences
α1<···< αn, and β1<···< βnof elements of H,
f(α1,···, αn) = f(β1,···, βn)⇔ ∀i∈S(αi=βi).
The usual proof establishing the existence of filters meeting countably many dense
sets shows:
Proposition 3.1. If (M,A)is a countable model of GBC,then there is a generic
filter Uover (M,A).
The following result reveals the key properties of generic ultrafilters.
Theorem 3.2. If (M,A)is a model of GBC + “Ord is weakly compact ”and U
is (M,A)-generic, then
1. Uis (M,A)-complete;
2. Uis (M,A)-iterable;
3. Uis (M,A)-canonically Ramsey.
Proof .(1): Given f∈ A with f:OrdM→κ, and κ∈OrdM, let
Df
1:= {Y∈P:fYis constant}.
Df
1is dense since OrdMbehaves like a regular cardinal in models (M,A) of GBC.
(2): For Xand Yin B, let us write X⊆∗Y(read “Xis almost contained in
Y”) if X\Yis bounded in OrdM(equivalently: X\Yis a set in M). Also, let “X
decides Y” abbreviate
“X⊆∗Yor X⊆∗OrdM\Y”.
For each g:OrdM→ {0,1}, with g∈ A, let
Dg
2:= {Y∈P:∀α∈OrdMYdecides Sg
α}.
7This notion of iterability is related to, but different from, its namesake in the theory of large
cardinals used in [Kan, p.249]. In the present context iterability provides a useful combinatorial
condition for the ultrapower construction to be iterated, using finite supports, along any linearly
ordered set. Our iterable ultrafilters are precisely the analogues of Kunen’s “M-ultrafilters” (as
in [Ku], or [Jec]) for models of GBC .
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 13
We observe that if every Dg
2is dense, then (2) holds. To show the density of
Dg
2suppose X∈P. We first claim that there is an A-coded sequence F=
DFα:α∈OrdMEsatisfying the following two properties:
(∗)∀α∈OrdMFα=Sg
α∩Xor Fα=X\Sg
α;
(∗∗)∀α∈OrdMT
δ<α
Fδis unbounded in X.
Argue within (M,A).For each s:α→ {0,1},define hFs
δ:δ < αiby:
Fs
δ:= Sg
δ∩X, if s(δ) = 1;
X\Sg
δ,if s(δ)=0.
Consider the subtree τof 2<Ord consisting of function s:α→ {0,1}such that
\
δ<α
Fs
δis unbounded in X.
It is easy to see that τhas nodes of every rank α∈Ord,because each level of τgives
rise to a partition of Xinto 2|α|pieces, and so by Power Set and Replacement, one
of the pieces must be a proper class since Xitself is a proper class. By weak com-
pactness of Ord,τhas a branch in A, which is the desired sequence hFα:α∈Ordi.
We can now define a proper class Y={yα:α < Ord}by transfinite induction
within (M,A) such that Yis almost contained in every Fαas follows:
•y0is the first element of F0;
•For α > 0, yαis the least member of T
δ≤α
Fδabove {yδ:δ < α}.
It is clear that Ydecides each Sg
α. Therefore Dg
2is dense.
(3): Suppose f: [OrdM]n→OrdM, where nis a standard natural number,
and f∈ A. Let
Df
3:= {Y∈P:fis canonical on Y}.
By Theorem 2.2, Df
3is dense.
Remark 3.2.1. In sharp contrast to the usual scenario in the theory of large
cardinals, aP-generic filter is never normal. This is because if f∈ A and for
every α∈OrdMf−1{β∈OrdM:β > α} ∈ P, then the following set Df
4forms a
dense subset of P:
Df
4:= {Y∈P:∃g:Y→1−1Ysuch that g∈ A and g < f on Y}.
We are now in a position to examine ultrapowers and their iterations.
Lemma 3.3. The following are equivalent for a countable model (M,A)of GBC :
1. (M,A)“Ord is weakly compact”.
2. There is a nonprincipal (M,A)-iterable ultrafilter U.
3. (M, S)S∈A has a proper conservative e.e.e. (N, S∗)S∈A.
Proof .(1) ⇒(2) follows from Proposition 3.1 and part (2) of Theorem 3.2.
For (2) ⇒(3),let (N, S∗)S∈A be the ultrapower of (M, S)S∈A modulo U. More
precisely, let Fbe the family of functions ffrom OrdMinto Msuch that fis
canonically coded by some element of A. For f∈ F, let [f]Ube the U-equivalence
class of fconsisting of members of Fthat agree with fon a member of U. Define
N:=(N, F ), and S∗for S∈ A as follows
N:= {[f]U:f∈ F};
14 ALI ENAYAT
[f]F[g] iff {α∈OrdM:f(α)E g(α)}∈U, where E=∈M;
[f]U∈S∗iff {α∈OrdM:f(α)∈S}∈U.
Thanks to the presence of a global well-ordering in A, the Lo´s Theorem for ul-
trapowers holds in this context. Consequently, if Uis a non-principal ultrafilter,
then Nis a proper elementary extension of M(with the obvious identification of
the U-equivalence classes of constant maps with elements of M). The (M,A)-
completeness property of U, coupled with the existence of a global well-ordering in
A, assures us that Nis an end extension of N. To verify the conservativity clause
one invokes the conservativity clause to show that A={cF∩M:c∈N}8.
To establish (3) ⇒(1), suppose τis an Ord-tree coded in A. We may assume
without loss of generality that τ= (OrdM, <τ), where the tree ordering <τsatisfies
(∗) (M, τ )∀α, β ∈Ord (α <τβ→α∈β).
Therefore, τis end extended by τ∗(in the sense that if α <τ∗β∈OrdMthen
α∈OrdM).Now let δ∈OrdN\OrdM, and define
B:= {α∈OrdM:α <τ∗δ}.
Since τ≺eτ∗and (∗) holds, Bforms a branch of τ, and by conservativity, B∈ A.
Note that the equivalence of (2) and (3), as well as (3) ⇒(1) do not require
the countability condition on (M,A).
For an (M,A)-iterable ultrafilter U, the fact that the U-based ultrapower does
not introduce new subsets of OrdMallows one to iterate the ultrapower formation
any finite number of times to obtain the finite iteration U ltU,n(M, S )S∈A. Indeed,
the finite iteration can also be obtained in one step by defining an ultrafilter Un
on OrdMn
. To do so, suppose X⊆OrdMn
,where Xis coded in A. By
definition, X∈ Un+1 iff
{α1:{(α2,···, αn+1 ) : (α1, α2,···, αn+1)∈X}∈Un}∈U.
(the iterability condition ensures that Un+1 is well-defined). Moreover, the process
of ultrapower formation modulo Ucan be iterated along any linear order Lto
yield the iterated ultrapower UltU,L(M, S)S∈A. There are two equivalent ways of
describing the isomorphism type of UltU,L(M, S )S∈A:
1. The model theoretic approach: This is essentially Gaifman’s iteration method in
his treatment of models of arithmetic [G]. Using an iterable ultrafilter Uone first
defines, for each positive natural number n, a complete n-type Γnover the model
(M, S)S∈A by defining Γn(x1,···, xn) as the set of formulas ϕ(x1,···, xn) such that
{(α1,···, αn):(M, S )S∈A ϕ(α1,···, αn)}∈Un.
Here ϕis a formula in the language L∗={∈} ∪ {S:S∈ A} (since for each
m∈M, {m}∈A,for all intents and purposes Lhas constant symbols for elements
of Mas well). Now augment the language L∗with a set of new constant symbols
{cl:l∈L}, and define ΨL:= {cl1∈cl2:l1<Ll2}and
TU,L:= ΨL∪ {ϕ(cl1,···, cl1) : ϕ(x1,···, xn)∈Γn(x1,···, xn)}.
8See [Kan, Lemma 19.1(c)] for a similar proof.
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 15
TU,Lturns out to be a complete Skolemized theory. Therefore UltU,L(M, S )S∈A can
be meaningfully defined as the prime model of TU,L.
2. The algebraic approach: This is how iterated ultrapowers are handled in set
theory `a la Kunen [Ku].Let I= [L]<ω = the family of finite subsets of L. One
builds a category Cwhose objects are models of the form NS,for S∈I,and whose
morphisms are maps
πS1,S2:NS1→NS2,with S1⊆S2∈I,
such that the following properties hold:
•N∅=M, and for S∈Iwith |S|=n > 1,NS=M(OrdM)n/Un;
•πS1,S2is an elementary embedding, whenever S1⊆S2∈I;
•πS1,S3=πS2,S3◦πS1,S2, whenever S1⊆S2⊆S3∈I.
In this approach, UltU,L(M, S)S∈A is defined as the direct limit of the category C.
One can then use the usual arguments from the theory of iterated ultrapowers (as
in [Jec], or [Kan]) to establish the following result.
Theorem 3.4. Suppose (N, S∗)S∈A := U ltU,L(M, S )S∈A,and l:= cN
l,where
(M,A)GBC, and Uis a nonprincipal (M,A)-iterable ultrafilter.
1. Elements of Nare of the form f∗(l1,···, ln),where f∈ A, and l1<···< ln;
2. for every L∗-formula ϕ(x1,···, xn),
Nϕ(l1,···, ln)iff {(α1,···, αn)∈OrdMn
:Mϕ(α1,···, αn)}∈Un;
3. {l:l∈L}is a set of order indiscernibles in (N, S∗)S∈A of order type L;
4. every automorphism hof Linduces an automorphism jhof (N, S ∗)S∈A defined
by
jh(f∗(l1,···, ln)) = f∗(h(l1),···, h(ln)).
Theorem 3.5. Suppose (M,A)GBC + “Ord is weakly compact ”, and let
hbe an automorphism of a linearly ordered set Lwith no fixed points. If Uis
(M,A)-generic, then the fixed point set of the automorphism jhof (N, S∗)S∈A
:= UltU,L(M, S )S∈A is precisely M.
Proof .Note that by part (2) of Theorem 3.2, Uis (M,A)-iterable and there-
fore Theorem 3.4 applies. Clearly jhfixes each a∈Msince the constant map
fa(x) = ais in A. To see that jhfixes no member of N\M, suppose that for some
f∗(l1,···, ln)∈N,
(1) f∗(h(l1),···, h(ln)) = f∗(l1,···, ln).
Since f∈ A,by Theorem 3.2 there is some H∈ U,and some S⊆ {1,···, n}such
that for all sequences α1<···< αnand β1<···< βnof elements of H,
(2) f(α1,···, αn) = f(β1,···, βn)⇔ ∀i∈S(αi=βi).
Moreover, since Hn∈ Un,
(3) (l1,···, ln)∈(H∗)n, whenever l1<L···<Lln.
(1), (2), and (3) together imply that S=∅.So fmust be constant on H, hence
f∗(l1,···, ln)∈M.
16 ALI ENAYAT
Proof of Theorem A.Start with a consistent completion Tof Z F C +Φ. By
Theorem 2.1 there is a countable Mmodel of Tthat expands to a model (M,A) of
GBC +“Ord is weakly compact”. Use Proposition 3.1 to fix some generic ultrafilter
Uover (M,A), and let
(N, S∗)S∈A := U ltU,Z(M, S)S∈A,
where Zis the ordered set of integers. Consider the automorphism
n7−→hn+ 1
of Z. By Theorem 3.5 jhis an automorphism of (N, S∗)S∈A whose fixed point set
is precisely M.
4. MAHLO CARDINALS FROM AUTOMORPHISMS
In this section we turn the table around and establish a “reversal” of Theorem
A. More specifically, in part (2) of Theorem B we show that over the weak fragment
EST +GW ∗of ZF +GW ∗(discussed in the preliminaries section), Z F C + Φ is
necessary for Theorem A. The proof of Theorem B relies on Theorem 2.1 and
Lemmas 4.1 through 4.5.
•Through this section, L:= {∈,C}, and N:= (N , F, C)ES T (L) +
GW ∗.
•For M⊆N, Mis the submodel (M, E , J) of N, where E:= F∩M2, and
J:= C∩M2.
•Mis a C-cut of N, if Nproperly C-end extends M(i.e., ∀a∈M∀b∈
N\M, a Cb) and N\Mhas no C-least member.
•Suppose Mis a C-cut of N.Mis a strong C-cut of N, if for each function
f∈Nwhose domain includes M, there is some sin N, such that for all
m∈M,
f(m)/∈Miff sCf(m).
This is the set theoretic analogue of the Kirby-Paris notion of strong cuts
[KP].
Lemma 4.1. Suppose Mis a C-cut of N.
1. ∀a∈N\M∃c∈Nsuch that N“c={x:xCa}”, and M⊆cF.
2. [∆0(L)-overspill ]If ϕ(x)is a ∆0(L)-formula such that every element of Mis
a solution of ϕ(x), then there is a solution of ϕ(x)in N\M.
Proof .(1) is an immediate consequence of GW ∗and the assumption that M
is C-end extended by N. To verify (2) assume the contrary and fix c∈Nsuch that
M⊆cF. By ∆0(L)-Separation, for some s∈N,
N“s={x∈c:¬ϕ(x)} 6=∅”.
Therefore shas a C-least element by GW , which must be the C-least member of
N\M, contradiction.
Theorem B. If jis an automorphism of Nwhose fixed point set Mis a C-initial
segment of N, and A:= {aF∩M:a∈N},then:
1. (M,A)GBC + “Ord is weakly compact ”;
2. MZF C + Φ.
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 17
Note that Theorem 2.1 shows that the second part of Theorem B follows from its
first part. The proof of part (1) of Theorem B will be presented after establishing
Lemmas 4.2 through 4.5. We begin verifying some basic closure properties of M.
•For the rest of this section jis an automorphism of Nwhose fixed point
set Mis a proper C-initial segment of N.
Lemma 4.2.
1. M⊆eN.
2. If a∈Nand aF⊆M,then a∈M.
3. If a∈M,b∈N, and Nb⊆a,then b∈M.
Proof .(1) is an immediate consequence of coupling the definition of a C-cut with
GW ∗. For (2), suppose a∈Nand aF⊆M. Then for every x∈N,
Nx∈a⇔Nx∈j(a).
Hence by Extensionality, j(a) = a, so a∈M. Finally, (3) follows easily from (2).
Lemma 4.3. Msatisfies ES T (L) + GW ∗+ Power Set.
Proof .Lemma 4.2 is our tool for verifying that Minherits ES T (L) + GW ∗
from N: Extensionality and Foundation are inherited by part (1); Pairs and Union
are inherited by part (2); ∆0(L)-Separation is inherited by part (3). Infinity holds
by coupling the assumption that Mforms a C-initial segment of N, with the fact
that ωNis first order definable in N(and is therefore fixed by every automorphism
of N). GW ∗is also easily verified by parts (1) and (2) of Lemma 4.2.
To verify Power Set in M, observe that by part (3) of Lemma 4.2, if a∈M
then all subsets of a(in the sense of N) are in M. By part (1) of Proposition 4.1,
there is some c∈Nsuch that M⊆cF, and by ∆0(L)-Separation in Nthe set
d:= {x∈c:x⊆a}exists in N. Since d∈Mby part (3) of Proposition 4.2, this
shows that Msatisfies Power Set.
Lemma 4.4 unveils the most important feature of M. It will be used in Lemma
4.5 to uniformly translate global predicates in (M, S)S∈A to local ones in N.
Lemma 4.4. Mis a strong C-cut of N.
Proof .Recall that by assumption Mis a C-initial segment of N. Clearly Mis
aC-cut of Nsince if ais the C-least element of N\M, then by a reasoning similar
to the proof of part (2) of Lemma 4.2, j(a) = a, which contradicts a /∈M. Next,
suppose fis the graph of a function in Nwhose domain includes M. Let g:= j(f),
and note that f /∈M,g /∈M, and f6=g. Therefore:
(1) ∀m∈M[f(m) = g(m)⇐⇒ f(m)∈M].
We wish to find s∈Nsuch that for all m∈M, f (m)/∈Miff sCf(m).Without
loss of generality there is some m0∈Mwith f(m0)/∈M. Fix c∈Nsuch that
M⊆cFand
Nc⊇(f∪(∪f)∪(∪ ∪ f)∪g∪(∪g)∪(∪ ∪ g)).
Let m∈Mwith m0Cm, and consider the ∆0(L)-formula ϕ(x) with parameters
m,fand g:
ϕ(x, m) := ∃v∈c[vCmand (x=f(v)6=g(v))].
18 ALI ENAYAT
Note that by ∆0(L)-Separation Nthinks that for every m∈Mwith m0Cm,
{x∈c:ϕ(x, m)}exists and has a C-least member h(m).By (1), for every m∈M
with m0Cm, h(m)∈N\M. Therefore,
(2) ∀m∈M[m0Cm⇒NmCh(m)].
Observe that the predicate ψ(m) expressing mCh(m) is a ∆0(L)-formula, because
all the quantifiers of ψcan be limited to c. So by ∆0(L)-overspill there is some
a∈N\Msuch that NaCh(a).The desired sdemonstrating the strength of M
in Nis h(a) since
(3) ∀m∈M[m0Cm⇒Nh(a)Eh(m)],
which, coupled with (2), implies
(4) ∀m∈M[h(a)Cm⇐⇒ f(m)/∈M].
Lemma 4.5. Let L∗={∈} ∪ {S:S∈ A}.For every L∗-formula
ϕ(−→
x , −→
S),
with free variables −→
xand parameters −→
Sfrom A,there is some ∆0(L)-formula
θϕ(−→
x , −→
b),
where −→
bis a sequence of parameters from N,such that for all sequences −→
aof
elements of Mof the same length as −→
x ,
(M, S)S∈A ϕ(−→
a , −→
S)iff Nθϕ(−→
a , −→
b).
Proof .Fix an element d∈Nwith M⊆dF. Since EST does not guarantee
the existence of Cartesian products, we first wish to show that for each standard
natural number nthere is some element cn∈Nsuch that M⊆(cn)Fand the
n-fold Cartesian product (cn)nexists in N. By Lemma 4.3, Power Set holds in
M, so for every standard natural number nand every x∈M, Mthinks that the
Cartesian product xnexists. Given a fixed standard natural number n, consider
the ∆0(L)-formula ψn(x) :
ψn(x) := ∃y∈d∃z∈d(“y={t∈d:tCx}” and z=yn).
Since every element of Mis a solution to ψn, by ∆0(L)-overspill there is some
solution aof ψnin N\M. The desired cnis {t∈d:tCa}.
We construct θϕby recursion on the complexity of ϕas follows:
•If ϕis an atomic formula of the form Si(v), where vis a term, then choose
b∈Nsuch that bF∩M=Si, and define θϕ:= (v∈b). For other atomic
formulas ϕ, θϕ:= ϕ.
•If ϕ=¬δ, then θϕ:= ¬θδ;
•If ϕ=δ1∨δ2, then θϕ:= θδ1∨θδ2;
•If ϕ=∃vδ(v , −→
x) and the length of −→
xis n, then consider the function
f(−→
x) defined in Non (cn+1)nby
f(−→
x) := the C-least v∈cn+1 such that θδ(v, −→
x),
if ∃v∈cn+1 θδ(v, −→
x); and 0 otherwise. Note that the graph of fis defined
by a ∆0(L)-formula within N, so fis coded in Nsince the n-fold Cartesian
product (cn+1)n+1 exists in Nas arranged earlier. By Lemma 4.4, there
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 19
is some s∈N, such that for all m∈M, f (m)∈Miff f(m)Es. Now
use GW ∗to find b∈Nsuch that Nb={x:xCs},and define:
θ∃vδ(v,−→
x):= ∃v∈b θδ(v, −→
x).
We are now in possession of sufficient machinery to present:
Proof of Theorem B.By Lemma 4.3 Msatisfies E ST (L)+ Power Set +
GW ∗. Hence we only need to verify:
(1) Σ∞(L∗)-Replacement, where L∗={∈} ∪ {S:S∈ A},and
(2) Ord is weakly compact.
To establish Σ∞(L∗)-Replacement, suppose that for some L∗-formula ϕ(x, y) (with
suppressed parameters), and some a∈M,
(∗) (M, S)S∈A ∀x∈a∃!y ϕ(x, y ).
We first wish to find a C-initial segment of Ncontaining Mon which the formula
θϕ(x, y) (whose parameters are also suppressed) defines the graph of a function
whose domain is a subset of a. This can easily be accomplished by ∆0(L)-overspill:
fix d∈Nwith M⊆dFand consider the ∆0(L)-formula ψ(v):
ψ(v) := ∀x∈a[∃≤1y∈d(yCvand θϕ(x, y))].
By (∗) and Lemma 4.5, every element of Mis a solution to ψ. Hence by ∆0(L)-
overspill there is some a∈N\Msatisfying ψ. Let c={t∈d:tCa}in the sense
of N. Note that:
if m0∈aE,m1∈cF,and Nθϕ(m0, m1), then m1∈M.
Hence
{y∈cF:N∃x∈a θϕ(x, y)}={y∈M:M∃x∈a ϕ(x, y)}.
By ∆0(L)-Separation in N, the left hand side of the above equation is the extension
of some element m∈N. But since mF⊆M, by part (2) of Lemma 4.2, m∈M.
This completes the verification of Σ∞(L∗)-Replacement in M.
We now know that (M, S)S∈A is a model of GBC. To show that Ord is weakly
compact in (M,A), we need to refine the reasoning used in the proof of (3) ⇒(1)
of Lemma 3.3 in order to use ∆0(L)-overspill instead of elementarity. Suppose τ
is an Ord-tree coded in A. Thanks to the existence of a global well-ordering in A
we may assume without loss of generality that τ= (M, <τ) for some relation <τ
of the form r∩M, where r∈N, and
(M, τ )∀x, y (x <τy→xCy).
Fix c∈Nwith M⊆cE. Let kbe the field of r, and consider the relational
structure τ∗:= (k, <τ∗)∈N. For x⊆k, let τ∗(x)=(x, x2∩<τ∗),and let θ(v)
be the ∆0(L)-formula:
∃z∈c[“z={y:yCv}” and “τ∗(z) is a tree” and ∀x, y ∈z(x <τ∗y→xCy)].
Clearly for all m∈M, Nθ(m). Therefore by ∆0(L)-overspill there is an element
of N\Msatisfying ϕ. This shows that there is some initial segment τof τ∗in N
whose field kcontains M, and <τis stronger than Con k. It follows that τdoes
not introduce any new elements below the elements of τ(in the sense of the tree
20 ALI ENAYAT
ordering).So we can choose t∈k\M, and define the desired branch B∈ A of τby
B:= {m∈M:m <τt}.
Remark 4.5. It is shown in [E-4] that, similar to Theorem A, Theorem 1.1 also
has a reversal: If Nis a model of the fragment I-∆0of P A (also known as bounded
arithmetic) that has an automorphism whose fixed point set is a proper initial
segment Mof N, then Mis a model of P A.
Remark 4.6. By a theorem of Kaye, Kossak, and Kotlarski [KKK, Theorem 5.6],
for a countable recursively saturated model Mof P A, the condition “Iis a strong
elementary cut of M” is equivalent to the condition “there is an automorphism of
Mwhose fixed point set is precisely I”. The analogue of this result for models of
set theory is established in [E-5].
5. CONSEQUENCES FOR N F U
In this section we reap the benefits of Theorems A and B for the Quine-Jensen
system of set theory N F U . The theory N F U was introduced by Jensen [Jen] as a
modification of Quine’s elegant formulation NF (New Foundations) [Q] of Russell’s
theory of types. N F is a first order theory formulated in the usual language of set
theory {∈} whose axioms consist of the stratifiable comprehension scheme and the
usual extensionality axiom. The stratifiable comprehension scheme is the collection
of sentences of the form “{x:ϕ(x)}exists”, provided there is an integer valued
function fwhose domain is the set of all variables occurring in ϕ, which satisfies
the following two requirements:
1. f(v) + 1 = f(w), whenever (v∈w) is a subformula of ϕ;
2. f(v) = f(w), whenever (v=w) is a subformula of ϕ.
Since the formula ϕ(x) := (x=x) is stratifiable, NF proves the existence of
a universal set, but the stratification requirement allows NF to avoid - at least
seemingly - the usual paradoxes of the existence of “large sets” such as Russell’s
and Burali-Forti’s. However, the formal consistency of NF relative to any ZF -style
system of set theory remains an open question.
Jensen’s variant N F U of N F is obtained by modifying the extensionality axiom
so as to allow urelements (hence the Uin N F U ). Inspired by Specker’s work [Sp]
on the equiconsistency of NF with the simple theory of types augmented by the
ambiguity scheme, Jensen made a breakthrough by establishing the consistency of
N F U relative to the fragment ZBQC9of Zermelo set theory consisting of EST +
Power Set. Jensen’s method, as refined by Boffa [Bo], shows that one can construct
a model of
N F U +:= NF U + Choice + Infinity,
starting from a model Mof ZBQC that has an automorphism jwith j(κ)≥(2κ)M
for some infinite cardinal κof M.In the other direction, Hinnion [Hi] showed that in
every model of Aof N F U +one can uniformly interpret a Zermelian structure ZA
of ZF C \{Power Set}, and a nontrivial endomorphism kof ZAonto a proper initial
9Zermelo set theory with bounded quantification was independently discovered by MacLane
who dubbed it ZBQC and championed it as a parsimonious foundation for mathematical practice
[Mac, p.373]. Curiously, ZB QC was also isolated by Ressayre [Re], in his work on the model
theory of weak systems of set theory (Ressayre calls this theory Bounded Set Theory, BST ).
Mathias’ [Mat] is an excellent source of information about ZB QC.
AUTOMORPHISMS, MAHLO CARDINALS, AND NFU 21
segment of ZA. Hinnion’s work on Zwas rediscovered by Holmes [Ho-1], whose
account prompted Solovay [So] to provide an alternative streamlined development of
Z. The endomorphism kcan be used to “unravel” ZAto a model (Z∗)Aof ZBQC
that has a nontrivial automorphism j. Consequently, the construction of models
of the radical system N F U is reduced to building appropriate automorphisms of
models of fragments of orthodox set theory. To summarize:
Theorem 5.1. (Jensen-Boffa-Hinnion) N F U +has a model iff there is a model M
of ZBQC that has an automorphism jsuch that for some infinite cardinal κof
M, (2κ)M≤j(κ).
The system N F U A is obtained by augmenting the theory NF U +with the
axiom10 “every Cantorian set is strongly Cantorian”. A set Xis said to be Canto-
rian if there exists a bijection between Xand the set of its singletons UCS(X) :=
{{x}:x∈X}.Strongly Cantorian sets, on the other hand, are sets Xfor which
the graph of the “obvious” bijection x7−→ {x}between Xand UC S(X) forms
a set. In the framework of the usual Zermelo-style systems of set theory, every
nonempty set is obviously strongly Cantorian. However, in N F U the universal set
Vand many other “large” sets are not Cantorian. Moreover, there are models of
N F U in which the set of finite cardinals provides an example of a Cantorian set
that fails to be strongly Cantorian. The following result shows that building models
of N F U A amounts to building appropriate automorphisms of models of the theory
ZBQC({∈,C}) + GW ∗(C).
Theorem 5.2. (Folklore). Let T0:= ZBQC({∈,C}) + GW ∗(C).
1. If AN F U A, then there is an initial segment Nof (Z∗)Asatisfying T0such
that the collection of Cantorian elements (CZ )Aof ZAform a C-cut of N,and
there is an automorphism jof Nwhose fixed point set is (C Z )A.
2. If NT0has an automorphism jfixing a C-cut Mof N,and for some infinite
cardinal κ, (2κ)M≤j(κ), then there is a model Aof N F U A such that Mis
isomorphic to (CZ)A.
Holmes introduced an extension N F U M of N F UA in [Ho-1] and proved in [Ho-
2] that N F U M is consistent relative to ZF C + “there is a measurable cardinal”.
This provided an upper bound for the consistency strength of N F U A until Solovay
established the unexpected calibrations of (1) the consistency strength of N F U A
in terms of Mahlo cardinals [1995, unpublished], and (2) the consistency strength
of the system N F U B in terms of weakly compact cardinals [So] (N F U B is an
intermediate system between NF U A and N F UM ). The next result uses Theorems
A and B to reveal an intimate connection between Z F C + Φ and N F U A.
Theorem 5.3. The following are equivalent for a theory T in the language {∈}:
1. Tis a consistent completion of ZF C + Φ.
2. There is a model Aof N F U A such that T=T h(CZ)A.
Proof .For (1) ⇒(2), suppose Tis a consistent completion of Z F C + Φ. By
Theorem A there is a model M= (M , E, C) of Tsuch that MZF (C) + GW ,
and Mhas a proper elementary C-end extension Nsuch that Mis the fixed point
set of some automorphism of N. So by part (2) of Theorem 5.2, there is a model
10This axiom was first considered by Henson [He].
22 ALI ENAYAT
Aof N F U A such that T=T h(CZ )A.(2) ⇒(1) is an immediate consequence of
coupling part (1) of Theorem 5.2 with Theorem B.
Corollary 5.4. (Solovay for LCZ , Holmes [Ho-2] for CZ)N F U A proves the exis-
tence of n-Mahlo cardinals in CZ.
Corollary 5.5. (Solovay) The following theories are equiconsistent:
1. T1:= ZF C +{“there is an n-Mahlo cardinal”:n∈ω}.
2. T2:= N F U A.
Proof .To see that Con(T1)⇒Con(T2), assume that T1has a model M. By
Proposition of Section 2, Z F C + Φ is finitely satisfiable in Mand therefore
ZF C + Φ is consistent. Now use (1) ⇒(2) of Theorem 5.3. The implication
Con(T2)⇒Con(T1) is an immediate consequence of (2) ⇒(1) of Theorem 5.3.
Our last corollary, which follows directly from Corollaries 2.1.1 and 2.1.2 and
Theorem 5.3, shows that if N F UA is consistent, then it cannot settle the truth
value of many statements, such as V=Lor Continuum Hypothesis, in CZ.
Corollary 5.6. Assume N F U A is consistent. If ψis a statement of set theory
satisfying either
1. ZF `“ψholds in L”, or
2. ZF C `“for some poset P, 1Pψ”,
then there is a model Aof N F U A such that (CZ)Aψ.
Remark 5.7. Part 1 of Theorem 5.2, Theorem B, and [E-2, Theorem 3.6] together
imply that there are countable models of ZF C + Φ that are not isomorphic to any
model of the form (CZ)A, with AN F U A. However, every countable recursively
saturated model of Z F C + Φ is of the form (CZ)A, for some AN F U A. This
can be verified by coupling Theorem 5.3 with the classical theorem of Barwise and
Ressayre [D, Theorem 4.59] on the resplendence property of countable recursively
saturated models. Conversely, it can be shown (using Theorem B, Theorem 5.2,
and [E-3, Theorem 4.5(i)]) that if AN F U A and ωAis nonstandard, then (CZ)A
is recursively saturated.
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