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On perturbations of continuous structures
Itaı̈ Ben Yaacov
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ON PERTURBATIONS OF CONTINUOUS STRUCTURES
hal-00259783, version 1 - 29 Feb 2008
ITAÏ BEN YAACOV
Abstract. We give a general framework for the treatment of perturbations of types
and structures in continuous logic, allowing to specify which parts of the logic may
be perturbed. We prove that separable, elementarily equivalent structures which are
approximately ω-saturated up to arbitrarily small perturbations are isomorphic up to
arbitrarily small perturbations (where the notion of perturbation is part of the data).
As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories
all of whose separable models are isomorphic up to arbitrarily small perturbations.
We also present a brief treatment of unbounded continuous structures, which is closer
in spirit to Henson’s logic for Banach space structures than that present in [BU]. We
introduce single point compactification as an alternative method for bringing such structures back into the setting of bounded continuous first order logic. This allows us to
obtain a proof of an unpublished theorem of Henson characterising theories of Banach
spaces which are separably categorical up to small perturbation of the norm as a special
case of the first result.
Introduction
In this paper we define what we call perturbation systems and study their basic properties. These are objects which formalise the intuitive notion of perturbing chosen parts
of a continuous logical structure by arbitrarily small amounts.
One motivation for this notion is an attempt to generalise an unpublished theorem of
C. Ward Henson, giving a Ryll-Nardzewski style characterisation of complete continuous
theories of pure Banach spaces which are separably categorical up to arbitrarily small
perturbation of the norm (but not of the underlying linear structure). Unfortunately
this result is stated in terms which are very specific to Banach space structures, and thus
only serves as a limited guide to the general result (in hind sight it is even somewhat
misleading). In Theorem 3.4 we give a general characterisation of countable complete
Date: 29th February 2008.
2000 Mathematics Subject Classification. 03C35,03C90,03C95.
Key words and phrases. continuous logic, metric structures, perturbation, categoricity.
Research partially supported by NSF grant DMS-0500172.
The author would like to thank the Isaac Newton Institute and the organisers of the programme on
Model Theory and Applications to Algebra and Analysis, during which this work was initiated.
The author would like to thank C. Ward Henson for helpful discussions.
The author would like to thank Hernando Tellez for a careful reading of the manuscript.
1
2
ITAÏ BEN YAACOV
continuous theories which are separably categorical up to arbitrarily small perturbation,
where the precise notion of perturbation is part of the given data alongside the theory.
There is another hurdle though, lying in the way we treat unbounded structures such
as Banach spaces in continuous logic. The method of splitting such a structure into a
many-sorted structure, each sort consisting of a bounded ball, as suggested in [BU], is
unsatisfactory when it is the norm (and therefore the partition into sorts) we wish to
perturb. For this purpose we define unbounded continuous signatures and structures,
and show that unbounded continuous logic is essentially the same as Henson’s logic
of positive bounded formulae. To avoid repeating the same work twice, we show how
to reduce problems concerning unbounded structures and theories to the bounded case
through the addition of a single point at infinity. In particular, Theorem 3.4 holds as
stated also for unbounded theories (Theorem 4.36).
With Henson’s permission we prove a result very similar to his as Corollary 4.39 to
Theorem 3.4 (or rather, to Theorem 4.36).
A second motivation comes from some open problems concerning automorphism group
of the separable model of an ω-categorical continuous theory. Such problems could be
addressed from a model-theoretic point of view as questions concerning the theory TA
(i.e., T with a generic automorphism, or even several non-commuting ones). Just as
the underlying metric of a continuous structure induces a natural metric on the space
of types, it also induces one on its automorphism group, namely the metric of uniform
convergence (and in both cases, if the structure is discrete, so are the induced metrics,
so they do not present anything interesting in the classical discrete setting). Now the
model theoretic counterpart of the consideration of small metric neighbourhoods of an
automorphism is the consideration of (M, σ) TA up to small perturbations of σ. While
the present paper does not contain any results in this direction, it did serve well as an
example towards the general setting, and in fact was the origin of the author’s interest
in perturbations.
A common feature to these two instances is that we only allow to perturb part of the
structure while keeping the rest untouched: In the first case the norm is perturbed while
the linear structure is untouched, while in the second it is only the automorphism that we
allow to perturb (and not the original structure). Thus a “notion of perturbation” should
say what parts of the structure can be perturbed, and by how much. Also, in order to
state a Ryll-Nardzewski style result concerning perturbations we need to consider on the
one hand perturbations of (separable) models, and on the other perturbations of types.
In Section 1 we compare these two notions (perturbations of structures and of types):
requiring them to be compatible yields the notion of a perturbation radius. In order to
speak of “arbitrarily small perturbation” we need to consider a system of perturbation
radii decreasing to the zero perturbation, which with some natural additional properties
yields the notion of a perturbation system.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
3
In Section 2 we study a variant of the notion of approximate ω-saturation which takes
into account a perturbation system, and show that separable models which are saturated
in this sense are also isomorphic up to small perturbation.
In Section 3 we prove the main result (Theorem 3.4) and discuss various directions in
which it may and may not be further generalised (Theorem 3.14 vs. Example 3.10).
In Section 4 we define unbounded signatures and structures, and show how to reduce
problems in unbounded continuous logic to the classical bounded case through the addition of a point at infinity. In particular, the main theorem holds as stated for unbounded
continuous theories (Theorem 4.36).
In Section 5 we conclude with a few questions concerning perturbations of automorphisms.
For an introduction to continuous logic we refer the reader to [BU].
1. Perturbations
1.1. Perturbation pre-radii. We start by formalising the notion of allowing structures
and types to be perturbed “by this much”. We start by defining perturbation pre-radii,
which tell us which types can be changed into which:
Definition 1.1. A perturbation pre-radius ρ is a family of closed subsets {ρn ⊆ Sn (T )2 }
containing the diagonals. If X ⊆ Sn (T ), then the ρ-neighbourhood around X is defined
as:
X ρ = {q : (∃p ∈ X) (p, q) ∈ ρn }.
Notice that if X is closed then so is X ρ .
We will consider mappings which perturb structures: they need not be elementary,
and are merely required to respect the perturbation pre-radius.
Definition 1.2.
(i) Let ρ be a perturbation pre-radius, M, N T . A partial ρperturbation from M into N is a partial mapping f : M 99K N such that for
every ā ∈ dom(f ):
tpN (f (ā)) ∈ tpM (ā)ρ .
If f is total then it is a ρ-perturbation of M into N. The set of all ρ-perturbations
of M into N is denoted Pertρ (M, N).
(ii) If f ∈ Pertρ (M, N) is bijective, and f −1 ∈ Pertρ (N, M), we say that f : M → N
is a ρ-bi-perturbation, in symbols f ∈ BiPertρ (M, N).
(iii) We say that two perturbation pre-radii ρ and ρ′ are equivalent, in symbols ρ ∼ ρ′ ,
if Pertρ (M, N) = Pertρ′ (M, N) for all M, N T . We say they are bi-equivalent,
in symbols ρ ≈ ρ′ , if BiPertρ (M, N) = BiPertρ′ (M, N) for all M, N T .
Note that ρ ∼ ρ′ =⇒ ρ ≈ ρ′ .
(iv) If ρ and ρ′ are two perturbation pre-radii, we write ρ ≤ ρ′ to mean that ρn ⊆ ρ′n
for all n (i.e., ρ is stricter than ρ′ ).
4
ITAÏ BEN YAACOV
Lemma 1.3. For every perturbation pre-radius ρ there exists a minimal perturbation preradius equivalent to ρ, denoted hρi, and a minimal perturbation pre-radius bi-equivalent
to ρ, denoted JρK.
If ρ = hρi we say that ρ is reduced. If ρ = JρK we say that ρ is bi-reduced.
T
T
Proof. One just verifies that hρi = {ρ′ : ρ′ ∼ ρ} and JρK = {ρ′ : ρ′ ≈ ρ} are perturbation pre-radii which are equivalent and bi-equivalent, respectively, to ρ.
1.3
Note that JρK ≤ hρi ≤ ρ, so if ρ is bi-reduced it is reduced.
Definition 1.4. Let ρ, ρ′ be perturbation pre-radii. We define their composition as the
pre-radius ρ′ ◦ ρ defined by:
(ρ′ ◦ ρ)n = {(p, q) : ∃r (p, r) ∈ ρn and (r, q) ∈ ρ′n }.
It may be convenient to think of a perturbation pre-radius as the graphs of a family
multi-valued mappings ρn : Sn (T ) → Sn (T ). In this case, our notion of composition
above is indeed the composition of multi-valued mappings.
Notice that we also obtain a composition mapping for perturbations:
◦ : Pertρ (M, N) × Pertρ′ (N, L) → Pertρ◦ρ′ (M, L).
The minimal perturbation pre-radius is id = {idn : n < ω}, where idn is the diagonal
of Sn (T ), i.e., the graph of the identity mapping. It is bi-reduced, ρ ◦ id = id ◦ρ = ρ for
all ρ, and an id-perturbation is synonymous with elementary embedding.
1.2. Perturbation radii. A perturbation pre-radius imposes a family of conditions saying which types may be perturbed to which. We may further require these conditions to
be compatible with one another:
Definition 1.5. A perturbation radius is a pre-radius ρ satisfying that for any two types
(p, q) ∈ ρn there exist models M and N and a ρ-perturbation f : M → N sending some
realisation of p to a realisation of q.
Notice that the identity perturbation pre-radius is a perturbation radius.
We now try to break down the notion of a perturbation radius into several technical
properties and see what each of them means.
Recall that a (uniform) continuity modulus is simply a mapping δ : (0, ∞) → (0, ∞),
and that a mapping between metric spaces f : (X, d) → (X ′ , d′ ) respects δ if for all ε > 0
and all x, y ∈ X:
d(x, y) < δ(ε) =⇒ d′ (f (x), f (y)) ≤ ε.
Such a mapping f is uniformly continuous if and only if it satisfies some uniform continuity modulus.
Definition 1.6. Let ρ be a perturbation pre-radius.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
5
(i) We say that ρ respects equality if
[x = y]ρ = [x = y].
(I.e., if (p, q) ∈ ρ2 , p x = y, then q x = y as well).
(ii) We say that ρ respects a continuity modulus δ if every ρ-perturbation does.
(iii) We say that ρ is uniformly continuous if it respects some continuity modulus δ.
(iv) We say that ρ respects a continuity modulus δ trivially if for all ε > 0:
[d(x, y) < δ(ε)]ρ ⊆ [d(x, y) ≤ ε].
Lemma 1.7. A perturbation pre-radius ρ respects equality if and only if there exists a
continuity modulus δ which ρ respects trivially.
Proof. Assume first that ρ respects δ trivially. For every ε > 0 we have δ(ε) > 0,
whereby [xT= y] ⊆ [d(x, y) < δ(ε)] and thus [x = y]ρ ⊆ [d(x, y) ≤ ε]. Therefore
[x = y]ρ ⊆ ε>0 [d(x, y) ≤ ε] = [x = y]. As the other inclusion is always true, we obtain
equality.
Conversely, assume that ρ respects no δ trivially. Then there exists some ε > 0 such
that for all δ > 0 there is some pair (pδ , qδ ) ∈ ρ2 such that pδ ∈ [d(x, y) < δ] and qδ ∈
[d(x, y) > ε]. Since S2 (T )2 is compact this sequence has an accumulation point (p, q) as δ
goes to 0. Since ρ2 is closed we have (p, q) ∈ ρ2 , and clearly (p, q) ∈ [x = y] × [d(x, y) ≥ ε]
as well, so [x = y]ρ 6= [x = y].
1.7
Lemma 1.8. Let ρ be a perturbation pre-radius, δ a continuity modulus. If ρ respects δ
trivially then it respects δ. Conversely, if ρ respects δ then hρi respects δ trivially.
In particular, if ρ is reduced then it respects δ if and only if it respects it trivially.
Proof. The first statement is straightforward. For the converse, let:
Xδ ={(p, q) ∈ S2 (T ) : (∀ε > 0)(d(x, y)p < δ(ε) → d(x, y)q ≤ ε)}
\
=
[d(x, y) ≥ δ(ε)] × S2 (T ) ∪ S2 (T ) × [d(x, y) ≤ ε] .
ε>0
Let ρ′ be obtained from ρ by replacing ρ2 with ρ2 ∩ Xδ . Notice that the identity mapping
of any model of T is a ρ-perturbation and must therefore respect δ, so Xδ contains the
diagonal and ρ′ is a perturbation pre-radius. Clearly ρ′ ∼ ρ, so hρ′ i = hρi, and ρ′ respects
δ trivially, whereby so does hρi.
1.8
Note that if ρ is uniformly continuous, M, N T , and A ⊆ M, then any partial
ρ-perturbation f : A → M is uniformly continuous. It therefore extends uniquely to
a mapping f¯: Ā → M. As ρ is given by closed sets, the completion f¯ is also a ρperturbation.
Lemma 1.9. A perturbation pre-radius ρ is a perturbation radius if and only if it is
uniformly continuous and reduced.
6
ITAÏ BEN YAACOV
Proof. Left to right is easy. For right to left, consider the family F = {(f, M, N) : M, N T, f ∈ Pertρ (M, N)}. Since ρ is uniformly continuous, ultraproducts of families of triplets
in F exist, and since ρ as a perturbation pre-radius consists of closed sets, F is closed
under ultraproducts. Define ρF by:
ρF ,n = {(tp(ā), tp(b̄)) : (f, M, N) ∈ F , ā ∈ M n , b̄ = f (ā)}.
Since F is closed under ultraproducts and contains all the identity mappings, ρF is a
perturbation pre-radius. It clearly satisfies ρF ≤ ρ, ρF ∼ ρ, and as ρ is reduced we
conclude that ρF = ρ.
On the other hand, it is clear from the construction of ρF that it is a perturbation
radius.
1.9
Proposition 1.10. A perturbation pre-radius ρ is equivalent to a perturbation radius if
and only if it is uniformly continuous, in which case hρi is the unique perturbation radius
equivalent to ρ.
Proof. Immediate from Lemma 1.9.
1.10
Recall that if p(x̄, ȳ) is a partial type, then the property ∃ȳ p(x̄, ȳ) (where the existential
quantifier varies over a sufficiently saturated elementary extension) is also definable by a
partial type.
Definition 1.11. A perturbation pre-radius ρ respects the existential quantifier ∃ if for
every partial type p(x̄, ȳ):
[∃ȳ p(x̄, ȳ)]ρ = [∃ȳ pρ (x̄, ȳ)].
Lemma 1.12. A perturbation pre-radius ρ respects ∃ if and only if for every two sufficiently saturated models M, N T , tuples ā ∈ M n , b̄ ∈ N n , and c ∈ M:
tp(b̄) ∈ tp(ā)ρ ⇐⇒ (∃d ∈ N) tp(b̄d) ∈ tp(āc)ρ .
Proof. Easy.
1.12
If σ is an n-permutation, it acts on Sn (T ) by σ ∗ (p(x<n )) = p(xσ−1 (0) , . . . , xσ−1 (n−1) ) (so
σ (tp(a<n )) = tp(aσ(0) , . . . , aσ(n−1) )).
∗
Definition 1.13. A perturbation pre-radius ρ is permutation-invariant if for every n,
and every permutation σ on n elements, ρn is invariant under the action of σ. In other
words, for every p, q ∈ Sn (T ):
(p, q) ∈ ρn ⇐⇒ (σ ∗ (p), σ ∗(q)) ∈ ρn .
Proposition 1.14. Let ρ be a perturbation pre-radius. Then the following are equivalent:
(i) ρ is a perturbation radius.
(ii) ρ respects =, ∃ and is permutation-invariant.
(iii) Whenever M, N T , ā ∈ M n , b̄ ∈ N n , and tp(b̄) ∈ tp(ā)ρ , there exist an
elementary extension N ′ N and a ρ-perturbation f : M → N ′ sending ā to b̄.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
7
Proof.
(i) =⇒ (ii). Straightforward.
(ii) =⇒ (iii). Let ā ∈ M and b̄ ∈ N be such that b̄ tp(ā)ρ . Let N ′ N realise every
type of finite tuples over finite tuples in N.
Since ρ respects ∃ and N ′ is sufficiently saturated, for every c̄ ∈ M there is d¯ ∈ N ′
such that b̄d¯ tp(āc̄)ρ . Since ρ respects equality, (ai 7→ bi ) ∪ (cj 7→ dj ) is a well-defined
mapping, call it f : M 99K N ′ . Since ρ respects ∃ and is permutation-invariant, f is a
partial ρ-perturbation.
Let I be the family of all partial ρ-perturbations f : M 99K N ′ where dom(f ) is finite
containing ā, and f (ā) = b̄. For any tuple c̄ ∈ M, let Jc̄ = {f ∈ I : c̄ ⊆ dom(f )}, and let
F ⊆ P(I) be the filter generated by the Jc̄ . By the argument above F is a proper filter,
and therefore extends to an ultrafilter U .
Q
Let N ′′ = N ′ U . Let g : M → N ′′ be given by g = f ∈I f /U . In other words, for every
c ∈ M we define g(c) ∈ N ′′ to be [cf : f ∈ I] ∈ N ′′ , where cf = f (c) if c ∈ dom(f ): since
J{c} is a large set, we need not care about cf for other values of f . Identifying N ′ with
its diagonal embedding in N ′′ we have N N ′ N ′′ , and clearly g(ā) = b̄.
Finally, for every (finite) tuple c̄ ∈ M we have g(c̄) = [c̄f : f ∈ I], where c̄f = f (c̄) for
every f in the large set Jc̄ . Since tp(c̄f ) ∈ tp(c̄)ρ for all f ∈ Jc̄ , and tp(c̄)ρ is a closed set,
we must have tp(g(c̄)) ∈ tp(c̄)ρ .
We conclude that g : M → N ′′ is a ρ-perturbation as required.
(iii) =⇒ (i). Clear.
1.14
It follows that the composition of perturbation radii is again one:
Lemma 1.15. If ρ, ρ′ are perturbation radii then ρ′ ◦ ρ is a perturbation radius as well.
′
′
Proof. Assume that q ∈ pρ ◦ρ . Then there is a type r ∈ pρ such that q ∈ r ρ . Let ā p in
M. Then there is a model N and ρ-perturbation f : M → N such that f (ā) r, and a
ρ′ -perturbation g : N → L such that g ◦ f (ā) q.
1.15
Recall from [Ben03] that the type-space functor of T is a the contra-variant functor
from N to topological spaces, sending an object n ∈ N to Sn (T ), and a mapping σ : n → m
to the mapping
σ∗ :
Sm (T )
→
Sn (T )
tp(ai : i < m) 7→ tp(aσ(i) : i < n).
We obtain the following elegant characterisation of perturbation radii:
Lemma 1.16. A perturbation pre-radius ρ is a perturbation radius if and only if for every
n, m ∈ N and mapping σ : n → m, the induced mapping σ ∗ : Sm (T ) → Sn (T ) satisfies
that for all p ∈ Sm (T ):
σ ∗ (pρ ) = σ ∗ (p)ρ .
Viewing ρ as the family of graphs of multi-valued mappings, we could write this property
more simply as σ ∗ ◦ ρm = ρn ◦ σ ∗ . Thus a perturbation pre-radius is a perturbation radius
if and only if it commutes with the type-space functor structure on {Sn (T ) : n < ω}.
8
ITAÏ BEN YAACOV
Proof. Assume that ρ is a perturbation radius, and let σ : n → m be a mapping. Let
p ∈ Sm (T ), q ∈ Sn (T ), and let a<m ∈ M realise p. Then each of q ∈ σ ∗ (pρ ) and
q ∈ σ ∗ (p)ρ is equivalent to the existence of a ρ-perturbation g : M → N such that
q = tp(g(aσ(i) ) : i < n).
Conversely, assume that σ ∗ (pρ ) = σ ∗ (p)ρ for all σ : n → m and p ∈ Sm (T ). When restricted to the special case where σ : 2 → 1 is the unique such mapping, this is equivalent
to ρ preserving equality; when restricted to the family of inclusions n ֒→ n + 1, this is
equivalent to ρ preserving ∃; and when restricted to the permutations of the natural numbers, this is equivalent to ρ being permutation-invariant. Therefore ρ is a perturbation
radius by Proposition 1.14.
1.16
Definition 1.17. We say that a perturbation radius (or pre-radius) is symmetric if
q ∈ pρ ⇐⇒ p ∈ q ρ .
Lemma 1.18. Assume that ρ is a symmetric perturbation radius, and let f ∈
Pertρ (M, N). Then there exist elementary extensions M ′ M, N ′ N, and a biperturbation f ′ ∈ BiPert(M ′ , N ′ ) extending f .
Proof. Since ρ is symmetric then f −1 : f (M) → M is a partial ρ-perturbation, and as
ρ is a perturbation radius it can be extended to a ρ-perturbation g : N → M ′ M.
Proceeding this way we may thus construct two elementary chains (Mi : i < ω) and
(Ni : i < ω) such that M0 = M, N0 = N, and two sequences of ρ-perturbations fi : Mi →
Ni and gi : Ni → Mi+1 such that f0 = f , gi ◦ fi = idMi , and fi+1 ◦ gi = idNi . Then
at the limit we obtain Mω M and Nω N, ρ-perturbations fω : Mω → Nω and
gω : Nω → Mω such that gω = fω−1 . Thus every ρ-perturbation can be extended by a
back-and-forth argument to a ρ-bi-perturbation fω ∈ BiPertρ (Mω , Nω ).
1.18
Lemma 1.19. A perturbation pre-radius ρ is a symmetric perturbation radius if and only
if it is uniformly continuous and bi-reduced.
Proof. If ρ is bi-reduced then it is reduced and symmetric, so one direction is by
Lemma 1.9. For the other, assume ρ is a symmetric perturbation radius. Let f ∈
Pertρ (M, N), and let f ′ ∈ BiPertρ (M ′ , N ′ ) extend it as in Lemma 1.18. Then f ′ ∈
BiPertJρK (M ′ , N ′ ) by definition, whereby f ′ ∈ PertJρK (M ′ , N ′ ) and f ∈ PertJρK (M, N).
Therefore JρK ∼ ρ, and as both are reduced they are equal.
1.19
Proposition 1.20. A perturbation pre-radius ρ is bi-equivalent to a symmetric perturbation radius if and only if it is uniformly continuous, in which case JρK is the unique
symmetric perturbation radius bi-equivalent to ρ.
Proof. Immediate from Lemma 1.19.
1.20
Finally, we may find the following observation useful:
Lemma 1.21. Let ρi be symmetric uniformly continuous perturbation pre-radii such
ρ1 ◦ ρ0 ≤ ρ2 . Then Jρ1 K ◦ Jρ0 K ≤ Jρ2 K.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
9
Proof. Let M0 , M1 T and f ∈ PertJρ1 K◦Jρ0 K (M0 , M1 ). Then every finite part of f can
be decomposed by definition into a partial Jρ0 K-perturbation followed by a partial Jρ1 Kperturbation. We can glue these together by an ultraproduct argument to obtain Mi′ Mi for i < 2 and M2′ T , such that f extends to f ′ : M0′ → M1′ , which in turn decomposes
into a Jρ0 K-perturbation g ′ : M0′ → M2′ followed by a Jρ1 K-perturbation h′ : M2′ → M1′ .
Since Jρi K are symmetric perturbation radii, we can construct through a back-andforth argument as in the proof of Lemma 1.18 extensions Mi′′ Mi′ for i < 3 and
g ′′ ∈ BiPertJρ0 K (M0′′ , M2′′ ), h′′ ∈ BiPertJρ0 K (M2′′ , M1′′ ). It follows that f ′′ = h′′ ◦ g ′′ ∈
Pertρ2 (M0′′ , M1′′ ) is bijective. Since ρ2 is assumed to be symmetric, f ′′ is a ρ2 -biperturbation and therefore a Jρ2 K-perturbation.
This shows that Jρ1 K ◦ Jρ0 K ≤ Jρ2 K.
1.21
1.3. Perturbation systems. A single perturbation radius gives us certain leverage at
perturbing types. But our goal is not to study perturbations by a single perturbation
radius, but rather by “arbitrarily small” perturbation radii, where the notion of a small
perturbation radius depends on the context. We formalise this through the notion of a
perturbation system:
Let R0 denote the family of perturbation pre-radii, and R denote the family of perturbation radii.
Definition 1.22. A perturbation pre-system is a mapping p : R+ → R0 satisfying:
T
(i) Downward continuity: If εn ց ε then p(ε) = p(εn ).
(ii) Symmetry: p(ε) is symmetric for all ε.
(iii) Triangle inequality: p(ε) ◦ p(ε′ ) ≤ p(ε + ε′ ).
(iv) Strictness: p(0) = id.
If in addition its range lies in R, then p : R+ → R is a perturbation system.
Given a perturbation (pre-)system p, we may define the perturbation distance between
two types p, q ∈ Sn (T ) as:
dp,n (p, q) = dp(p, q) = inf{ε ≥ 0 : (p, q) ∈ pn (ε)}.
Notice that by strictness and the triangle inequality this is indeed a [0, ∞]-valued metric,
where infinite distance means one type can under no circumstances be perturbed one
into the other.
Lemma 1.23. Let p be a perturbation pre-system. Then the family of metrics (dp,n : n <
ω) has the following properties:
(i) For every n, the set {(p, q, ε) ∈ Sn (T )2 × R+ : dp,n (p, q) ≤ ε} is closed.
(ii) If p is a perturbation system, then for every n, m < ω and mapping σ : n → m, the
induced mapping σ ∗ : Sm (T ) → Sn (T ) satisfies for all p ∈ Sm (T ) and q ∈ Sn (T ):
dp,m (p, (σ ∗ )−1 (q)) = dp,n (σ ∗ (p), q).
(Here we follow the convention that dp,m(p, ∅) = inf ∅ = ∞.)
10
ITAÏ BEN YAACOV
Conversely, given a family of metrics with values in [0, ∞] satisfying the first property,
and defining pn (ε) = {(p, q) ∈ Sn (T )2 : dp,n(p, q) ≤ ε}, we obtain that p is a perturbation
pre-system, and it is a perturbation system if and only if the second property is satisfied
as well.
Proof. This is merely a reformulation:
– Symmetry, triangle inequality and strictness correspond to each dp,n being a metric;
– Downward continuity corresponds to the set {(p, q, ε) ∈ Sn (T )2 × R+ : dp,n (p, q) ≤ ε}
being closed; and
– Each of the p(ε) being a perturbation radius corresponds to dp,m(p, (f ∗ )−1 (q)) =
dp,n (f ∗ (p), q), by Lemma 1.16.
1.23
We say that two perturbation systems p and p′ are equivalent if the perturbation
metrics dp and dp′ are uniformly equivalent on each Sn (T ).
We say that a perturbation pre-system p respects equality if p(ε) does for all ε > 0.
In this case, by Proposition 1.20 we can define Jp(ε)K = JpK(ε) to be the symmetric perturbation radius generated by p(ε). By Lemma 1.21, JpK satisfies the triangle inequality.
One can verify that JpK satisfies downward continuity, and it is clearly symmetric and
strict, so it is a perturbation system. As expected, we call JpK the perturbation system
generated by p.
If L consists of finitely many predicate symbols, a natural perturbation system for
L is the one allowing to perturb all symbols by “a little”. In order to construct it we
first define a perturbation pre-system p by letting p(ε) be the (symmetric) perturbation
pre-radius allowing the distance symbol d to change by a multiplicative factor of e±ε ,
and every other symbol to change by ±ε. Then p respects equality, and thus generates a
perturbation system JpK. Similarly, if L is an expansion of L0 by finitely many symbols,
we might want to require that all symbols of L0 be preserved precisely, while allowing
the new symbols to be perturbed as in the previous case.
A particularly interesting example of the latter kind is the case of adding a generic
automorphism to a stable continuous theory. Consider for example the case of infinite
dimensional Hilbert spaces: If σ, σ ′ ∈ U(H), then (H, σ ′ ) is obtained from (H, σ) by
a small perturbation of the automorphism (which keeps the underlying Hilbert space
unmodified) if and only if the operator norm kσ − σ ′ k is small. Thus the notion of
perturbation brings into the realm of model theory the uniform convergence topology
on automorphism groups of structures. We will say a little more about this in the last
section.
In case of a classical (i.e., discrete) first order theory T in a finite language, there are
no non-trivial perturbation systems. Indeed, let p be a perturbation system, and let P
be an n-ary predicate symbol. Let
XP = ([P (x̄)] × [¬P (ȳ)]) ∪ ([¬P (x̄)] × [P (ȳ)]) ⊆ Sn (T )2 .
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
11
Then XP ∩ pn (0) = ∅, but X is compact, so there is εP > 0 such that XP ∩ pn (εP ) = ∅.
Replacing function symbols with their graphs we may assume the language is purely relational, and as we assumed the language to be finite we have can define ε0 = min{εP : P ∈
L} > 0. By the construction every p(ε0 )-perturbation is an elementary mapping, that
is to say that a small enough perturbation, according to p, is not a perturbation at all.
Another way of sating this is that p is equivalent to the identity perturbation system.
In short, structures in a finite discrete language cannot really be perturbed. The same
argument holds if we have a pair of languages L0 ⊆ L, where we only allow to perturb
symbols in L r L0 which are finite in number.
Thus, the notion of perturbation is a new feature of continuous logic which essentially
does not exist in discrete logic.
This last statement is of course not 100% correct, as there was a finiteness assumption.
Indeed, let L = {Ei : i < ω} and let T be the theory saying that each Ei is an equivalence
relation with two equivalence classes, and every intersection of finitely many equivalence
classes of distinct Ei ’s is infinite. This is a classical example of a theory which is not
ω-categorical, but every restriction of T to a finite sub-language is. For ε > 0, let p(ε) be
the symmetric perturbation radius generated by requiring Ei to be fixed for all i < 1/ε,
and p(0) = id. Then p is a perturbation system, and “a model of T up to a small pperturbation” is the same as “a model of T restricted to a finite sub-language”. Thus T
is p-ω-categorical in the sense of Section 2 below.
2. Saturation up to perturbation
For the rest of this section, fix a perturbation system p.
Notation 2.1. If p(x) is any partial type and ε ≥ 0 then p(xε ) denotes the partial type
∃y (p(y) ∧ d(x, y) ≤ ε).
We define p(xε , y δ , . . .) similarly. We follow the convention that the metric on finite
tuples is the supermum metric, so if x̄ is a (finite) tuple of variables then p(x̄ε ) means
p(xε0 , xε1 , . . .).
This notation can (and will) be used in conjunction with previous notation. If p(x̄) is
a partial type and ρ a perturbation radius then pρ (x̄) is also a partial type, so we can
make sense of pρ (x̄ε ): ā pρ (x̄ε ) if and only if there are b̄ such that d(ā, b̄) ≤ ε and
b̄ pρ . Similarly, ā p(x̄ε )ρ if and only if there are b̄ and c̄ such that b̄ p, d(b̄, c̄) ≤ ε,
and tp(ā) ∈ tp(c̄)ε . The difference between the two examples is that in the first we first
perturb p and then allow the realisation to move a little, while in the second we do it the
other way around. Since ρ is uniformly continuous, this does not make much difference,
as for all ε > 0 and δ = δρ (ε) > 0:
[pρ (x̄δ )] ⊆ [p(x̄ε )]ρ ,
[p(x̄δ )]ρ ⊆ [pρ (x̄ε )].
Definition 2.2. A structure M is p-approximately ω-saturated if for every finite tuple
ā ∈ M, type p(x, ā) ∈ S1 (ā) and ε > 0, the partial type pp(ε) (xε , āε ) is realised in M.
12
ITAÏ BEN YAACOV
Lemma 2.3. The definition of p-approximate ω-saturation, which was given for a single
free variable x, implies the same property with any finite tuple of variables x<n .
Proof. We do this by induction on n. For n = 0 there is nothing to prove, so we assume
for n and prove for n + 1.
So let p(x≤n , ā) ∈ Sn+1 (ā) for some finite tuple ā ∈ M, where p(x≤n , ȳ) is a complete type without parameters, and let ε > 0. We need to find in M a realisation for
pp(ε) (xε≤n , āε ).
First, find δ > 0 such that [d(x, y) ≤ δ]p(ε) ⊆ [d(x, y) ≤ ε/2], so in particular, δ ≤ ε/2.
Let q(x<n , ȳ) = p(x≤n , ȳ)↾(x<n ,ȳ) . By the induction hypothesis we can realise q p(δ) (xδ<n , āδ )
in M: that is to say that we can find b<n ∈ M, and b′<n , ā′ possibly outside M, such that
d(b′<n ā′ , b<n ā) ≤ δ and q p(ε/2) (b′<n , ā′ ). Since p(ε/2) is a perturbation radius we can
find b′n (still, possibly outside M) such that pp(ε/2) (b′≤n , ā′ ). Thus in particular:
pp(ε/2) (b′n , bδ<n , āδ ).
Let r(x) = tp(b′n /b<n , ā). Using p-approximate ω-saturation, find bn ∈ M such that
ε/2 ε/2
r p(ε/2) (bn , b<n , āε/2 ). That is to say that there exist d≤n , c̄ such that
d(b≤n ā, d≤n , c̄) ≤ ε/2,
r p(ε/2) (d≤n , c̄),
From which we conclude that:
pp(ε/2) (dn , dδ<n , c̄δ )p(ε/2) ,
ε/2
pp(ε) (dn , d<n , c̄ε/2 ),
ε
ε
pp(ε) (bε/2
n , b<n , ā ),
pp(ε) (bε≤n , āε ).
2.3
Remark 2.4. Lemma 2.3 can be restated as saying that p-approximate ω-saturation is
invariant under the adjunction of the sort of n-tuples (with the supremum metric).
It follows that it is also invariant under the adjunction of any imaginary sort (with
the natural metric), as well as of the sort of ω-tuples (with the metric d(a<ω , b<ω ) =
P
2−n−1 d(an , bn )). Thus the following results can be extended to ω-tuples and imaginary sorts as well.
Lemma 2.5. Assume that M is p-approximately ω-saturated. Then for every finite tuple
ā ∈ M, type p(x̄, ā) ∈ Sn (ā) and ε > 0, pp(ε) (x̄, āε ) is realised in M.
Proof. By Lemma 2.3 we may assume that x and a are singletons.
Let εi = (1 − 2−i )ε, and choose δi > 0 small enough so that:
(i) δi ≤ 2−i−2 ε.
(ii) [d(x, y) ≤ δi ]p(ε) ⊆ [d(x, y) ≤ 2−i ].
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
13
(iii) [d(x, y) ≤ εi]p(δi ) ⊆ [d(x, y) ≤ εi + 2−i−2 ε].
Notice that the second is possible since p(ε) is uniformly continuous. The third is possible by a compactness argument using the facts that [d(x, y) ≤ εi + 2−i−2 ε] contains a
neighbourhood of [d(x, y) ≤ εi ], and
\
[d(x, y) ≤ εi ] = [d(x, y) ≤ εi ]p(0) =
[d(x, y) ≤ εi ]p(δ) .
δ>0
Let us also agree that δ−1 = ∞.
δ
We now choose a sequence bi ∈ M such that pp(εi) (bi i−1 , aεi ):
– Since δ−1 = ∞ and p(x, a) is consistent, any b0 ∈ M will do.
– Let bi be given. Then possibly outside M there exists c such that d(c, bi ) ≤ δi−1 and
pp(εi ) (c, aεi ). Let q(x, y, z) = tp(c, bi , a). By the saturation assumption there exists
i
bi+1 ∈ M such that q p(δi ) (bδi+1
, bδi i , aδi ). We know that q(x, y, z) ⊢ pp(εi ) (x, z εi ), so:
−i−2 ε
q p(δi ) (x, y, z) ⊢ pp(εi +δi ) (x, z εi +2
q p(δi ) (xδi , y δi , z δi ) ⊢ pp(εi+1 ) (xδi , z
−i−2
) ⊢ pp(εi+1) (x, z εi +2
εi +2−i−2 ε+δi
)
) ⊢ pp(εi+1) (xδi , z εi+1 ).
i
Thus pp(εi+1 ) (bδi+1
, aεi+1 ) as required.
We also know that q(x, y, z) ⊢ d(x, y) ≤ δi−1 . It follows that q p(δi ) (x, y, z) ⊢ d(x, y) ≤
2−i+1 (except when i = 0), so d(bi , bi+1 ) ≤ 2−i+1 + 2δi ≤ 2−i−1 (4 + ε), so (bi : i < ω)
is a Cauchy sequence in M and therefore converges to some b ∈ M. For all i < j < ω
we have pp(ε) (bδji , aε ), so pp(ε) (bδi , aε ) for all i < ω, and as δi → 0 we conclude that
pp(ε) (b, aε ), as required.
2.5
Proposition 2.6. Assume that M is p-approximately ω-saturated. Then for every finite
tuple ā ∈ M, type p(x̄, ā) ∈ Sn (ā) and ε > 0 there are b̄, ā′ ∈ M such that:
(i) d(ā, ā′ ) ≤ ε.
(ii) pp(ε) (b̄, ā′ ).
Proof. Let
q(x̄, ȳ, ā) := p(x̄, ȳ) ∧ ȳ = ā.
By Step II there are b̄, ā′ ∈ M such that q p(ε) (b̄, ā′ , āε ). Since q(x̄, ȳ, z̄) implies that
ȳ = z̄, so does q p(ε), so p(b̄, ā′ ) and d(ā′ , ā) ≤ ε, as required.
2.6
Proposition 2.7. Any two elementarily equivalent separable p-approximately ω-saturated
structures are p-isomorphic.
Proof. Let M ≡ N be two separable p-approximately ω-saturated models, and let ε > 0
be given. Let M0 = {ai : i < ω} and N0 = {bi : i < ω} be countable dense subsets of M
and N, respectively.
Define for convenience εi = (1 − 2−i )ε for all i < ω. As p(ε) is uniformly continuous,
we may also choose δi > 0 such that [d(x, y) ≤ δi ]p(ε) ⊆ [d(x, y) ≤ 2−i−1 ] (so in particular,
δi ≤ 2−i−1 ).
14
ITAÏ BEN YAACOV
We will construct a sequence of mappings fi : Ai → N and gi : Bi → M, where Ai ⊆ M
and Bi ⊆ N are finite, such that:
(i) A0 = B0 = ∅, and for i > 0:
Ai+1 = a≤i ∪ Ai ∪ gi (Bi )
Bi+1 = b≤i ∪ Bi ∪ fi+1 (Ai+1 ).
(ii) For all c ∈ Ai : d(c, gi ◦ fi (c)) ≤ δi .
(iii) For all c ∈ Bi : d(c, fi+1 ◦ gi (c)) ≤ δi .
(iv) For each i, fi is a p(ε2i )-perturbation and gi is a p(ε2i+1 )-one.
We start with f0 = ∅, which is 0-as we assume that M ≡ N.
Assume that fi is given. Then Ai is given, and is finite by the induction hypothesis,
and this determines Bi which is also finite. Fix enumerations for Ai and Bi as finite
tuples, and let p(x̄, ȳ) = tpN (Bi , f (Ai )).
As fi is a p(ε2i )-perturbation, there is a type q(x̄, ȳ) ∈ pp(ε2i ) such that q(x̄, Ai ) is
consistent. By p-approximate ω-saturation of M there are tuples Bi′ , A′i ⊆ M such that
−2i−1 ε)
d(Ai , A′i ) ≤ δi and M q(Bi′ , A′i )p(2
. Then gi : Bi 7→ Bi′ is p(ε2i+1 )-elementary, so it
will do.
We construct fi+1 from gi similarly.
We now have for all c ∈ Ai :
d(c, gi ◦ fi (c)) ≤ δi =⇒ d(fi+1 (c), fi+1 ◦ gi ◦ fi (c)) ≤ 2−i−1
=⇒ d(fi+1 (c), fi (c)) ≤ 2−i .
Therefore the sequence of mappings fi converges to a mapping f : A → N, where A =
S
Ai . As fi is an p(ε)-perturbation for all i so is f . As M0 ⊆ A we have Ā = M, so f
extends uniquely to a p(ε)-perturbation f¯: M → N. An p(ε)-perturbation ḡ : N → M
is constructed similarly.
Finally, for i < j < ω choose k ≥ j such that 2−k+2 ≤ δj . Then:
¯ i )) ≤ d(ai , ḡ ◦ fk (ai )) + 2−j
d(ai , ḡ ◦ f(a
≤ d(ai , gk+1 ◦ fk (ai )) + 2−j+1 + 2−j
≤ 2−j + 2−j+1 + 2−j ≤ 2−j+2 .
By letting j → ∞ we see that ḡ ◦ f¯ is the identity on M0 , and therefore on M. Similarly
f¯ ◦ ḡ = idN .
2.7
3. Categoricty up to perturbation
We now turn to the proof of a Ryll-Nardzewski style characterisation of separable
categoricity up so small perturbations.
Definition 3.1. We say that a theory T is p-κ-categorical if every two models M, N T
such that kMk = kNk = κ are p-isomorphic.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
15
Remark 3.2. It is not difficult to verify a general converse to Proposition 2.7, i.e., that if
p is a perturbation system and M and N are p-isomorphic then M ≡ N. Thus Vaught’s
Test holds just as well for perturbed categoricity: if T has no compact models and is
p-κ-categorical for some κ > |L| then T is complete.
Lemma 3.3. A complete countable theory T is p-ω-categorical if and only if all separable
models of T are p-approximately ω-saturated.
Proof. Right to left follows from 2.7.
Conversely, assume that T is p-ω-categorical, and let M T be separable. Let ā ∈ M n ,
q(x̄) = tp(ā), and let q(y, ā) ∈ S1 (ā), where q(y, x̄) ∈ Sn+1 (T ) is a complete pure type.
Let also ε > 0, and δ = δρ (ε) > 0.
By the downward Löwenheim-Skolem theorem there exists a separable model N T
such that for every n-tuple b̄ ∈ N, if b̄ satisfies pp(ε/2) then q p(ε/2) (y, b̄δ ) is realised in N.
By assumption there exists a p(ε/2)-isomorphism f : M → N. Then b̄ = f (ā) pp(ε/2) ,
and let c ∈ N be such that q p(ε/2) (c, b̄δ ). Letting d = f −1 (c) we get q p(ε) (d, āε ).
Thus M is p-approximately ω-saturated.
3.3
We observe that if p is a perturbation system, then the topology on Sn (T )Tinduced by
dp is finer than the logic topology. Indeed, if U is a neighbourhood of p, then ε>0 pp(ε) =
{p} ⊆ U, and by compactness we must have pp(ε) ⊆ U for some ε > 0.
Unlike what the statement of Corollary 4.39 might lead one to think, this topology
may be too fine: for example, in case of the identity perturbation (i.e., no perturbation
allowed at all), dp,n is a discrete metric (with values in {0, ∞}), while the standard RyllNardzewski theorem for continuous logic does consider a much coarser topology, namely
that induced by the metric d. We therefore need to combine the two metrics:
Theorem 3.4. Let T be a complete countable theory, p a perturbation system for T .
Then the following are equivalent:
(i) The theory T is p-ω-categorical.
(ii) For every n < ω, finite ā, p ∈ Sn (ā) and ε > 0, the set [pp(ε) (x̄ε , āε )] has nonempty interior in Sn (ā).
(iii) Same restricted to n = 1.
Proof.
(i) =⇒ (ii). Assume there is some finite tuple ā, n < ω and p(x̄, ā) ∈ Sn (ā),
such that for some ε > 0 the set [pp(ε) (x̄ε , āε )] has empty interior in Sn (ā). Then it
is nowhere dense in Sn (ā), and can be omitted in a dense subset of some separable
model (M, ā) Tā . Therefore [pp(ε) (x̄, āε )] is omitted in M, which is therefore not papproximately ω-saturated. Therefore T cannot be p-ω-categorical.
(ii) =⇒ (iii). Clear.
(iii) =⇒ (i). We will show that every M T is p-approximately ω-saturated. Indeed,
let ā ∈ M be a finite tuple and p(x, ā) ∈ S1 (T ). As [pp(ε) (xε , āε )] has non-empty interior
in Sn (ā) is must be realised in M.
3.4
16
ITAÏ BEN YAACOV
We may wish to combine the two metrics in a single one. While one may try to achieve
this trough various general approaches for the combination of two metrics, the specific
situation in which we find ourselves suggest a specific construction as the “natural” one.
Fix m < ω, and let Lc̄ = L ∪ {ci : i < m}, where each ci is a new distinct constant
symbol. Let Tc̄ be the (incomplete) Lc̄ -theory generated by T . We extend p into a
perturbation system pc̄ for Tc̄ as explained in the end of Section 1, i.e., by allowing the
constant symbols to move a little. It is more convenient to think in terms of a relational
language, in which each of the constants ci is represented by a unary predicate giving
the distance to ci . We therefore define p0c̄ (ε) to be the perturbation pre-radius which, for
p, q ∈ Sn (Tc̄ ), allows to perturb p to q if and only if:
(i) q↾L ∈ (p↾L )p(ε) ; and:
(ii) For all i < m and j < n: |d(xj , ci )p − d(xj , ci )q | ≤ ε.
It is easy to verify that p0c̄ is a uniformly continuous perturbation pre-system, which
generates a perturbation system pc̄ = Jp0c̄ K. Thus pc̄ can be roughly described as allowing
to perturb models of T according to p, and the move the new constants (i.e., change the
distance to them) a little as well. If c̄ = ∅ we changed nothing: p∅ = p.
By definition of the bi-reduct J·K, we have for all ε > 0, and (M, ā), (N, b̄) Tc̄ :
BiPertp0c̄ (ε) ((M, ā), (N, b̄)) = BiPertpc̄ (ε) ((M, ā), (N, b̄)) =
f ∈ BiPertp(ε) (M, N) : (∀e ∈ M, i < m) |dM (e, ai ) − dN (f (e), bi )| ≤ ε .
The space S0 (Tc̄ ) is the set of completions of Tc̄ and can be naturally identified with
Sm (T ). We define a metric d˜p,m on Sm (T ) as the image of dpc̄ under this identification.
Equivalently:
Definition 3.5. For p, q ∈ Sn (T ), we define d˜p(p, q) as the infimum of all ε for which
there exist models M, N T , ā ∈ M n and b̄ ∈ N n and a mapping f : M → N such that:
(i) ā p and b̄ q.
(ii) f ∈ BiPertp(ε) (M, N).
(iii) For all i < n and c ∈ M: |dM (c, ai ) − dN (f (c), bi )| ≤ ε.
Alternatively, we may wish to restrict pc̄ to a specific completion of Tc̄ . Any such
completion is of the form Tā = Th(M, ā), where M T and ā ∈ M n . Let us denote the
restriction of pc̄ to Tā by pā .
Of course, once we have constructed pā , we can construct d˜pā as above, and it follows
immediately from the definitions that:
Lemma 3.6. The construction p 7→ d˜p commutes with the addition of parameters, in the
sense that for all ā, b̄ and c̄, if |b̄| = |c̄| then:
d˜pā (tp(b̄/ā), tp(c̄/ā)) = d˜p(tp(b̄, ā), tp(c̄, ā)).
In an arbitrary metric space (X, d), let Bd (x, ε) denote the closed ε-ball around a point
x. The following result characterise the topology defined by d˜p:
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
17
Lemma 3.7. Fix n < ω and a finite tuple ā ∈ M T . The metric d˜pā is coarser (i.e.,
smaller) on Sn (ā) than both d and dpā , and finer than the logic topology.
Also, for every p(x̄, ā) ∈ Sn (ā), the family {[pp(ε) (x̄ε , āε )] : ε > 0} forms a base of d˜pā neighbourhood for p(x̄, ā).
Proof. Let us start by showing that for every ε > 0 there is ε′ > 0 such that:
′
′
′
[pp(ε ) (x̄ε , āε )] ⊆ Bd˜p ā (p(x̄, ā), ε) ⊆ [pp(ε) (x̄ε , āε )].
Let us first consider the case without parameters. The set {(q(x, y), q ′(x, y)) ∈
′
S2 (T ) : |d(x, y)q − d(x, y)q | ≥ ε/2} is closed and disjoint of the diagonal, so by com′
pactness there is ε′ > 0 such for all (q, q ′) ∈ p2 (ε′ ): |d(x, y)q − d(x, y)q | ≤ ε/2. We may
of course assume that ε′ ≤ ε/2, and the first inclusion follows. The second inclusion is
immediate from the definition of d˜p.
The case over parameters ā follows from the case without parameters and the fact that
by Lemma 3.6:
Bd˜p ā (p(x̄, ā), ε) = q(x̄, ā) ∈ Sn (ā) : q(x̄, ȳ) ∈ Bd˜p (p, ε) .
Finally, let K ⊆ Sn (T ) be closed in the logic topology and q ∈
/ K. Then there is ε > 0
such that q p(ε) ∩ K = ∅, and since q p(ε) is closed in the logic topology there is also ε′ > 0
′′
′′
′
such that [q p(ε) (x̄ε )] ∩ K = ∅. Letting ε′′ = min{ε, ε′} we see that [q p(ε ) (x̄ε )] ∩ K = ∅.
Therefore K is d˜p-closed. This shows that d˜p refines the logic topology. It is clearly
coarser than both d and dp. Substituting Tā for T in the last argument we get the case
with parameters.
3.7
Thus we can restate Theorem 3.4 as:
Theorem 3.8. Let T be a complete countable theory. Then the following are equivalent:
(i) The theory T is p-ω-categorical.
(ii) For every n < ω, finite ā, p ∈ Sn (ā) and ε > 0, the ε-ball Bd˜p ā (p, ε) has nonempty interior in the logic topology on Sn (ā).
(iii) Same restricted to n = 1.
The statement of the result in terms of non-empty interior may sound a little weird,
as the non-perturbed Ryll-Nardzewski theorem tells us that T is ω-categorical if and
only if the metric d coincides with the logic topology. In order to explain this apparent
discrepancy let us make a few more observations.
First, the coincidence of the logic topology with the metric d˜p is a sufficient condition
for T to be p-ω-categorical. In this case it suffices to check Sn (T ) alone (i.e., no need to
consider parameters).
Proposition 3.9. Assume that T is countable and complete, and d˜p coincides with the
logic topology on Sn (T ) for all T . Then T is p-ω-categorical.
18
ITAÏ BEN YAACOV
Proof. Let M T , and let ā ∈ M, p(x, ā) ∈ S1 (ā), and ε > 0. By assumption
Bd˜p (p(x, ȳ), ε) is a neighbourhood of p, so there is a formula ϕ(x, ȳ) such that
p ∈ [ϕ = 0] ⊆ [ϕ < 1/2] ⊆ Bd˜p (p(x, ȳ), ε′) ⊆ [pp(ε) (xε , ȳ ε )].
Therefore [ϕ(x, ā) < 1/2] is a non-empty open subset of S1 (ā) (as it contains p(x, ā)),
and is therefore realised in M. Thus pp(ε) (xε , āε ) is realised in M.
3.9
In case we only know that Bd˜p (p, ε) has non-empty interior, which needs not necessarily
contain p, it may happen that no q(x, ȳ) in this interior is consistent with r(ȳ) = tp(ā),
so pulling up to S1 (ā) we may end up with an empty set. This is why in the proof of
Theorem 3.4 we need to consider parameters.
The reader (that’s you) may think that the condition in Proposition 3.9 is more natural
and useful than that in Theorem 3.4, and will of course be perfectly right in doing so.
The reader might further think that the author (that’s me) should have proved an “if and
only if” version of Proposition 3.9 and scrap Theorem 3.4. Unfortunately, that would be
going a little too far, as we can show with a counter-example. Roughly speaking, this
example says that if the “if and only if” variant of Proposition 3.9 were true, we could
prove Vaught’s no-2-models theorem, which fails in continuous first order logic.
Example 3.10. Let T be the theory of atomless Lp -Banach lattices for some fixed
p ∈ [1, ∞), studied in [BBH]. It is known that T is ω-categorical, all of its separable models being isomorphic to Lp [0, 1]. However, if f is any positive function of norm 1 (which
determines tp(f )), say f = χ[0,1] , then Tf has precisely two non-isomorphic separable
models, namely (Lp [0, 1], χ[0,1] ) and (Lp [0, 2], χ[0,1]). (Allowing to perturb the new named
constant we see that Tf is perturbationally ω-categorical, but that’s not what we are looking at). Let g be another positive function of norm 1 such that f ∧ g = 0 (this determines
tp(f, g)). Then again, Tf,g has precisely two separable models, (Lp [0, 2], χ[0,1] , χ[1,2] ) and
(Lp [0, 3], χ[0,1] , χ[1,2] ).
Let p be the identity perturbation system for Tf , and thus pg is the perturbation system
for Tf,g that allows to perturb g while preserving all the rest untouched. Then the two
models above are pg -isomorphic, so Tf,g is pg -ω-categorical. Let πn : Sn (Tf,g ) → Sn (Tf )
be the reduct projection. As p is the identity perturbation on Tf , d˜p = d on Sn (Tf ).
Therefore, if U ⊆ Sn (Tf ) is d-open then πn−1 (U) ⊆ Sn (Tf,g ) is d˜pg -open.
But Tf is not ω-categorical, so the metric d defines a non-compact topology on Sn (T ),
whereby d˜pg defines a non-compact topology on Sn (Tf,g ), which in particular cannot
coincide with the logic topology, even though Tf,g is pg -ω-categorical.
This may nevertheless leave our hypothetical reader with a somewhat bitter taste,
since unperturbed ω-categoricity is characterised by the coincidence of the logic topology
with the metric, and thus does not seem to be follow as a special case from Theorem 3.4.
To see that it actually does, we need to explore some further properties perturbation
metrics may have.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
19
Let us start by recalling properties of the standard metric d on Sn (T ). We observe in
[BU] that the metric d has the following properties:
(i) It refines the logic topology.
(ii) If F = [p(x̄)] ⊆ Sn (T ) is closed, then so is F ε = {p : d(p, F ) ≤ ε} = [p(x̄ε )].
(iii) For every injective σ : n → m, p ∈ Sn (T ), q ∈ Sm (T ):
d(p, σ ∗ (q)) = d(σ ∗−1 (p), q).
A perturbation metric has all these properties as well, and in fact satisfies the last one
also for σ which is not injective. One last interesting property of (Sn (T ), d) is analogous
to the second property:
Lemma 3.11. If U ⊆ Sn (T ) is open, then so is U <ε = {p : d(p, U) < ε}.
Proof. It suffices to show this for a basis of open sets, i.e., for sets of the form U =
[ϕ(x̄) < ε]. But then U <ε = [inf ȳ (ϕ(ȳ) ∨ d(x̄, ȳ)) < ε] is open.
3.11
For lack of a better name, let us call provisionally a metric on a topological space open
if it satisfies the property of Lemma 3.11.
Definition 3.12. Let p be a perturbation system for T .
(i) We say that p is open if dp is open on Sn (T ) for all n.
(ii) We say that p is weakly open if for all ε > 0 and n < ω there is δ > 0 such that
for every open set U ⊆ Sn (T ):
◦
˜
˜
U dp <δ ⊆ U dp <ε .
˜
(Where U dp <δ = {p : d˜p(p, U) < δ}.)
Lemma 3.13. Let p be a perturbation system.
(i) p is weakly open if and only if for every ε > 0 and n < ω there is δ > 0 such
that for every open U ⊆ Sn (T ):
◦
˜
U p(δ) ⊆ U dp <ε .
(ii) If p is open then it is weakly open.
(iii) If d˜p is open on Sn (T ) for all n then p is weakly open.
Proof.
˜
(i) For one direction use the fact that U p(δ/2) ⊆ U dp <δ . For the other, assume
◦
˜
that U p(δ) ⊆ U dp <ε/2 and δ < ε/2. Then since the metric d is open:
◦ d<δ
◦
◦
˜
˜
˜
˜
U dp <δ ⊆ (U p(δ) )d<δ ⊆ U dp <ε/2
⊆ (U dp <ε/2 )d<δ ⊆ U dp <ε .
(ii) We use the criterion from the previous item:
◦
◦
˜
U p(ε/2) ⊆ U dp <ε = U dp <ε ⊆ U dp <ε .
(iii) Immediate from the definition.
3.13
20
ITAÏ BEN YAACOV
Theorem 3.14. Let T be a complete countable theory, p a weakly open perturbation
system. Then T is p-ω-categorical if and only if for every n, d˜p coincides with the logic
topology on Sn (T ).
Proof. Right to left is by Proposition 3.9, so we prove left to right.
Assume that T is p-ω-categorical. Fix p ∈ Sn (T ) and ε > 0. Then by definition there
◦
˜
˜
is δ > 0 such that for every open set U ⊆ Sn (T ): U dp <δ ⊆ U dp <ε/2 . We may also
assume that δ < ε.
Let U = Bd˜p (p, δ/2)◦, so U 6= ∅ by Theorem 3.4. Then:
◦
◦
◦
˜
˜
˜
p ∈ U dp <δ ⊆ U dp <ε/2 ⊆ Bd˜p (p, δ/2)dp <ε/2 ⊆ Bd˜p (p, ε)
Therefore Bd˜p (p, ε) is a logic neighbourhood of p for all p and ε > 0. Since d˜p refines the
logic topology, they must coincide.
3.14
Example 3.15. The identity perturbation system is open.
Corollary 3.16. In the case where p is the identity perturbation system, we obtain from
Theorem 3.14 the unperturbed Ryll-Nardzewski theorem: if T is complete and countable,
then it is ω-categorical if and only if d (which coincides with d˜p) coincides with the logic
topology on Sn (T ), for all n < ω.
Example 3.17. Let ā = M T , and let p = tp(ā) be isolated (i.e., d(x̄, p) is a definable
predicate). Let p be the identity perturbation for T , and pā as above be a perturbation
system for Tā allowing to move the named parameter. Then pā is open.
Proof. Exercise.
3.17
Of course, in Example 3.10 the type of the new parameter tp(g/f ) was not isolated.
4. Unbounded structures, non-compact theories and their
compactification
While we tend to claim that continuous first order logic generalises Henson’s logic
for Banach space structures (see for example [HI02]), this statement should be taken
with a grain of salt. After all, a Banach space cannot be a continuous structure, since
a continuous structure is by definition bounded. What we actually do (e.g., in [BU,
Example 4.5] and the discussion that follows it) is decompose a Banach space into a multisorted structure, where each closed ball of radius n < ω forms a sort, and thus on each
sort or tuple of sorts everything is bounded and can be rescaled into [0, 1]. Furthermore,
we can actually rescale all such sorts into the sort of the unit ball, which therefore suffices
as a single sorted structure. This translation from Banach space structures in Henson’s
logic to “unit ball structures” in continuous logic preserves such notions as elementary
classes and extensions, of type-definability of subsets of the unit ball, etc.
This technique allowed us so far to reduce every question about Banach space structures in Henson’s sense to continuous logic. However, this may fail when dealing with
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
21
perturbations. Indeed, this approach would require the set of all elements of norm ≤ 1
to remain the same, precluding perturbations of the norm we may be interested in (this
is not an issue for Example 3.10, though, since there the norm is not perturbed).
So another approach is needed.
4.1. Unbounded structures (and non-compact theories). We would like to allow
unbounded structures, while at the same time making sure that bounded parts of the
structure behave as ordinary bounded continuous structures. The “bounded parts” of a
structure are given by means of a gauge.
Definition 4.1. Let (X, d) be a metric space, ν : X → R any function. We define
X ν≤r = {x ∈ X : ν(x) ≤ r} and similarly X ν≥r , X ν<r , etc.
(i) We call X ν≤r and X ν<r the closed and open ν-balls of radius r in X, respectively.
(ii) We say that ν is a gauge on (X, d), and call the triplet (X, d, ν) a (ν-)gauged
space if ν is 1-Lipschitz in d and every ν-ball (of finite radius) is bounded in
d. Note that this implies that the bounded subsets of (X, d) are precisely those
contained in some ν-ball.
(iii) Let δ : (0, ∞) → (0, ∞) be a continuity modulus, and f : (X, d) → (X ′ , d′ ) a
mapping of metric spaces. We say that f respects δ ν-uniformly if for all ε > 0:
x, y ∈ X ν<ε
−1
∧ d(x, y) < δ(ε) =⇒ d′ (f (x), f (y)) ≤ ε.
While respecting a given δ ν-uniformly depends on the choice of ν, the fact that some
δ is respected ν-uniformly does not:
Fact 4.2. Let (X, d, ν) be a gauged space. Then a function f : (X, d, ν) → (X ′ , d′ ) is
uniformly continuous on every bounded set if and only it respects some continuity modulus
δ ν-uniformly.
Remark 4.3. We could have given a somewhat more general definition, replacing the
1-Lipschitz condition with the weaker condition that the gauge ν should be bounded
and uniformly continuous on every bounded set. This does not cause any real loss of
generality, since in that case we could define
d′ (x, y) = d(x, y) + |ν(x) − ν(y)|.
Then a set is bounded in (X, d) if and only if it is bounded in (X, d′ ), and on every
bounded set the two metrics are uniformly equivalent. (And of course, ν is 1-Lipschitz
with respect to d′ .)
Definition 4.4. An unbounded continuous signature is defined as a continuous signature,
with the following addition:
(i) On every sort S there is a distinguished unary predicate symbol νS called the
gauge symbol (which will usually be just denoted ν as the sort is always clear
from the context).
22
ITAÏ BEN YAACOV
(ii) To every n-ary symbol s there is an associated continuous function βs : (R+ )n →
R+ , usually increasing, which we call the bound function of s.
(iii) Uniform continuity moduli are specified for each symbol as usual (but see below
concerning their satisfaction).
Definition 4.5. Let L be an unbounded signature. An (unbounded) L-structure M is
defined as a bounded structure, with the following differences:
(i) Predicate symbols are no longer required to have bounded range, and may take
any value in R+ . Instead, we only require that if P is an n-ary predicate symbol
and ā ∈ M n then:
P M (ā) ≤ βP (ν M (a0 ), . . . , ν M (an−1 )).
(ii) Similarly, if f is an n-ary function symbol then:
ν M (f M (ā)) ≤ βf (ν M (a0 ), . . . , ν M (an−1 )).
(iii) The gauge symbol ν M is 1-Lipschitz. All other symbols respect their prescribed
uniform continuity moduli ν M -uniformly.
(iv) Bounded structures are allowed to be empty.
Thus, restricted to a ν-ball, everything is bounded and uniformly continuous as in
bounded continuous logic, and closed ν-balls are moreover complete.
P
As the most common bound function for n-ary symbols is the sum β(x̄) = i<n xi ,
we call it the standard bound function.
Remark 4.6. If the language contains a constant symbol 0 then the formula ν ′ (x) = d(x, 0)
′
can act as an alternative gauge. Indeed, if r ∈ R+ then M ν ≤r ⊆ M ν≤r+ν(0) , since ν is 1′
Lipschitz, and conversely M ν≤r ⊆ M ν ≤βd (r,ν(0)) by definition of an unbounded structure.
Thus we can pass between ν-balls and ν ′ -balls in a way which depends only on L.
In most cases, ν will indeed be equal to d(x, 0).
Example 4.7. Let L be a standard continuous signature, i.e., all predicate symbols take
values in [0, 1]. Let L′ be the unbounded signature obtained from L by adding a gauge
symbol νS for each sort S, and by setting βs to be the constant 1 for all symbols.
Then every L-structure M can be naturally viewed as an unbounded L′ -structure by
interpreting all gauges as the constant 1.
Example 4.8 (Banach spaces). We would like to view Banach spaces as unbounded structures. Let L = {0, +, mr : r ∈ Q}, where mr is unary scalar multiplication by r. We view
kxk as shorthand for d(x, 0), and take it to be the gauge. Let βmr be multiplication by
|r|, and for every other symbol s let βs be standard. Then every real Banach space is
naturally an (unbounded) L-structure.
This can be extended to additional structure on the Banach space. For example a
complex Banach space also has a function symbol for multiplication by i, while a Banach
lattice is given by binary function symbols ∨, ∧ (again the bound functions are standard).
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
23
Connectives are continuous functions from (R+ )n to R+ , or any convenient family
of such functions which is dense in the compact-open topology, i.e., in the topology of
. y, x + y, x/2}
uniform convergence on every compact set. We will use the system {1, x −
which generates such a dense set through composition (while alternative systems may be
. in what follows).
legitimate, we will always require the presence of 1 and −
When defining quantifiers we need to be careful. First, supx ϕ could be infinite, and
even if ϕ is bounded we may still have witnesses with arbitrarily high gauges, causing
trouble with ultraproducts and compactness. In Henson’s logic of positive bounded
formulae [HI02], where the truth values are True/False, one gets around this by restricting
quantifiers to bounded balls (and then again, one needs to play around with the radii
of the balls when considering approximations. . . ) If we tried to do the same thing with
continuous quantifiers we could run into trouble if, say, supν(x)<r ϕ < supν(x)≤r ϕ. We
will follow a different path, looking for the simplest syntactic conditions on a formula ϕ
that ensure that inf x ϕ and supx ϕ are semantically legitimate. This approach will allow
us nonetheless to recover approximate versions of bounded quantifiers later on.
We define formulae by induction, and at the same time we define whether a formula
is (syntactically) eventually constant in a variable x and/or bounded.
• Atomic formulae are defined as usual.
– If ϕ is atomic and x does not appear in ϕ, then ϕ is eventually constant in x.
– No atomic formula is bounded.
• – Every combination by connectives of formulae which are bounded (eventually
constant in x) is bounded (eventually constant in x).
. ψ is bounded for any ψ and ϕ −
. ν(x) is eventually
– If ϕ is bounded then ϕ −
constant in x.
• If ϕ is eventually constant in x then inf x ϕ and supx ϕ are formulae (but not
otherwise).
– If ϕ is bounded or eventually constant in y (for any variable y) then so are
supx ϕ and inf x ϕ.
Terms are interpreted in the usual manner. We interpret formulae in structures with
two minor twists. The first twist is that we define ϕ(ā) where ā ∈ M ∪ {∞}, allowing
. ν(∞) is zero
ai = ∞ if ϕ is eventually constant in x. If ψ(x̄) is bounded then ψ(ā) −
regardless of ā. All the other cases follow the usual rules. The second twist is that when
interpreting quantifiers we also take the value at infinity into account:
sup ϕ(x, ā) = sup ϕ(b, ā) : b ∈ M ∪ {∞}
x
inf ϕ(x, ā) = inf ϕ(b, ā) : b ∈ M ∪ {∞} .
x
We leave it to the reader to observe that formulae which are syntactically bounded
are indeed bounded in this interpretation. Similarly, if ϕ(x, ȳ) is syntactically eventually
constant in x then it is constant in x when ν(x) is large enough, equal to ϕ(∞, ȳ). Our
interpretation makes therefore sense.
24
ITAÏ BEN YAACOV
Considering the values at infinity is superfluous for unbounded structures which are
of course the kind of structures one would usually consider in unbounded logic. Still, in
some pathological situations bounded and even empty structures may arise. For things to
run smoothly in such situations it is necessary to take the value at infinity into account.
See Remark 4.10 below.
Notice that with our choice of connectives every formula is equivalent to one in prenex
normal form. Notice also that for every formula ϕ(x̄) we can construct by induction a
bound function βϕ (x̄) in a natural manner (we do not claim in any way that it is an
optimal bound function – it is just the syntactically obvious one).
The formula 1 is both bounded and constant. It follows that for natural (or even
. (k −
. ϕ) is bounded.
dyadic) constant k the formula ϕ ∧ k = k −
Let us see how to obtain approximate versions of the bounded quantifier supν(x)≤r ϕ.
We will assume ϕ to be bounded with (minimal) syntactic bound k. We observe that
. (ν(x) −
. r) is bounded as well (for dyadic r). It is further equivalent to (ϕ + (ν(x) ∧
ϕ−
.
r)) − ν(x) which is both bounded and eventually constant in x. It follows that for every
. m(ν(x) −
. r) is equivalent to one which is bounded and
natural m > 0 the formula ϕ −
eventually constant in x. Let 0 < r < r ′ , and find the least m such that we can write
r ≤ s = ℓ2−m < (ℓ + 1)2−m ≤ r ′ , and choose the minimal possible s. Define:
′
. k2m (ν(x) −
. s),
ϕ↓x≤r,r = ϕ −
′
. (k −
. ϕ)↓x≤r,r′ .
ϕ↑x≤r,r = k −
′
Both formulae are bounded and eventually constant in x. Moreover, ϕ↓x≤r,r ≤ ϕ, it
′
coincides with ϕ when ν(x) ≤ r and is equal to zero when ν(x) ≥ r ′ . Thus supx ϕ↓x≤r,r ,
′
′
which we will abbreviate by supr,r
x ϕ, is also a formula, approximating (when r is close
x≤r,r ′
r,r ′
approximates
enough to r) the expression supν(x)≤r ϕ. Similarly, inf x ϕ = inf x ϕ↑
inf ν(x)≤r ϕ.
We may further extend these abbreviations to the case where ϕ is not bounded by
′
′
truncating it at 1, defining ϕ↓x≤r,r = (ϕ ∧ 1)↓x≤r,r (and proceeding as above). This will
′
s,s′
only be used in conditions of the form supr,r
x inf y . . . ϕ = 0, whose satisfaction does not
depend on our particular choice of constant at which we truncate.
Now let L be an unbounded signature, and let {Mi : i ∈ I} be a family of L-structures
and U an ultrafilter on I. Let N0 be the set:
n
o
Y
N0 = (ai ) ∈
Mi : lim ν Mi (ai ) < ∞ .
U
For a function symbol f or predicate symbol P , and arguments (ai ), (bi ), . . . ∈ N0 , define:
f N0 (ai ), (bi ), . . . = (f Mi (ai , bi , . . .))
P N0 (ai ), (bi ), . . . = lim P Mi (ai , bi , . . .).
U
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
25
Note that by definition of N0 , the values of P Mi (ai , bi , . . .) are bounded on a large set of
indexes, so limU P Mi (ai , bi , . . .) ∈ R+ . It is now straightforward verification that N0 is an
L-pre-structure, i.e., that it verifies all the properties of a structure with the exception
that dN0 might be a pseudo-metric and needs not be complete.
Q Then the completion
N = N̂0 is called the ultraproduct of the Mi modulo U , denoted Mi /U . As we quotient
by “d((ai), (bi )) = 0”, the image in N of (ai ) ∈ N0 will be denoted [ai ]. (Compare with
the construction of ultraproducts of Banach spaces in [HI02] and of bounded continuous
structures in [BU].)
By induction on the structure of a formula we prove:
Q
Fact 4.9 (Loś’s Theorem). For every formula ϕ(x̄) and [ai ], [bi ], . . . ∈ Mi /U :
ϕ([ai ], [bi ], . . .)
Q
Mi /U
= lim ϕ(ai , bi , . . .)Mi .
U
Proof. Mostly as for bounded logic. When calculating inf x ϕ(x, ai , bi , . . .) witnesses can
always be assumed to be taken from a uniformly bounded ν-ball or to be the point at
infinity. One of the cases must hold on a large set of indexes and a corresponding witness,
either finite in the same bounded ν-ball or ∞, exists in the ultraproduct.
4.9
Remark 4.10. Loś’s Theorem may fail if our semantic interpretation does not take the
. ν(x)). Let
value at infinity into account. For example, consider the sentence ϕ = inf x (1 −
Mn be the structure consisting of two points, ν(an,0 ) = 0, ν(an,1 ) = n. Then ϕMn = 0
for
all n ≥ 0. On the other hand the ultraproduct contains a single point, ν(a0 ), so
Q
Mn /U
ϕ
= 1.
Worse still, if Mn consisted only of an,1 then their ultraproduct would be empty. This
can also be obtained with unbounded structures, for example Mn = E r B(n) where E
is a Banach space and B(n) is its open ball of radius n. (These and other pathological
examples were pointed out to the originally over-optimistic author by C. Ward Henson.)
Definition 4.11. Say that a family of conditions Σ = {ϕi ≤ ri : i ∈ λ} is approximately
finitely satisfiable if for every finite w ⊆ λ and ε > 0, the family Σ0 = {ϕi ≤ ri +ε : i ∈ w}
is satisfiable.
Corollary 4.12. If a set of sentential conditions (i.e., conditions without free variables)
is approximately finitely satisfied in a family of structures, then it is satisfied in some
ultraproduct of these structures.
Proof. Standard.
4.12
Corollary 4.13 (Bounded compactness for unbounded continuous logic). Let L be an
unbounded signature, r ∈ R+ , and let Σ be a family of conditions in the free variables
x<n . Then Σ ∪ {ν(xi ) ≤ r : i < n} is satisfiable of and only if it is approximately finitely
satisfiable.
As usual, a theory is a set of sentential conditions. The complete theory of a structure
M, elementary equivalence and elementary embeddings are defined as usual.
26
ITAÏ BEN YAACOV
Corollary 4.14. Two structures M and N are elementarily equivalent if and only if M
embeds elementarily into an ultrapower of N.
Proof. One direction is clear. For the other we observe that if M and N are elementarily
equivalent, then the elementary diagram of M is approximately finitely satisfiable in
N.
4.14
We could prove an analogue of the Shelah-Keisler theorem that if N and M are elementarily equivalent then they have isomorphic ultrapowers. We will give a more elementary
proof of a lesser result, which will suffice just as well later on:
Lemma 4.15.
(i) Two models M and N are elementarily equivalent if and only if
there are sequences M = M0 M1 . . . and NS= N0 N1S . . . where each
Mn+1 (Nn+1 ) is an ultrapower of Mn (Nn ) and n<ω Mn ≃ n<ω Nn (so their
completions are isomorphic as well).
(ii) A class of structures K is elementary if and only if it is closed under elementary
equivalence and ultraproducts.
Proof. For the first item, right to left by the elementary chain lemma, which is proved as
usual. For left to right, assume that M ≡ N. Then there is an ultrapower N1 = N U and
an elementary embedding f0 : M → N1 . Then (M, M) ≡ (N1 , f0 (M)) (in a language with
′
all elements of M named) so there exists an ultrapower M1 = M U and an elementary
embedding g0 : N1 → M1 such that g0 ◦ f0 = idM . Proceed in this manner to obtain the
sequences.
The second item is standard.
4.15
It is easily verified that any theory is logically equivalent to one which only consists
of conditions of the form ϕ = 0. A universal theory is one which only consists of conditions of the form supx̄ ϕ(x̄) = 0 where ϕ is quantifier-free (and bounded, and eventually
constant in each xi ). Observe that:
• We may express ∀x̄ ϕ(x̄) = 0 by the universal axiom scheme supx̄n,n+1 ϕ(x̄) = 0.
• If t and s are terms we can express ∀x̄ t = s by ∀x̄ d(t, s) = 0.
. ϕ = 0.
• If ϕ and ψ are formulae we can express ∀x̄ ϕ ≥ ψ by ∀x̄ ψ −
Example 4.16. We can continue Example 4.8 and give the (universal) theory of the class
of Banach spaces:
hUniversal equational axioms of a vector spacei
∀x skxk ≤ kmr (x)k ≤ s′ kxk
s, s′ dyadic, s ≤ |r| ≤ s′
∀xy kx + yk ≤ kxk + kyk
∀xy d(x, y) = kx + m−1 (y)k.
Example 4.17 (Measure algebras). Let L = {0, ∨, ∧, r}, where 0 is a constant symbol,
∨, ∧, r are binary function symbols. We use µ(x) as shorthand for d(x, 0), and take it
to be the gauge. All symbols are 1-Lipschitz and all the bound functions are standard.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
27
The universal theory of measure algebras (which are the topic of [Fre04]) consists of:
hUniversal equational axioms of relatively complemented distributive latticesi
∀xy µ(x) + µ(y) = µ(x ∧ y) + µ(x ∨ y)
µ(0) = 0
∀xy d(x, y) = µ(x r y) + µ(y r x).
We can further say that a measure algebra is atomless by the axiom scheme:
|µ(x ∧ y) − µ(x)/2| = 0.
supn,n+1
inf n+1,n+2
y
x
(This holds indeed in every atomless measure algebra: the only witnesses relevant to
supn,n+1
have measure ≤ n + 1, and for those there are witnesses y such that µ(y) ≤
x
(n + 1)/2 ≤ n + 1. Conversely, if this holds then the algebra is atomless.)
We can define type spaces more or less as usual:
Definition 4.18. Fix an unbounded signature L.
(i) Given an n-tuple ā, we define its type as usual as the set of all L-conditions in
the variables x<n satisfied by ā.
(ii) A complete n-type (in L) is the type of some n-tuple. By Corollary 4.13, this is
the same as a maximal finitely consistent set of conditions p(x<n ) such that for
some r ≥ 0 we have ν(xi ) ≤ r ∈ p for all i < n.
(iii) The set of all n-types is denoted Sn . The set of all n-types containing a theory
T (equivalently: realised in models of T ) is denoted Sn (T ).
(iv) For every condition s in the free variables x<n , [s]Sn (T ) (or just [s], if the ambient
type space is clear from the context) denotes the set of types {p ∈ Sn (T ) : s ∈ p}.
(v) The family of all sets of the form [s]Sn (T ) forms a base of closed sets for the logic
topology on Sn (T ). It is easily verified to be Hausdorff.
W
For each n < ω, we can define ν : Sn (T ) → R by ν(p) = i<n ν(xi )p . With this
definition, (Sn (T ), d, ν) is a gauged space. Applying previous definitions we have:
\
_
Snν≤r (T ) = [ν(xi ) ≤ r] = [( ν(xi )) ≤ r].
i<n
i<n
By Corollary 4.13, Snν≤r (T ) is compact. If Sn (T ) = Snν≤r (T ) for some r, then Sn (T ) is
compact. Conversely, if Sn (T ) is compact for n ≥ 1, then ν is necessarily bounded on
ν≤r
models of T , so there is some r such that T ⊢ supx ν(x)∧(r+1) ≤ r and Sm (T ) = Sm
(T )
for all m < ω. In this case all the other symbols are also bounded in models of T , so up
to re-scaling everything into [0, 1] we are in the case of standard continuous first order
logic.
S
In the non-compact case we still have Sn (T ) = r Snν≤r (T ). Thus each p ∈ Sn (T ) there
is r such that p ∈ Snν≤rW
(T ), and Snν≤r+1 (T ) is a compact neighbourhood of p (since it
contains the open set [( ν(xi )) < r + 1]). Therefore Sn (T ) is locally compact.
28
ITAÏ BEN YAACOV
4.2. On the relation with Henson’s positive bounded logic. We sketch out here
how unbounded continuous logic generalises, in an appropriate sense, Henson’s logic of
approximate satisfaction positive bounded formulae in Banach space structures. For this
purpose we will assume familiarity with the syntax and semantics of Henson’s logic (see
for example [HI02]).
The classical presentation of Henson’s logic involves a purely functional signature LH
with a distinguished sort for R. There is no harm in assuming that the distinguished sort
only appears as the target sort of some function symbols (otherwise we can add a second
copy and a single function symbol for the identity mapping into the copy, and treat the
copy as the distinguished sort). Also, there is no harm in replacing R with R+ .
We can therefore define an unbounded continuous signature L by dropping the distinguished sort and replacing all function symbols into it with R+ -valued predicate symbols.
As every sort is assumed to be normed, we identify ν with k · k. While a signature in
Henson’s logic does not specify continuity moduli and their likes, every class under consideration implies continuity moduli and bounding functions for the symbols (or else
the logic would fail to describe them), and we can use those. It is a known fact that
there exists a (universal) LH -theory, call it T0 , whose models are precisely the structures
satisfying these continuity moduli and bounds.
From now on by “structure” we mean a model of T0 , or equivalently a L-structure
(as these can be identified). The ambiguity concerning whether a structure is a Henson
or unbounded continuous structure is further justified by the fact that the definitions
of isomorphism and ultraproducts in either logic coincide. As we can moreover prove
Lemma 4.15 for Henson’s logic just as well, we conclude:
Fact 4.19. A class of structures K is elementary in Henson’s logic if and only if it is
elementary in unbounded continuous logic.
S
Fact 4.20. Let X = n<ω Xn be a topological space where each Xn is closed and Xn+1
is a neighbourhood of Xn . Then a subset F ⊆ X is closed if and only if F ∩ Xn is for all
n.
As the space of n-types is just the space of complete theories with n new constant
symbols, we conclude:
Theorem 4.21. Two n-tuples in a structure have the same type in one logic if and only
if they have the same type in the other, and this identification induces a homeomorphism
SLn H (T0 ) ≃ SLn .
Proof. The first statement is by Fact 4.19. Also, a set X ⊆ Sk·k≤r
is closed if and only if
n
the class {(M, ā) : tp(ā) ∈ X} is elementary: the bounds on the norm are needed since
we need to impose bounds on the norms of constant symbols. It follows from Fact 4.19
that the bijection SLn H (T0 ) ≃ SLn is a homeomorphism when restricted to Sk·k≤n
. Now use
n
k·k<r
Fact 4.20 and the fact that Sk·k≤r
is
compact
and
S
is
open
in
both
topologies
to
n
n
conclude that this is a global homeomorphism.
4.21
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
29
This can be restated as:
Theorem 4.22. For every set Σ(x̄) of LH -formulae there exists a set Γ(x̄) of Lconditions, and for every set Γ(x̄) of L-conditions there exists a set Σ(x̄) of LH -formulae,
such that for every structure M and ā ∈ M:
M a Σ(ā) ⇐⇒ M Γ(ā).
Remark 4.23. In Henson’s logic, the bounded quantifier ∀≤r x (∃≤r x) mean “for all (there
exists) x such that kxk ≤ r”. Thus Henson’s logic coincides with unbounded continuous
logic of normed structures where ν = k · k. One may generalise Henson’s logic to allow
an arbitrary ν and obtain full equivalence of the two logics.
For the benefit of the reader who finds this proof a little too obscure, let us give one
direction explicitly. We know that every in Henson’s logic is equivalent to one in prenex
form. Thus, every LH -formula can be assumed to be of the form:
∀≤r0 x0 ∃≤r1 x1 . . . ϕ(x̄, ȳ)
Where ϕ is a positive Boolean combination of atomic formulae of the form ti (x̄, ȳ) ≥ ri
or ti ≤ ri , and every term ti can be identified with an atomic L-formula. Replacing ti
. r or with r −
. t , we may assume all these atomic formulae are of the form
with ti −
i
i
i
ti ≤ 0. As (ti ≤ 0) ∧ (tj ≤ 0) ⇐⇒ (ti ∨ tj ) ≤ 0 and (ti ≤ 0) ∨ (tj ≤ 0) ⇐⇒ (ti ∧ tj ) ≤ 0,
we can find a single t such that ϕ(x̄, ȳ) is equivalent to t ≤ 0. We thus reduced to:
∀≤r0 x0 ∃≤r1 x1 . . . t(x̄, ȳ) ≤ 0 .
We can view t as a quantifier-free L-formula, in which case the above holds approximately
if and only if the following holds for all ε > 0:
suprx00−ε,r0 inf x≤r1 1 ,r1 +ε . . . t(x̄, ȳ) ∧ 1 ≤ 0.
Thus the approximate satisfaction of a LH -formula, and therefore of a partial type, are
equivalent to the satisfaction of a partial type in L.
4.3. Compactification. As we mentioned earlier, the multi-sorted approach to unbounded structures allows us to reduce many issues concerning unbounded structures
to their well-established analogues in bounded continuous logic, but this does not work
well for perturbations if we wish to perturb ν itself.
We could of course generalise everything we did to the unbounded case, but that would
be extremely tedious and boring. Instead, we will look for a reduction to the bounded
case in which the entire structure fits in a single sort.
One naı̈ve way would be to choose a continuous function mapping R+ into [0, 1], say
x
θ(x) = x+1
, and apply it to all the predicate symbol: for every L-structure M we define
θ
M as having the same underlying set, and for every predicate symbol P we define
θ
P M (ā) = θ(P M (ā)). It can be verified that for x, y ≥ 0: θ(x + y) ≤ θ(x) + θ(y): for
30
ITAÏ BEN YAACOV
example, this is true when x = 0, and the partial derivative with respect to x of the left
θ
hand side is smaller. It follows that dM is a metric:
θ
dM (a, b) = θ(dM (a, b)) ≤ θ(dM (a, c) + dM (c, b))
θ
θ
≤ θ(dM (a, c)) + θ(dM (c, b)) = dM (a, b) + dM (a, b).
θ
Of course dM needs not be a complete metric, so we obtain new elements. Similarly,
if T θ = Th{M θ : M T }, then we have a natural embdding of Sn (T ) in Sn (T θ ), and it
can be verified that the latter is the Stone-Čech compactification of the former. This is
essentially the same thing as allowing ∞ as a legitimate truth value (as θ extends to a
homeomorphism [0, ∞] → [0, 1]). As usual with Stone-Čech compactification, this adds
too many new types and while it does work, it is quite unmanageable. In fact even the
following is not clear (to the author):
Question 4.24. Is every model of T θ of the form M θ , where M T ?
As we pointed out earlier, the type spaces are locally compact, so we might try and
look for a single-point compactification. More precisely, we will add a unique “point at
infinity” to the structures, such that a sequence (ai : i < ω) in the original structure
converges to ∞ if and only if ν(ai ) → ∞. Of course the modified structures will not
be compact, but they will be bounded and complete, and the type spaces Sn (T ) will be
compactified (although with the exception of n = 1 this will not be by a single point).
There is one drawback to this approach, namely that this can only be done in a purely
relational language (indeed, when performing this process on a Banach space we must
give up the function symbol + as we cannot give meaning to ∞ + ∞).
The trick is as follows: First, we define a new language
L∞ = {d} ∪ {Pϕ n-ary predicate symbol : ϕ(x<n ) ∈ Lω,ω }.
Note that the language is purely relational, and we introduce a new distance relation (as
Pd(x,y) will not be a metric). We will not specify at this point the uniform continuity
moduli, but we do show below that such moduli can be chosen that would fit our purpose.
For every L-structure M we define an L∞ -structure M ∞ . Its domain is the set M ∪
{∞}. For elements coming from M we interpret the symbols as follows:
∞
PϕM (ā)
∞
dM (a, b)
θ ◦ ϕ(ā)M
P
1 + i<n ν M (ai )
θ ◦ dM (a, c) θ ◦ dM (b, c) −
:c∈M
= sup 1 + ν M (a)
1 + ν M (b) =
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
31
With the convention that ν(∞) = ∞, this extends naturally to:
∞
PϕM (. . . , ∞, . . .)
d
M∞
(a, ∞)
Clearly, dM
M
ν (ai ) → ∞.
∞
=0
= sup
θ ◦ dM (a, c)
:c∈M
1 + ν M (a)
is a metric. Moreover, if (ai : i < ω) ⊆ M then ai →M
∞
∞ if and only if
∞
Lemma 4.25. The bijection ι : (M, dM ) → (M, dM ) is ν-uniformly continuous in both
′
∞
directions, and for every r ′ > r the ν-ball M ν≤r contains a uniform dM -neighbourhood
of M ν≤r .
∞
Proof. Saying that ι is ν-uniformly continuous is (by definition of dM ) the same as
saying that θ◦d(x,y)
is ν-uniformly continuous in the argument x, which is true as it is a
1+ν(x)
continuous combination of ν-uniformly continuous predicate symbols.
Assume now that a ∈ M ν≤r . Then for every b ∈ M we have, by substituting c = b in
∞
the definition of dM :
θ(dM (a, b))
θ(dM (a, b))
∞
dM (a, b) ≥
≥
,
1 + ν(a)
1+r
Whereby:
∞
dM (a, b) ≤ θ−1 ((1 + r)dM (a, b)).
As limxց0 θ−1 (x) = 0, it follows that ι−1 is ν-uniformly continuous.
Also, for every r ′ > r:
θ(r ′ − r)
′
ν≤r
⊆ Bd (M ν≤r , r ′ − r) ⊆ M ν≤r .
BdM ∞ M ,
1+r
4.25
Proposition 4.26. For every L-structure M, M ∞ as defined above is an L∞ -structure.
That is to say that M ∞ is complete, and that we can complete the definition of L∞
choosing uniform continuity moduli for its symbols which are satisfied in every M ∞ .
Proof. Consider an L-formula of the form ϕ(x, ȳ) and a corresponding predicate symbol
Pϕ (x, ȳ) in L∞ . Let ε > 0 be given, and we need to find δ > 0 such that for all a, b ∈ M ∞ :
∞
dM (a, b) ≤ δ =⇒ sup |Pϕ (a, x̄) − Pϕ (b, x̄)|M
∞
≤ ε.
x̄
First, if ν(a), ν(b) ≥ 1/ε − 1 (where ν(∞) = ∞) then the above is satisfied regardless of
∞
∞
dM (a, b). Otherwise, without loss of generality we have ν(a) < 1/ε − 1, so if dM (a, b)
∞
is small enough we have ν(b) ≤ 1/ε. As ι−1 is uniformly continuous on (M ν≤1/ε , dM )
we conclude using the fact that ϕ and ν are uniformly continuous on (M ν≤1/ε , dM ).
∞
Since the results of Lemma 4.25 are uniform in M, the continuous predicates PϕM are
uniformly continuous, and uniformly so in all M.
32
ITAÏ BEN YAACOV
Let us now show completeness. Let (ai : i < ω) be a Cauchy sequence in M ∞ . If
ν(ai ) → ∞ then ai → ∞ (again, ν(∞) = ∞). Otherwise, there is r such that ai ∈ M ν≤r
infinitely often. Passing to a sub-sequence, we may assume that the entire sequence fits
inside M ν≤r . We now conclude using Lemma 4.25 and completeness of (M, dM ). 4.26
It is straightforward to verify that the point-at-infinity construction commutes with
the ultraproduct construction, as everything is continuous:
Y
∞ Y
Mi /U
=
Mi∞ /U .
Mi
(In particular, all the tuples
dropped
Q were
Q (ai ) such that limU ν (aMii) = ∞, which
∞
Mi /U .) Also,
during the construction of
Mi /U , satisfy [ai ] = [∞ ] = ∞ in
assume that M is an L-structure, Ñ is an L∞ -structure, and Ñ M ∞ . Then we can
recover an L-structure N on Ñ r {∞} such that Ñ = N ∞ , and then necessarily N M.
Proposition 4.27. Assume K is an elementary class of L-structures, and let
K∞ = {M ∞ : M ∈ K}.
Then K∞ is elementary.
Proof. Assume K is elementary. Then, by the arguments above, K∞ is closed under
ultraproducts, isomorphism and elementary substructures. It is therefore elementary.
4.27
By Proposition 4.27 we may replace every (unbounded) L-theory T with its (bounded)
single point compactification (or maybe boundedification?) T ∞ = ThL∞ (Mod(T )∞ ). By
naming constants we further see that L-types of tuples in M are in bijection with L∞ types of tuples in M ∞ r {∞} (i.e., in M again, but this time viewed as a subset of a
L∞ -structure).
Given a tuple ā ∈ M ∞ , let w = w(ā) = {i < n : ai 6= ∞}. Then we may identify
M∞
tp S (ā) with the pair (w, tpM (a∈w )). We can therefore express the set of types Sn (T ∞ )
as w⊆n {w} × S|w| (T ). For w ⊆ n, r ∈ R+ and ϕ(x∈w ) ∈ L, define:
(
)
w ⊆ v ⊆ n, ϕq < 1/2,
Vn,w,r,ϕ = (v, q(x∈v )) ∈ Sn (T ∞ ) : ^
.
ν(xi )q > r
i∈vrw
∞
Given a type (w, p) ∈ Sn (T ), one can verify that the family of all sets of the form Vn,w,r,ϕ
where ϕp = 0 forms a base of neighbourhoods for (w, p). In particular, the natural
inclusion Sn (T ) ֒→ Sn (T ∞ ), consisting of sending p 7→ (n, p), is an open topological
embedding. In case T is complete (so | S0 (T )| = 1), this embedding for n = 1 is indeed
a single point compactification of S1 (T ) obtained by adding the type at infinity.
Once we understand types we know what saturation means. Among other things we
have:
Lemma 4.28. An L-structure M is approximately ω-saturated if and only if M ∞ is.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
33
Proof. Follows from the facts that there is a unique point at infinity, which belongs to
∞
M ∞ , and that in the neighbourhood of every other point dM and dM are equivalent.
4.28
Finally, we point out that the theory T is bounded to begin with if and only if the
point at infinity in models of T ∞ is isolated, in analogy with what happens when one
attempts to add a point at infinity to a space which is already compact.
4.4. Perturbations of unbounded structures. We may now define perturbation radii
and systems for unbounded theories by a correct reduction to the bounded case. We fix
an unbounded theory T and its compactification T ∞ .
Definition 4.29. A perturbation pre-radius for T is defined as for a bounded theory,
i.e., as a family ρ = {ρn ⊆ Sn (T ) : n < ω} containing the diagonals. We define X ρ ,
Pertρ (M, N), BiPertρ (M, N), hρi, JρK as usual.
Let ρ be a perturbation pre-radius for T . We can always extend it to a perturbation
radius ρ∗ for T ∞ by:
∞
ρ∞
n = ((w, p), (w, q)) ∈ Sn (T ) : w ⊆ n, (p, q) ∈ ρ|w| .
Clearly, this is a perturbation pre-radius for T ∞ . Conversely, if ρ′ is a perturbation preradius for T ∞ then its restriction to S(T ), denoted ρ′ ↾S(T ) , is a perturbation pre-radius
for T , and as the inclusion Sn (T ) ⊆ Sn (T ∞ ) is open we have the identity:
ρ∞ ↾S(T ) = ρ.
Also, as every f ∈ Pertρ (M, N) extends to f ∪ (∞ 7→ ∞) ∈ Pertρ∞ (M ∞ , N ∞ ), we also
have hρ∞ i↾S(T ) ≥ hρi.
We define perturbation radii for T directly by reduction to T ∞ :
Definition 4.30.
(i) Let ρ′ be a perturbation pre-radius for T ∞ . We say that ρ′
separates infinity if for all f ∈ Pertρ′ (M ∞ , N ∞ ) and a ∈ M ∞ :
a = ∞ ⇐⇒ f (a) = ∞.
(ii) A perturbation pre-radius ρ for T is a perturbation radius if ρ∞ is a perturbation
radius for T ∞ which separates infinity.
We define a perturbations pre-system for T as a decreasing family p of perturbation
pre-radii satisfying downward continuity, symmetry, triangle inequality and strictness as
in Definition 1.22. It is a perturbation system if p(ε) is a perturbation radius for all ε,
i.e., if p∞ is a perturbation system separating infinity for T ∞ .
The first thing is to characterise perturbation radii as in Section 1, and establish more
precisely the relation between perturbations of T and of T ∞ .
Definition 4.31. Let ρ a perturbation pre-radius for T .
34
ITAÏ BEN YAACOV
(i) We say that ρ respects infinity if for all r ∈ R+ there exists r ′ ∈ R+ such that
[ν(x) ≥ r ′ ]ρ ⊆ [ν(x) ≥ r]
[ν(x) ≤ r]ρ ⊆ [ν(x) ≤ r ′ ].
and
(ii) We define when ρ respects equality, respects ∃, or is permutation-invariant as in
the bounded case.
Proposition 4.32. Let ρ be a perturbation pre-radius for T . The the following are
equivalent:
(i) ρ is a perturbation radius.
(ii) ρ respects infinity, and for every n, m ∈ N and mapping σ : n → m, the induced
mapping σ ∗ : Sm (T ) → Sn (T ) satisfies that for all p ∈ Sm (T ):
σ ∗ (pρ ) = σ ∗ (p)ρ .
(I.e., σ ∗ ◦ ρm = ρn ◦ σ ∗ as multi-valued functions).
(iii) ρ respects ∞, =, ∃, and is permutation-invariant.
(iv) ρ∞ separates ∞, respects = and ∃ and is permutation-invariant.
Proof.
(i) =⇒ (ii). Assume ρ is a perturbation radius, so ρ∞ is a perturbation radius
respecting infinity. If ρ does not respect infinity, then by definition of ρ∞ we will have
in ρ∞
1 a pair (p, q) where p is the type of a finite elements and q = tp(∞) or vice versa,
contradicting the assumption on ρ∞ .
∞
∗
Since ρ∞ is a perturbation radius, for all σ : n → m we have in S(T ∞ ): σ ∗ ◦ρ∞
m = ρn ◦σ .
As ρ∞ also separates infinity we can restrict this to S(T ) and obtain σ ∗ ◦ ρm = ρn ◦ σ ∗ .
(ii) =⇒ (iii). By restricting to the case where σ is the mapping 2 → 1, n ֒→ n + 1, or
a permutation of n ∈ N.
(iii) =⇒ (iv). By a mirror-image to the argument above, if ρ respects ∞ then ρ∞
must separate ∞.
We claim that since ρ respects ∞ and ∃ and is permutation-invariant, we have for all
n ∈ N:
∞
ρ∞
n = ((w, p), (w, q)) ∈ Sn (T ) : w ⊆ n, (p, q) ∈ ρ|w|
(i.e., the right hand side is a closed set). Indeed, assume we have pairs ((wi , pi ), (wi, qi ))
for i ∈ I and U is an ultrafilter on I, and let ((v, p), (u, q)) = limU ((wi , pi), (wi , qi )). We
need to show that v = u and (p, q) ∈ ρ|v| . First, as there are finitely many possibilities
for wi ⊆ n we may assume that wi = w ⊆ n for all i. Then we might as well assume
w = n throughout.
For s ⊆ n, let psi and qis be the restrictions of pi and qi , respectively, to x∈s . As ρ
respect ∃ and is permutation-invariant, (psi , qis ) ∈ ρ|s| . As ρ respects infinity we have:
{k}
k∈
/ v ⇐⇒ pi
{k}
→U tp(∞) ⇐⇒ qi
→U tp(∞) ⇐⇒ k ∈
/ u.
Therefore v = u, and as ρ|v| is closed (p, q) = limU (pvi , qiv ) ∈ ρ|v| . This proves our claim.
It is now immediate that as ρ respects = and ∃ and is permutation-invariant, the same
holds of ρ∞ .
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
(iv) =⇒ (i). Since then ρ∞ is a perturbation radius.
35
4.32
Corollary 4.33. Giving of a perturbation system p for T is the same as giving a family
{dp,n : n < ω}, each df p,n being a metric on Sn (T ) for which the infinite distance is
admissible, such that:
(i) For every n, the set {(p, q, ε) ∈ Sn (T )2 × R+ : dp,n (p, q) ≤ ε} is closed.
(ii) For every r ∈ R+ there is r ′ ∈ R+ such that if p, q ∈ S1 (T ) and dp,1(p, q) ≤ r,
then
ν(x)p ≥ r ′ =⇒ ν(x)q ≥ r
(iii) For every n, m < ω and mapping σ : n → m, the induced mapping σ ∗ : Sm (T ) →
Sn (T ) satisfies for all p ∈ Sm (T ) and q ∈ Sn (T ):
dp,m(p, (f ∗ )−1 (q)) = dp,n (f ∗ (p), q).
(Here we follow the convention that dp,m(p, ∅) = inf ∅ = ∞.)
Similarly, the giving of a perturbation pre-system is the same as a family of metrics
satisfying the first condition alone.
Proof. Same as Lemma 1.23, where the new condition corresponds to the requirement
that every p(ε) respect infinity.
4.33
Let us fix a perturbation system p for T , and let p∞ be the corresponding perturbation
system for T ∞ . As for plain approximate ω-saturation, we have
Lemma 4.34. A model M T is p-approximately ω-saturated if and only if M ∞ is
p∞ -approximately ω-saturated.
Proof. As for Lemma 4.28.
4.34
In particular, and two separable p-approximately ω-saturated models of T must be
p-isomorphic.
Similarly:
Lemma 4.35. Two models M, N T are p-isomorphic if and only if M ∞ and N ∞ are
p∞ -isomorphic.
The theory T is p-ω-categorical if and only if T ∞ is p∞ -ω-categorical.
We conclude that Theorem 3.4 holds as stated:
Theorem 4.36. Let T be a complete countable unbounded theory, p a perturbation system
for T . Then the following are equivalent:
(i) The theory T is p-ω-categorical.
(ii) For every n < ω, finite ā, p ∈ Sn (ā) and ε > 0, the set [pp(ε) (x̄ε , āε )] has nonempty interior in Sn (ā).
(iii) Same restricted to n = 1.
36
ITAÏ BEN YAACOV
Proof. The idea is to reduce to Theorem 3.4. Most of the reduction is in the preceding
results: T is complete if and only if T ∞ is, T is p-ω-categorical if and only if T ∞ is
p∞ -ω-categorical, etc. The last thing to check is that the property
(∗)
p(x̄, ā) ∈ Sn (ā), ε > 0
[pp(ε) (x̄ε , āε )]◦ 6= ∅
=⇒
holds for T, p if and only it holds for T ∞ , p∞ .
Indeed, assume first (∗) holds for T ∞ , p∞ . Let ā ∈ M T , p(x̄, ā) ∈ Sn (ā). Then
ā can be viewed also as a tuple in M ∞ T ∞ , and we can identify p(x̄, ā) with a type
∞
∞
p∞ (x̄, ā) ∈ SLn (ā). Then pp(ε) (x̄, ȳ) and p∞p (ε) (x̄, ȳ) coincide more or less by definition,
∞
and fit in Sν≤r (T ) for some r ∈ R+ . It is not true that pp(ε) (x̄ε , ȳ ε) and p∞p (ε) (x̄ε , ȳ ε )
coincide since in the metrics on models of T and T ∞ differ. But as everything fits inside
some ν-ball, and the two metrics are uniformly equivalent on every ν-ball, we can still
find ε′ > 0 such that
∞ (ε′ )
[p(ε)(x̄ε , ȳ ε )]◦ ⊇ [p∞p
′
′
(x̄ε , ȳ ε )]◦ 6= ∅.
∞
For the converse, consider a finite tuple ā ∈ M ∞ T ∞ , and a type p(x̄, ā) ∈ SLn (ā).
As ∞ is definable in T ∞ (it is the unique element satisfying P1 (x) = 0, for example) we
never need it as a parameter, so we may assume that ā ∈ M. Assume first that p(x̄, ā)
says that all xi are finite as well. Then in fact p(x̄, ā) ∈ SLn (ā), and we conclude as above
by the uniform equivalence of the metric. In the general case we may need to write p(x̄, ȳ)
as (w, q) where w ⊆ |x̄, ȳ|, and q ∈ S|w| (T ). Then q is a type of finite elements and is
taken care of by the previous case, while the infinite coordinates are taken care of by the
∞
∞
fact that ∞ is definable, so [dL (x, ∞) < ε] defines an open set in SL (ā).
4.36
The discussion at the end of Section 3, and in particular the characterisation of pω-categoricity for an open perturbation system p by coincidence of topologies (Theorem 3.14), can be transferred to an unbounded theory T via reduction to T ∞ in precisely
the same way.
4.5. An example: Henson’s categoricity theorem. Let T0 be the (unbounded) theory of pure Banach spaces as given in Example 4.16.
Definition 4.37. Let E and F be Banach spaces (i.e., models of T0 ). Say that a mapping
f : E → F is an ε-isomorphism if it is an isomorphism of the underlying vector spaces,
and satisfies in addition:
∀v ∈ E
e−ε kvk ≤ kf (v)k ≤ eε kvk.
Definition 4.38. Let ā ∈ E0 T0 . Define the Banach-Mazur distance between two
types p, q ∈ Sn (ā), denoted dBM,n (p, q), as the minimal ε > 0 such that there exist
models (E, ā), (F, ā) Th(E0 , ā), and tuples b̄ ∈ E, c̄ ∈ F realising p and q, respectively,
and an ε-isomorphism f : E → F fixing ā and sending b̄ to c̄. If no such ε > 0 exists then
dBM,n (p, q) = ∞.
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
37
The following result is very similar to an unpublished result communicated to the
author by C. Ward Henson and published here with his permission:
Corollary 4.39 (Henson, unpublished). Let T be a complete theory of Banach spaces
with no additional structure (i.e., a completion of T0 ). Then the following are equivalent:
(i) If E and F are two separable models of T , then for every ε > 0 there exists an
ε-isomorphism (i.e., a bijective ε-embedding) from E to F .
(ii) For n < ω and finite tuple ā ∈ E T , let S∗n (ā) be the space of types of n-tuples
which are linearly independent over ā. Then every Banach-Mazur ball in S∗n (ā)
has non-empty interior in the logic topology on S∗n (ā).
Proof. First we observe that the Banach-Mazur distance defines a perturbation system
BM by Corollary 4.33. Therefore, by Theorem 4.36, the first condition is equivalent to
the one saying that for all ε > 0 and p(x̄, ā) ∈ Sn (ā): [pBM (ε) (x̄ε , āε )]◦ 6= ∅ in Sn (ā).
We need to show that this is equivalent to the second condition. Since the BanachMazur perturbation preserves linear dependencies we may drop superfluous parameters
and always assume that the tuple ā is linearly independent. Thus, if p(x̄, ā) ∈ S∗ (ā) then
p(x̄, ȳ) ∈ S∗ (T ).
Observe also that S∗n (ā) is a dense open subset of Sn (ā) (indeed, it is metrically dense
there in the usual metric on types). It follows that a subset X ⊆ S∗n (ā) has the same
interior in Sn (ā) and in S∗n (ā), so we may simply speak of its interior. Moreover, a subset
X ⊆ Sn (ā) has non-empty interior if and only if X ∩ S∗n (ā) has.
For left to right, let us show that if p ∈ S∗nP
(T ) and ε > 0 then there exists δ > 0 such
δ
BM (ε)
n
that [p(x̄ )] ⊆ [p
]. So let Λ = {λ ∈ F :
|λi| = 1}, i.e., the (compact) space of all
formal linear combinations of n variables of k·k1 -norm 1, and let s = min{kλ(x̄)kp(x̄) : λ ∈
sε
Λ} > 0. We claim that δ = 2n
> 0 will do.
δ
Indeed, let q ∈ [p(x̄ )]. Let E be a model, b̄, c̄ ∈ E such that b̄ p, c̄ q and
kbi P
− ci k ≤ δ for all i < n. For i < n define a linear functional ηi : Span(b̄) → F by
ηi ( λj bj ) = λi . Then kηi k ≤ s−1 , and by the Hahn-Banach Theorem we may extend
them to P
η̃i : E → F such that kη̃i k ≤ s−1 . Define a linear operator S : E → E by
S(x) =
i η̃i (x)(bi − ci ). Then a simple calculation shows that S(bi ) = bi − ci and
kSk ≤ ε/2. Assuming ε was small enough to begin with (which we may), I − S is
invertible, its inverse being I + S + S 2 + . . .. Finally, for all v ∈ E:
e−ε kvk ≤ (1 − ε/2)kvk ≤ kv − S(v)k ≤ (1 + ε/2)kvk ≤ eε kvk.
We conclude that I − S is an ε-automorphism sending b̄ to c̄, so q ∈ pBM (ε) .
Re-choosing our numbers we find ε/2 > δ > 0 such that that [p(x̄δ )] ⊆ [pBM (ε/2) (x̄)],
so [p(x̄δ )]BM (δ) ⊆ [pBM (ε) (x̄)]. As the former has non-empty interior so does the latter
(in Sn (T ) as well as when restricted to S∗n (T )). When considering parameters we have
p(x̄, ā) ∈ S∗n (ā) such that p(x̄, ȳ) ∈ S∗n+m (T ), so we find δ > 0 such that [p(x̄δ , ȳ δ )]BM (δ) ⊆
[pBM (ε) (x̄, ȳ)], and thus [p(x̄δ , āδ )]BM (δ) ⊆ [pBM (ε) (x̄, ā)], concluding as above.
38
ITAÏ BEN YAACOV
For the other direction, let us show that for all p ∈ Sn (T ) and ε > 0, [pBM (ε) (x̄ε )]◦ 6= ∅.
Assume first that p ∈ S∗n (T ). Then [pBM (ε) ]◦ 6= ∅ in S∗n (T ), and therefore in Sn (T ), as
S∗n (T ) is open in Sn (T ). In case p ∈
/ S∗n (T ) we need to be more delicate. Up to a
permutation of the variables we may assume
V that p is of the form p(x<m , y<k ), where
∗
m + k = n, q(x̄) = p↾x̄ ∈ Sm (T ), and p ⊢ i<k (yi = λi (x̄)) for some linear combinations
λi .
Then we know there is a formula ϕ(x̄) such that ∅ 6= [ϕ < 1/2] ⊆ q BM (ε) . Then in
Sn (T ) we have:
\
∅ 6= [ϕ(x̄) < 1/2] ∩ [d(yi , λi(x̄)) < ε]
i<k
⊆ [p
BM (ε)
ε
(x̄, ȳ )]
BM (ε)
⊆ [p
(x̄ε , ȳ ε)].
T
Indeed, if p′ ∈ [ϕ(x̄) < 1/2] ∩ i<k [d(yi , λi (x̄)) < ε], then there is p′′ ∈ [p′ (x̄, ȳ ε )] such
V
′′
that p′ ↾x̄ = p′′ ↾x̄ , and p′′ ⊢ i<k (yi = λi (x̄)). As ϕ(x̄)p < 1/2, we have p′′ ↾x̄ ∈ q BM (ε) .
We by variable-invariance we may find p′′′ ∈ (p′′ )BM (ε) such that p′′′ ↾x̄ = q. As the linear
structure
is left untouched by the Banach-Mazur perturbation we must have p′′′ (x̄, ȳ) ⊢
V
′′′
i<k (yi = λi (x̄)), so in fact p = p, as required.
The case with parameters is proved identically (with each yi being equal to a linear
combination of x̄ and ā).
4.39
5. Perturbations of automorphisms
We conclude with a few problems concerning perturbations of automorphisms which
motivated the author’s initial interest in perturbations, and which the author therefore
finds worthy of future study.
One such problem comes from the study of the properties of automorphism groups
of classical (i.e., discrete) countable structures, and in particular of ones whose first
order theory is ω-categorical, viewed as topological groups. There are many results of
Lascar, Shelah, Kechris, Rosendal (and possibly others) concerning such groups, and
one naturally asks what of these results can be generalised to the automorphism groups
of separable continuous structures (with a separably categorical theory). The simplest
instance would be the unitary group U(H) where H is a separable infinite-dimensional
Hilbert space. Indeed, U(H) is a polish group in the pointwise convergence topology,
also known as the strong operator topology, but it is fairly quickly clear that none of the
properties that Kechris and Rosendal were looking for (e.g., existence of ample generics)
can possibly occur there: this group is just way too big.
That is definitely not a new phenomenon: we already know that the type spaces
of a continuous theory are “too big”. Thus, in order to define properly notions such as
superstability and local ϕ-stability, one needs to take into account some underlying metric
structure. In fact, having stared at those spaces long enough one gets the impression that
ON PERTURBATIONS OF CONTINUOUS STRUCTURES
39
these are not merely spaces with two distinct structures, a topological one and a metric
one, but in fact these are metric spaces with additional topological structure (in the case
of the discrete metric, these are simply ordinary topological spaces). Single points in the
underlying metric space are just “too small” to see, so one must always consider balls of
some positive radius.
As in the case of the space of types, there is a natural metric on the automorphism
group of a continuous structure which would indeed be the discrete metric in the classical
case, namely the metric of uniform convergence. For example, in the case of U(H), this
is the operator norm metric (we only consider some bounded ball of H as the actual
structure, so this is indeed uniform convergence). So for example, one can (and should)
restate the question of the existence of ample generics as follows:
Question 5.1. Let M be a separable structure, and view G = Aut(M) as a topological
group (pointwise convergence topology) on top of a metric space (uniform convergence).
Under what assumptions on M can we find, for each n < ω, a tuple ḡ ∈ Gn such that
for every ε > 0, the G-conjugacy class of the ε-ball (in the sense of the metric) around ḡ
is co-meagre (in the sense of the topology)? In other words can we find ḡ such that the
metric closure of the orbit of ḡ is co-meagre?
In particular, can one prove this is the case if Th(M) is ω-categorical and ω-stable?
For the special case of U(H) this essentially follows from [Dav96, Theorem II.5.8].
What about the automorphism group of the unique separable atomless probability algebra?
Another question leading to similar considerations is raised by Berenstein and Henson
[BH]. In this paper they consider the theory of probability algebras with an automorphism, and ask whether it is superstable (equivalently, supersimple, as the theory is
known to be stable). The author answered this negatively in [Ben], in striking contrast
with the classical situation (i.e., discrete logic) where if T is superstable then TA is supersimple: see Chatzidakis and Pillay [CP98] for the case where TA exists as a first order
theory. This may seem surprising at first, but in fact it shouldn’t: after all, when dealing with superstability (or supersimplicity) in continuous logic one always allows moving
things by arbitrarily small positive distances, so in light of the considerations above, when
dealing with supersimplicity in TA one should also allow arbitrarily small (yet non-zero)
modifications of the automorphism:
Question 5.2. Let T be a superstable continuous theory. Let
Tσ = T ∪ {“σ is an automorphism”}.
Assume furthermore that Tσ has a model companion TA . Is TA supersimple up to small
perturbations of σ?
And actually:
Question 5.3. What should it mean precisely for a theory to be supersimple up to small
perturbations?
40
ITAÏ BEN YAACOV
Regarding the last question it should be pointed out that there are several natural candidates for the definition of “a is independent up to distance ε from B over A” (denoted
usually aε ⌣
| A B). While these notions of approximate independence are not equivalent,
they all give rise to the same notion of supersimplicity (see for example in [Ben06]). In
a stable theory this is further equivalent to Iovino’s definition of superstability (which
goes through counting the metric density character of types). One should therefore look
for a notion of supersimplicity/superstability up to small perturbations which shares this
robustness.
References
[BBH]
[Ben]
[Ben03]
[Ben06]
[BH]
[BU]
[CP98]
[Dav96]
[Fre04]
[HI02]
Itaı̈ Ben Yaacov, Alexander Berenstein, and C. Ward Henson, Model-theoretic independence in
the Banach lattices Lp (µ), submitted.
Itaı̈ Ben Yaacov, Probability algebras with an automorphism are not superstable, unpublished
notes.
, Positive model theory and compact abstract theories, Journal of Mathematical Logic 3
(2003), no. 1, 85–118.
, On supersimplicity and lovely pairs of cats, Journal of Symbolic Logic 71 (2006), no. 3,
763–776.
Alexander Berenstein and C. Ward Henson, Model theory of probability spaces with an automorphism, submitted.
Itaı̈ Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, Transactions of the AMS, to appear.
Zoé Chatzidakis and Anand Pillay, Generic structures and simple theories, Annals of Pure and
Applied Logic 95 (1998), 71–92.
Kenneth R. Davidson, C ∗ -algebras by example, Fields Institute Monographs, vol. 6, American
Mathematical Society, Providence, RI, 1996.
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C. Ward Henson and José Iovino, Ultraproducts in analysis, Analysis and Logic (Catherine
Finet and Christian Michaux, eds.), London Mathematical Society Lecture Notes Series, no.
262, Cambridge University Press, 2002.
Itaı̈ Ben Yaacov, Université de Lyon, Université Lyon 1, Institut Camille Jordan,
CNRS, UMR 5208, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex,
France
URL: http://math.univ-lyon1.fr/~begnac/