Vaught’s conjecture on analytic sets^{∗}

Greg Hjorth ^{†}

June 12, 2000

Abstract

Let G be a Polish group. We characterize when there is a Polish space X with a continuous G-action and an analytic set (that is to say, the Borel image of some Borel set in some Polish space) A ⊂ X having uncountably many orbits but no perfect set of orbit inequivalent points.

Such a Polish G-space X and analytic A exist exactly when there is a continuous, surjective homomorphism from a closed subgroup of G onto the infinite symmetric group, S_{∞}, consisting of all permutations of ℕ equipped with the topology of pointwise convergence.

§0 Prehistory

The Vaught conjecture stands as a major problem in mathematical logic and may be stated as follows:

0.1 Conjecture (Original Vaught conjecture) Let T be a first order theory in a countable language. Then either T has ≤ ℵ_{0} many non-isomorphic countable models or it has 2^{ℵ_{0}} many non-isomorphic countable models.

The study of this conjecture and the proof or refutation of its specializations and generalizations – found in papers such as [1], [5], [10], [13], [18], [19], [20] – has almost become a research area in its own right.

One natural extension of the Vaught conjecture is to classes of models defined by countably infinitary sentences in the sense of [16].

0.2 Conjecture (Vaught conjecture for infinitary logic) Let σ ∈ L_{ω_{1},ω}. Then either σ has ≤ ℵ_{0} many non-isomorphic countable models or it has 2^{ℵ_{0}} many non-isomorphic countable models.

∗1991 Mathematics Subject Classification. Primary 03E15. Key words and phrases. Polish group. Group actions.

Topological Vaught conjecture.

†Research partially supported by NSF grant DMS 96-22977

G. Hjorth 2

0.2, like 0.1, remains open. Under the assumption of the continuum hypothesis both these conjectures are trivially true, and for this reason it is often customary to consider a strengthened version which has the same spirit as the original. To phrase this stronger version we need to equip the collection of all models whose underlying set is ℕ with a topology, whose basic open sets consist of all M for which

M ⊧ φ(a_{1}, a_{2}, ..., a_{n}),

for some first order formula φ and a⃗ = a_{1}, ..., a_{n} ∈ ℕ. It is known from [9] that the space of all structures or models on ℕ in that given topology forms a Polish space. The collection of models satisfying a given theory T is then closed inside this space, and thus again a Polish space.

0.3 Conjecture Let T be a first order theory in a countable language L. Then either T has ≤ ℵ_{0} many non-isomorphic countable models or there is a perfect set P included in the above described space of L-structures modeling T for which any two models in P are non-isomorphic.

0.4 Conjecture Let σ ∈ L_{ω_{1},ω}. Then either σ has ≤ ℵ_{0} many non-isomorphic countable models or there is a perfect set P included in the space of L-structures which model σ for which any two models in P are non-isomorphic.

Since any perfect set has size 2^{ℵ_{0}}, 0.3 and 0.4 clearly imply 0.1 and 0.2 respectively. More subtle is the fact that 0.3 and 0.4 are equiprovable over ZFC with their counterparts 0.1 and 0.2. A proof of this further fact can be found in [20].

It was demonstrated in [22] that the isomorphism invariant Borel subsets of the space of L-structures on ℕ are exactly those defined by some infinitary sentence σ ∈ L_{ω_{1},ω}. Thus 0.4 asserts that any invariant Borel set in our space of countable structures has either only countably many orbits or it has perfectly many orbits, in the sense of there being a perfect set of pairwise non-isomorphic models. Thus we might be led to asking whether the Vaught conjecture can hold even on analytic sets.

0.5 Definition Let X be a Polish space. A ⊂ X is analytic, or Σ ∼_{1}^{1}, if there is some other Polish space Y and some Borel function

f : Y → X

and some Borel B ⊂ Y with

f[B] = A.

0.6 Fact There are countable languages L ⊂ L^{∗} and a first order L^{∗} theory S such that the class

A_{S}

of L-structures on ℕ admitting an expansion to a model of S has uncountably many nonisomorphic countable models but not perfectly many non-isomorphic models. (See [2] for a discussion of this and related results.)

Since the process of assigning to each L^{∗} structure on ℕ its reduction to an L structure is Borel, we obtain the failure of Vaught’s conjecture on Σ ∼_{1}^{1} sets: We can have an analytic subset of the space of countable L structures having uncountably many but not perfectly many non-isomorphic models.

A second direction in which we might try to extend the Vaught conjecture is to consider general actions of Polish groups. Here it is important to note that the Polish group S_{∞}, consisting of all permutations of the natural numbers in the topology of pointwise convergence, acts continuously on any such space of L structures with underlying set ℕ, and moreover its orbit equivalence relation is exactly the isomorphism relation on these countable structures.

0.7 Conjecture (Topological Vaught conjecture) Let G be a Polish group acting continuously on a Polish space X. Then either X has only countably many orbits under this action or it has perfectly many orbits.

This conjecture just for the group S_{∞} implies 0.3, and hence 0.1, since the subspace of models for T is a closed subset of the space of structures on ℕ. By a clever choice of the topology on our space of countable structures it can also be shown to imply 0.4. More recently Becker and Kechris in [4] have shown that 0.4 is in fact equivalent to the topological Vaught conjecture for the Polish group S_{∞}.

0.7, like 0.1-0.4, remains open. It has however been proved for various classes of Polish groups:-

0.8 Theorem (Folklore) All locally compact Polish groups satisfy Vaught’s conjecture, in the sense that if G is a locally compact Polish group acting continuously on a Polish space X then either |X/G| ≤ ℵ_{0} or there is a perfect set of points with different orbits (and hence |X/G| ≥ 2^{ℵ_{0}}).

0.9 Theorem (Sami; see [18]) Abelian Polish groups satisfy Vaught’s conjecture.

0.10 Theorem (Hjorth, Solecki; [13]) Invariantly metrizable and nilpotent Polish groups satisfy Vaught’s conjecture.

0.11 Theorem (Becker; [2]) Complete left invariant metric groups satisfy Vaught’s conjecture.

As discussed in [2], this implies the Vaught conjecture for solvable groups, as well as the version obtained at 0.10.

In each of these cases the result was shortly or immediately after extended to analytic sets.

0.12 Definition For G a Polish group let TVC(G, Σ ∼_{1}^{1}) be the assertion that whenever G acts continuously on a Polish space X and A ⊂ X is analytic then either |A/G| ≤ ℵ_{0} or there is a perfect set of orbit inequivalent points in A.

Thus we have TVC(G, Σ ∼_{1}^{1}) for G in each of the collections groups mentioned in 0.8-0.11 above. But it is known that TVC(S_{∞}, Σ ∼_{1}^{1}) fails; A_{S} from 0.6 provides a counterexample.

In this paper we show that the presence of S_{∞} is a necessary condition for TVC(G, Σ ∼_{1}^{1}) to fail:

0.13 Theorem If G is a Polish group for which the Vaught conjecture fails on analytic sets then there is a closed subgroup of G that has S_{∞} as a continuous homomorphic image.

From the failure of TVC(G, Σ ∼_{1}^{1}) we may obtain the homomorphism onto S_{∞}. The converse of 0.13 is known and follows by 2.3.5 of [4]. Thus we have an exact characterization of TVC(G, Σ ∼_{1}^{1}). If as widely suspected the Vaught conjecture should fail for S_{∞} then this would as well characterize the groups for which the topological Vaught conjecture holds.

0.13 implies the earlier results at 0.8-0.11. For instance one can use [8] to show that no group having S_{∞} as its continuous homomorphic image can be given a compatible complete left invariant metric.

§1 Preliminaries

This paper uses much the same techniques as [11]. The notation is also similar to [11], though in this regard [4] probably provides a better reference to the notation relating to Polish group actions.

In other respects we follow the usual set theoretic conventions, as can be found in [14]. Perhaps the most alarming of these is to use ω for

ℕ = {0, 1, 2, 3, ...}.

A basic tool in the descriptive set theory of Polish group actions is provided by the Effros lemma.

1.1 Lemma (Effros; see [7] or 2.2.2 [4]) Let G be a Polish group acting continuously on a Polish space X (in other words, let X be a Polish G-space). For x ∈ X we have [x]_{G} ∈ Π ∼_{2}^{0} if and only if

G → [x]_{G},
g ↦ g · x

is open.

1.2 Corollary Let G be a Polish group and X a Polish G-space. Suppose that [x]_{G} is Π ∼_{2}^{0}. Then for all V containing the identity we may find open U containing x such that for all

x′ ∈ U ∩ [x]_{G} and U′ ⊂ X open

[x]_{G} ∩ U′ ∩ U ≠ ∅

implies that there exists g ∈ V such that

g · x′ ∈ U′.

Proof. Choose W an open neighborhood of the identity with W^{−1} = W and W^{2} ⊂ V. Then by 1.1 let U be an open set with U ∩ [x]_{G} = W · x.

For all x′ ∈ [x]_{G} ∩ U and open U′ with [x]_{G} ∩ U′ ∩ U ≠ ∅ we choose some x̂ ∈ [x]_{G} ∩ U′ ∩ U. Since x′, x̂ ∈ U ∩ [x]_{G}, the assumption on U provides g_{0}, g_{1} ∈ W with

g_{0} · x = x′,
g_{1} · x = x̂.

Then g_{1}g_{0}^{−1} is as required. □

1.3 Definition Let X be a Polish space and B a basis. Let L(B) be the propositional language formed from the atomic propositions ẋ ∈ U, for U ∈ B. Let L_{∞,0}(B) be the infinitary version, obtained by closing under negation and arbitrary disjunction and conjunction. F ⊂ L_{∞,0}(B) is a fragment if it is closed under subformulas and the finitary Boolean operations of negation and finite disjunction and finite conjunction and it includes all atomic propositions.

For a point x ∈ X and φ ∈L_{∞0}(B), we can then define x ⊧ φ by induction in the usual fashion, starting with

x ⊧ ẋ ∈ U

if in fact x ∈ U, then x ⊧ ⋁_{i∈Λ}(φ_{i}) if there is some i ∈ Λ with x ⊧ φ_{i}, and x ⊧ ¬φ if x does not model φ. For 𝕍[H] a generic extension of 𝕍 and x, φ ∈ 𝕍 we have 𝕍 ⊧ (x ⊧ φ) if and only if 𝕍[H] ⊧ (x ⊧ φ).

In the case that X is a Polish G-space and V ⊂ G open we may also define the Vaught transform φ^{ΔV} by induction on the logical complexity of φ, mimicking the definition of the usual Vaught transforms for Borel sets. For this purpose let (V_{i})_{i∈ℕ} be an enumeration of the non-empty basic open subsets of G. (ẋ ∈ U)^{ΔV} is just

⋁_{U′∈B′} ẋ ∈ U′,

where B′ is the set of basic open sets U′ for which there is some i with V_{i} · U′ ⊂ U and V_{i} ⊂ V.

( ⋁_{i∈Λ} φ_{i})^{ΔV}

is

⋁_{i∈Λ}(φ_{i}^{ΔV}).

(¬φ)^{ΔV}

is

⋁{¬(φ^{ΔV_{n}}) : V_{n} ⊂ V }.

In any generic extension in which φ is hereditarily countable

x ⊧ φ^{ΔV}

if and only if

∃^{∗}g ∈ V (g · x ⊧ φ)

(where ∃^{∗} is the categoricity quantifier “there exists non-meagerly many”).

In general φ^{ΔV} depends on the choice of B_{0} and B. Our notation suppresses this dependence.

Note that if φ ∈ L_{∞,0}(B) and g ∈ G is such that B is fixed set wise under g translation, then we can canonically define gφ with the property that

x ⊧ gφ

if and only if g^{−1} · x ⊧ φ. We define gφ by induction the complexity of φ, the base case given by g(ẋ ∈ U) being the sentence ẋ ∈ g · U, and the inductive steps through the infinitary boolean operations carried out in the usual way.

In what follows there is a frequent peril that we may confuse the usual name for the generic object – Ġ – with the customary name for the group – G. For this reason I will generally resist explicit mention of the generic. If σ is a term, existing in our ground model, then σ̇ will denote the effect of applying the term to the generic object in the generic extension. At times we will wish to consider product forcing, and then I will use Ġ_{l} × Ġ_{r} to name the generic object on a product forcing ℙ × ℚ. Similarly to avoid confusion with an open set, we use the embellished font 𝕍, rather than the plain V, to indicate the universe of all sets.

d_{X} will always denote a complete metric on X and d_{G} a complete metric on the group G. G_{x} indicates the stabilizer of x, that is to say

G_{x} = {g ∈ G : g · x = x}.

1.4 Lemma Let X be a Polish G-space. ℙ a forcing notion, p ∈ ℙ a condition, σ a ℙ-term. Suppose that B is a countable basis for X and B_{0} a countable basis for G. Suppose that G_{0} is a countable dense subgroup of G and B is closed under G_{0} translation and that B_{0} is closed under left and right G_{0} translation. Then there is a formula φ_{0} and a fragment F_{0} containing φ_{0} so that:

(i)

{{x ∈ X : x ⊧ ψ^{ΔV} } : ψ ∈ F_{0}, V ∈ B_{0}}

provides a basis for a topology τ_{0}(F_{0}), and in any generic extension in which F_{0} becomes countable (X, τ_{0}(F_{0})) is a Polish G-space; (ii) φ_{0} absolutely describes the equivalence class (if any) indicated by the triple (ℙ, p, σ), in the sense that in any generic extension 𝕍[H] we find

𝕍[H] ⊧ ∀x ∈ X(p ⊩_{ℙ} xE_{G}σ̇ ⇔ x ⊧ φ_{0}).

Proof. Note that if F is any fragment and τ(F) is the topology with basis {{x ∈ X : x ⊧ ψ} : ψ ∈ F }, then τ(F) includes the original topology in virtue of F including all the atomic propositions.

Claim(1): If F is a fragment, then in any generic extension in which F becomes countable, τ(F) generates a Polish topology.

Proof of claim. Recall from 13.2 [15] that whenever we have a Polish topology τ_{Y} on a Polish space Y and A ⊂ Y a τ_{Y} -open set, then the minimal topology obtained from τ_{Y} by adding A as a clopen set gives a topology on Y equivalent to taking the disjoint union of (A, τ_{Y}) and (Y \ A, τ_{Y}), and is therefore again Polish. Recall also that if τ_{i} is an increasing sequence of Polish topologies on Y, then the minimal topology including the union is again Polish, since it corresponds to the diagonal in the resulting space

∏_{i∈ω}(Y, τ_{i}).

Iterating these two operations one shows by induction on the logical complexity of the infinitary sentences in F that τ(F) is Polish in any generic extension in which F becomes countable. (Claim(1)□)

Let F_{0} be a fragment closed under the Vaught transforms (ψ ↦ ψ^{ΔV} for V ⊂ G basic open) and such that for each ψ ∈ F_{0} and g ∈ G_{0} we have a corresponding gψ ∈ F_{0} with the property that through all generic extensions

(x ⊧ gψ) ⇔ (g^{−1} · x ⊧ ψ).

By the Becker-Kechris theorem on changing topologies, as found at [3] or as in §5.2 and the proof of 7.1.3 of [4], we have that

{{x ∈ X : x ⊧ ψ^{ΔV} } : ψ ∈ F_{0}, V ∈ B_{0}}

generates a topology τ_{0}(F_{0}), with (X, τ_{0}(F_{0})) a Polish G-space whenever F_{0} becomes countable. Since the sets of the form

{x ∈ X : x ⊧ ψ},

ψ ∈ F_{0}, form an algebra closed under G_{0} translation, it is easily seen that the collection {{x ∈ X : x ⊧ ψ^{ΔV} } : ψ ∈ F_{0}, V ∈ B_{0}} provides a basis for the topology τ_{0}(F_{0}).

In what follows it is crucial that the properties of most interest are absolute, and thus we can be ambiguous about the generic extension in which we make an evaluation. In particular, for any x appearing in any generic extension, the assertion p ⊩_{ℙ} xE_{G}σ̇ depends solely on x, and not the generic extension in which we consider this question:-

Claim(2): If 𝕍[H_{1}] ⊂ 𝕍[H_{1}][H_{2}] are two generic extensions of 𝕍 and x ∈ 𝕍[H_{1}], then

𝕍[H_{1}] ⊧ p ⊩_{ℙ} xE_{G}σ̇

if and only if

𝕍[H_{1}][H_{2}] ⊧ p ⊩_{ℙ} xE_{G}σ̇.

Proof of claim. Otherwise we may find H ⊂ ℙ that is generic for both models, and such that the generic extensions 𝕍[H_{1}][H] and 𝕍[H_{1}][H_{2}][H] disagree on xE_{G}σ[H], with a violation of absoluteness of Σ ∼_{1}^{1}. (Claim(2)□)

Claim(3): There is a generic extension of 𝕍[H] and x ∈ 𝕍[H] with

𝕍[H] ⊧ p ⊩_{ℙ} xE_{G}σ̇

if and only if

𝕍 ⊧ (p, p) ⊩_{ℙ×ℙ} σ[Ġ_{l}]E_{G}σ[Ġ_{r}].

Proof of claim. Clearly if (p, p) ⊩_{ℙ×ℙ} σ[Ġ_{l}]E_{G}σ[Ġ_{r}] then we can let H ⊂ ℙ be generic below p, and obtain some x = σ[H] in 𝕍[H] with

𝕍[H] ⊧ p ⊩_{ℙ} xE_{G}σ̇.

Conversely, if we have 𝕍[H] ⊧ p ⊩_{ℙ} xE_{G}σ̇ in some generic extension 𝕍[H] of 𝕍 and q_{1}, q_{2} ≤ p with

𝕍 ⊧ (q_{1}, q_{2}) ⊩_{ℙ×ℙ} σ[Ġ_{l}]¬E_{G}σ[Ġ_{r}],

then we may choose H_{1} × H_{2} a 𝕍[H]-generic filter on ℙ × ℙ below (q_{1}, q_{2}). Then

σ[H_{1}]¬E_{G}σ[H_{2}],

by assumption on (q_{1}, q_{2}), and hence for some i ∈ {1, 2} we have

σ[H_{i}]¬E_{G}x,

contradicting assumption on p. (Claim(3)□)

Claim(4): The set of x for which

p ⊩_{ℙ} xE_{G}σ̇

is Σ ∼_{1}^{1} in any generic extension in which (P(ℙ))^{𝕍} (=_{df} Power set of ℙ as calculated in 𝕍) becomes countable.

Proof of claim: It suffices by claim(3) to show that if

𝕍 ⊧ (p, p) ⊩_{ℙ×ℙ} σ[Ġ_{l}]E_{G}σ[Ġ_{r}]

then for any two H_{1}, H_{2} ⊂ ℙ meeting all the dense open sets in 𝕍 we have

σ[H_{1}]E_{G}σ[H_{2}].

But given H_{1}, H_{2} we can go to a third generic extension 𝕍[H_{3}] where H_{3} ⊂ ℙ is 𝕍[H_{1}] and 𝕍[H_{2}] generic below p, and thus σ[H_{3}]E_{G}σ[H_{i}] for both i = 1 and i = 2. (Claim(4)□)

Thus we may find a transfinite sequence (ψ_{α})_{α∈ Ord} of infinitary propositions in L_{∞0}(B) such that in any extension 𝕍[H] in which |P(ℙ)^{𝕍}| ≤ ℵ_{0} and for any x ∈ 𝕍[H] we have

(p ⊩_{ℙ} xE_{G}σ̇) ⇔ ∃α < ω_{1}^{𝕍[H]}(x ⊧ ψ_{α}).

This is obtained either by appeal to the usual decomposition of Σ ∼_{1}^{1} into Borel sets or by letting ψ_{α} be chosen canonically so that for all x in any generic extension

(x ⊧ ψ_{α}) ⇔ (L_{α}(G, X, σ, (P(ℙ))^{𝕍}, x) ⊧ p ⊩_{ℙ} xE_{G}σ̇).

Claim(5): In any generic extension 𝕍[H] in which (P(ℙ))^{𝕍} becomes countable and for any x ∈ 𝕍[H] with p ⊩_{ℙ} xE_{G}σ̇ we have some α < ω_{1}^{𝕍[H]} with

∃^{∗}g ∈ G(g · x ⊧ ψ_{α}).

Proof of claim. Since for any particular α < ω_{1}^{𝕍[H]} the displayed sentence is absolute, we may as well assume MA_{ℵ_{1}}. By invariance of the set in question we have that for all g ∈ G there will be some α ∈ ω_{1}^{𝕍[H]} with

g · x ⊧ ψ_{α}.

By MA_{ℵ_{1}}, for some α the set of g for which g · x ⊧ ψ_{α} must be non-meager. (Claim(5)□)

So far there is no guarantee that any ψ_{α} is non-trivial; nothing yet has ruled against us never having the situation that p ⊩_{ℙ} xE_{G}σ̇.

Claim(6): If there is some generic extension in which

𝕍[H] ⊧ p ⊩_{ℙ} xE_{G}σ̇

then there is some α for which φ_{0} = (ψ_{α})^{ΔG} satisfies (ii) from the statement of the lemma.

Proof of claim. Following claim(4) and applying the absoluteness of Σ ∼_{1}^{1} we can assume that 𝕍[H] arises by simply collapsing (P(ℙ))^{𝕍} to ℵ_{0}. Following claim(5) let us choose some q ∈ Coll(ω, (P(ℙ))^{𝕍}) and α < (|P(ℙ)|^{+})^{𝕍} such that

q ⊩_{Coll(ω,(P(ℙ))^{𝕍})} ∃x∃^{∗}g(g · x ⊧ ψ_{α}).

Now suppose that 𝕍[H_{1}] is any generic extension with some x_{1} ∈ 𝕍[H_{1}] such that

𝕍[H_{1}] ⊧ (p ⊩_{ℙ} x_{1}E_{G}σ̇).

Then we may choose H_{2} ⊂ Coll(ω, (P(ℙ))^{𝕍}) to be 𝕍[H_{1}]-generic below q. Thus by assumption on q we may find some x_{2} ∈ 𝕍[H_{2}] with

𝕍[H_{2}] ⊧ ∃^{∗}g ∈ G(x_{2} ⊧ ψ_{α}),

and thus in particular

𝕍[H_{2}] ⊧ (p ⊩_{ℙ} x_{2}E_{G}σ̇).

And then by claim(2)

𝕍[H_{1}][H_{2}] ⊧ (p ⊩_{ℙ} x_{2}E_{G}σ̇),
𝕍[H_{1}][H_{2}] ⊧ (p ⊩_{ℙ} x_{1}E_{G}σ̇).

So if we let H ⊂ ℙ be 𝕍[H_{1}][H_{2}]-generic below p we obtain

𝕍[H_{1}][H_{2}][H] ⊧ x_{2}E_{G}σ[H],

𝕍[H_{1}][H_{2}][H] ⊧ x_{1}E_{G}σ[H].

Thus x_{1}E_{G}x_{2}.

But (ψ_{α})^{ΔG} is absolute between forcing extensions, and thus

𝕍[H_{1}][H_{2}][H] ⊧ ∃^{∗}g ∈ G(g · x_{2} ⊧ ψ_{α}),

and then by invariance

𝕍[H_{1}][H_{2}][H] ⊧ ∃^{∗}g ∈ G(g · x_{1} ⊧ ψ_{α}).

One final application of absoluteness gives

𝕍[H_{1}] ⊧ (x_{1} ⊧ ψ_{α}^{ΔG}).
(Claim(6)□)
□

Nothing in the statement of the lemma rules out (ii) holding trivially. It could just be the case that in every extension 𝕍[H_{0}] we have 𝕍[H_{0}] ⊧ ∀x(p ⊩_{ℙ} x¬E_{G}σ̇). However in the non-trivial case where there is a generic extension 𝕍[H_{0}] satisfying ∃x(p ⊩_{ℙ} xE_{G}σ̇) it will follow by absoluteness for Σ ∼_{1}^{1} that for any generic extension 𝕍[H_{1}] in which F_{0} becomes countable we must have 𝕍[H_{1}] ⊧ ∃x(x ⊧ φ_{0}), and therefore 𝕍[H_{1}] ⊧ ∃x(p ⊩_{ℙ} xE_{G}σ̇).

1.5 Lemma Let G be a Polish group, X a Polish G-space, A ⊂ X a Σ ∼_{1}^{1} set displaying a counterexample to TVC(G, Σ ∼_{1}^{1}) – so that A/G has uncountably many orbits, but no perfect set of E_{G}-inequivalent points.

Then for each ordinal δ there is a sequence (ℙ_{α}, p_{α}, σ_{α})_{α<δ} so that:

(0) for each α < δ

p_{α} ⊩_{ℙ_{α}} σ̇_{α} ∈ A;

(i) for each α < δ

(p_{α}, p_{α}) ⊩_{ℙ_{α}×ℙ_{α}} σ_{α}[Ġ_{l}]E_{G}σ_{α}[Ġ_{r}];

(ii) for each α < β < δ

(p_{α}, p_{β}) ⊩_{ℙ_{α}×ℙ_{β}} ¬(σ_{α}[Ġ_{l}]E_{G}σ_{β}[Ġ_{r}]).

Proof. Claim(1): If ℙ is any forcing notion, σ a ℙ-term for an element of A, then there is p ∈ ℙ such that (p, p) ⊩_{ℙ×ℙ} σ[Ġ_{l}]E_{G}σ[Ġ_{r}].

Proof of claim. It is known from [6], [21], and recorded in many places, that whenever we have a Σ ∼_{1}^{1} equivalence relation F on a Polish space Y for which there is no perfect set of inequivalent reals, then for all ℙ ⊩ τ̇ ∈ Y we may find some p with

(p, p) ⊩_{ℙ×ℙ} τ[Ġ_{l}]F τ[Ġ_{r}].

Taking π : Y ↠ A to be the continuous map witnessing A ∈ Σ ∼_{1}^{1} we may let F be the pullback of E_{G}. Since A has no perfect set of E_{G}-inequivalent reals, Y has no perfect set of F -inequivalent reals. Thus if σ is a term for an element of A then we let τ be a term for an element of Y with

ℙ ⊩ π(τ̇) = σ̇,

and we obtain as above some p ∈ ℙ with

(p, p) ⊩_{ℙ×ℙ} τ[Ġ_{l}]F τ[Ġ_{r}],
∴ (p, p) ⊩_{ℙ×ℙ} σ[Ġ_{l}]E_{G}σ[Ġ_{r}].
(Claim(1)□)

Claim(2): In every generic extension of 𝕍 we have |A/G| ≥ ℵ_{1}.

Proof of claim (compare also [18]): We use the fact that every orbit in X is Borel in any code for its stabilizer. (See 7.1.2 [4].)

Let F(G) be the standard Borel space of closed subsets of G in the Effros Borel structure. Let S ⊂ X × F(G) be Π ∼_{1}^{1} with S = {(x, G_{x}) : x ∈ X}. Let

E ⊂ X × X × F(G)

be Π ∼_{1}^{1} such that

(x, y, G_{x}) ∈ E

if and only if xE_{G}y. Then the uncountability of A/G becomes the Π ∼_{2}^{1} statement

∀(x_{i}, F_{i})_{i∈ω} ∈ (X × F(X))^{ω}∃y ∈ A(∃i(¬S(x_{i}, F_{i})) ∨ ∀i(¬E(x_{i}, y, F_{i}))),

and is therefore absolute. (Claim(2)□) Thus we may successively choose ℙ_{α} and σ_{α} such that

ℙ_{α} ⊩ σ̇_{α} ∈ A

and for all β < α and q ∈ ℙ_{α}

(q, p_{β}) ⊩_{ℙ_{α}×ℙ_{β}} ¬(σ_{α}[Ġ_{l}]E_{G}σ_{β}[Ġ_{r}]).

Refining to some p_{α} ∈ ℙ_{α} with

(p_{α}, p_{α}) ⊩_{ℙ_{α}×ℙ_{α}} σ_{α}[Ġ_{l}]E_{G}σ_{α}[Ġ_{r}]

we finish. □

Note that in the situation of 1.5 we may find for each α < δ some corresponding φ_{0,α} for the forcing notion ℙ_{α} below p_{α} as in 1.4. We are guaranteed that this be non-trivial, in the sense that if H ⊂ ℙ_{α} is 𝕍-generic below p_{α}, and x = σ_{α}[H], then

p_{α} ⊩_{ℙ} xE_{G}σ̇_{α},
∴ x ⊧ φ_{0,α}.

On the other hand, if x_{0} ∈ 𝕍[H_{0}] is an element of some generic extension 𝕍[H_{0}] and x ⊧ φ_{0,α} and H_{1} ⊂ ℙ_{α} is any 𝕍-generic with x_{1} = σ_{α}[H_{1}], then we may choose H_{2} ⊂ ℙ_{α} that is generic for both 𝕍[H_{1}] and 𝕍[H_{2}]. Thus

x_{0}E_{G}σ_{α}[H_{2}]

by assumption on φ_{0,α}. Moreover

x_{1}E_{G}σ_{α}[H_{2}]

by the assumption ℙ_{α}, p_{α}, σ_{α}. Hence x_{0}E_{G}x_{1}.

In this sense we can associate to (ℙ_{α}, p_{α}, σ_{α})_{α<δ} a corresponding sequence (φ_{0,α})_{α∈δ}, each φ_{0,α} providing an ∞-Borel code for the indicated equivalence class [σ_{α}[H]]_{G} whenever H ⊂ ℙ_{α} is 𝕍-generic below p_{α}.

§2 Proof

2.1 Definition U is a regular open set if U equals the interior of its closure:

(\overline{U})^{o} = U

For A a set let RO(A) = (\overline{A})^{o}.

Note that RO(A) is always regular open.

2.2 Lemma Let G be a Polish group. For V_{0}, V_{1} ⊂ G regular open sets,

{g ∈ G : V_{0} · g = V_{1}}

is a closed subset of G.

Proof. The set in question arises as the intersection of the closed sets

\{g ∈ G : V_{0} · g ⊂ \overline{V_{1}}\}

and

\{g ∈ G : \overline{V_{0}} · g ⊃ V_{1}\}.
□

I need that the reader is willing to allow that we may speak of an ω-model of set theory containing a Polish space, group, action, Borel set, and so on, provided suitable codes exist in the well founded part. I will lean on the usual set theoretical identifications, and speak of open sets and Borel sets being “in” an ω-model, when really it is only the case that the model has some suitable subset of the natural numbers which codes them.

Illfounded ω-models are essential to the arguments below. The structure of the argument is to use the large sequence of forcing notions layed out in 1.5 to obtain a suitable ω-model with generating indiscernibles of order type (ℚ, <). Thus we can inject Aut(ℚ, <) into the automorphisms of the ω-model, and then using a kind of “back-and-forth” argument at 2.3 obtain for each automorphism some corresponding element of the group G. Since wellfounded models are rigid and since we want automorphisms in great number, the passage to an ω-model occupies a necessary step in the proofs.

Let ZFC^{∗} be some large fragment of ZFC; at the very least strong enough to prove all the lemmas of §1, but weak enough to admit a finite axiomatization.

The next lemma gives the connection between automorphisms of an ω-model and the corresponding elements of G. Here it should be understood that we do everything honestly: We are supposing that the M-generic H exists in 𝕍, our ground model, and conclude with ĝ again in the ground model. We are working inside a single class model of ZFC which sees not only M and π but also X, G, G_{0}, ℙ, F_{0}, B, B_{0}, φ_{0}, and H.

2.3 Lemma Let M be an ω-model of ZFC^{∗}. Let X, G, G_{0}, ℙ, F_{0}, B, B_{0}, φ_{0} satisfy 1.4 inside M. Suppose

π : M ≅ M

is an automorphism of M fixing X, G, G_{0}, ℙ, F_{0}, φ_{0}, and all elements of B and B_{0}. Suppose H ⊂ Coll(ω, F_{0}) is M-generic and x ∈ X^{M[H]} with

M[H] ⊧ (x ⊧ φ_{0}).

Then there exists ĝ ∈ G so that for all ψ ∈ F_{0} and V ∈ B_{0}

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ ψ^{ΔV})})ĝ^{−1} = RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ π(ψ)^{ΔV})}).

Proof. It suffices to find g_{0}, g_{1} ∈ G so that

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ ψ^{ΔV})})g_{0}^{−1} = RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ π(ψ)^{ΔV})})g_{1}^{−1}

for all ψ and V.

Let ℙ_{0} be the forcing notion Coll(ω, F_{0}) in M. Fixing d_{G} a complete metric on G we build h_{i}, h′_{i} ∈ G_{0}, ψ_{i}, ψ′_{i} ∈ F_{0}, W_{i}, W′_{i}, V_{i} ∈ B_{0} so that

(i) W_{i}, W′_{i} are open neighborhoods of the identity, with

W_{i+1} ⊂ W_{i},
W′_{i+1} ⊂ W′_{i},
d_{G}(W_{i}) < 2^{−i},
d_{G}(W′_{i}) < 2^{−i};

(ii) π(ψ_{i}) = ψ′_{i}; (iii) h_{2i} = h_{2i+1}; for all g ∈ W_{2i+1}h_{2i}

d_{G}(g, h_{2i}) < 2^{−i};

(iv) h′_{2i+1} = h′_{2i+2}; for all g ∈ W_{2i+2}h′_{2i+1}

d_{G}(g, h′_{2i+1}) < 2^{−i};

(v) h_{i+1} ∈ W_{i}h_{i} and h′_{i+1} ∈ W_{i}h′_{i}; (vi) M[H] ⊧ (h_{i} · x ⊧ (ψ_{i})^{ΔV_{i}}); (vii) M[H] ⊧ (h′_{i} · x ⊧ (ψ′_{i})^{ΔV_{i}}); (viii) M^{ℙ_{0}} satisfies that for all y_{0}, y_{1} ∈ X all ψ ∈ F_{0}, and all V ∈ B_{0}, if

y_{0} ⊧ φ_{0} ∧ (ψ_{i})^{ΔV_{i}}

y_{1} ⊧ φ_{0} ∧ (ψ_{i})^{ΔV_{i}} ∧ ψ^{ΔV}

then

y_{0} ⊧ ((ψ_{i})^{ΔV_{i}} ∧ ψ^{ΔV})^{ΔW_{i}};

(ix) conversely M^{ℙ_{0}} satisfies that for all y_{0}, y_{1} ∈ X, ψ ∈ F_{0}, V ∈ B_{0}, if

y_{0} ⊧ φ_{0} ∧ (ψ′_{i})^{ΔV_{i}}
y_{1} ⊧ φ_{0} ∧ (ψ′_{i})^{ΔV_{i}} ∧ ψ^{ΔV}

then

y_{0} ⊧ ((ψ′_{i})^{ΔV_{i}} ∧ ψ^{ΔV})^{ΔW_{i}}.

Actually (ix) follows from (viii), (ii), and the elementarity of π.

Before verifying that we may produce h_{i}, h′_{i} ∈ G_{0}, ψ_{i}, ψ′_{i} ∈ F_{0}, W_{i}, V_{i} ∈ B_{0} as above, let us imagine that it is already completed and see how to finish. Using (iii), (iv), and (v) we may obtain

g_{0} = lim_{i→∞}h_{i}

and

g_{1} = lim_{i→∞}h′_{i}.

It suffices to check that for all ψ ∈ F_{0}, V ∈ B_{0}, and

g ∈ RO({h ∈ G_{0} : M[H] ⊧ (h · x ⊧ ψ^{ΔV})})g_{0}^{−1}

we have

g ∈ \overline{(\{h ∈ G_{0} : M[H] ⊧ (h · x ⊧ π(ψ)^{ΔV})\})}g_{1}^{−1}

(the converse implication will be exactly symmetric). Fix g in

RO({h ∈ G_{0} : M[H] ⊧ (h · x ⊧ ψ^{ΔV})})g_{0}^{−1}.

By assumption on g there are arbitrarily large i and h ∈ G_{0} with h(h_{i})^{−1} arbitrarily close to g and M[H] ⊧ (h · x ⊧ ψ^{ΔV}).

Thus replacing h(h_{i})^{−1} with ĝ for sufficiently large i we may choose a sufficiently small open neighbourhood W ∈ B_{0} of the identity and ĝ ∈ G_{0} sufficiently close to g so that W ĝW_{i} is an arbitrarily small neighbourhood of g and

M[H] ⊧ (ĝh_{i} · x ⊧ ψ^{ΔV})
∴ M[H] ⊧ (h_{i} · x ⊧ (ψ^{ΔV})^{ΔW ĝ})

hence, as witnessed by y = h_{i} · x,

M^{ℙ_{0}} ⊧ ∃y(y ⊧ φ_{0} ∧ (ψ_{i})^{ΔV_{i}} ∧ (ψ^{ΔV})^{ΔW ĝ}),
∴ M^{ℙ_{0}} ⊧ ∃y(y ⊧ φ_{0} ∧ (ψ′_{i})^{ΔV_{i}} ∧ (π(ψ)^{ΔV})^{ΔW ĝ})

by (ii) and elementarity of π,

∴ M[H] ⊧ (h′_{i} · x ⊧ ((π(ψ)^{ΔV})^{ΔW ĝ})^{ΔW_{i}})

by (ix). So there exists some ḡ ∈ W ĝW_{i} such that

M[H] ⊧ (ḡh′_{i} · x ⊧ π(ψ)^{ΔV}).

By letting d_{G}(W ĝW_{i}) → 0 and h′_{i} → g_{1} we get

g ∈ \overline{\{h ∈ G_{0} : M[H] ⊧ (h · x ⊧ π(ψ)^{ΔV})\}}g_{1}^{−1},

as required.

We are left to hammer out the sequences.

Suppose that we have ψ_{j}, ψ′_{j}, W_{j}, V_{j}, h_{j}, h′_{j} for j ≤ 2i. Immediately we may set h_{2i+1} = h_{2i} and find W_{2i+1} ⊂ W_{2i} giving (iii), and then by 1.2 and 1.4(i) we can produce ψ_{2i+1} and V_{2i+1} satisfying (viii) and such that

M[H] ⊧ h_{2i} · x =_{df} h_{2i+1} · x ⊧ (ψ_{2i+1})^{V_{2i+1}}.

Then by considering that π is elementary

M^{ℙ_{0}} ⊧ ∃y(y ⊧ φ_{0} ∧ π(ψ_{2i})^{ΔV_{2i}} ∧ π(ψ_{2i+1})^{ΔV_{2i+1}}).

Thus by (ix) we may find h′ ∈ G_{0} ∩ W_{2i} so that

M[H] ⊧ (h′h′_{2i} · x ⊧ π(ψ_{2i})^{ΔV_{2i}} ∧ π(ψ_{2i+1})^{ΔV_{2i+1}}).

In other words, by (ii), if we let ψ′_{2i+1} = π(ψ_{2i+1}) then

M[H] ⊧ (h′h′_{2i} · x ⊧ (ψ′_{2i})^{ΔV_{2i}} ∧ (ψ′_{2i+1})^{ΔV_{2i+1}}).

Taking h′_{2i+1} = h′h′_{2i} we complete the transition from 2i to 2i + 1.

The further step of producing ψ_{2i+2}, ψ′_{2i+2}, W_{2i+2}, h_{2i+2}, V_{2i+2} and h′_{2i+2} is completely symmetrical. □

2.4 Definition S_{∞} divides a Polish group G if there is a closed subgroup H < G and a continuous onto homomorphism

π : H ↠ S_{∞}.

By Pettis’ lemma, as at 1.2.6 of [4], any Borel homomorphism between Polish groups must be continuous.

2.5 Lemma S_{∞} divides Aut(ℚ, <), the automorphism group of the rationals equipped with the usual linear ordering.

Proof. Let {D_{i} : i ∈ ω} be a partition of ℚ into countably many dense subsets.

Now let H be the group of all σ ∈ Aut(ℚ, <) such that for all i, j ∈ ω, all q_{1}, q_{2} ∈ D_{i}

σ(q_{1}) ∈ D_{j} ⇔ σ(q_{2}) ∈ D_{j}

and

σ^{−1}(q_{1}) ∈ D_{j} ⇔ σ^{−1}(q_{2}) ∈ D_{j}.

For any σ ∈ H we define π_{σ} ∈ S_{∞} by π_{σ}(i) = j if and only if σ[D_{i}] = D_{j}. The usual backand-forth arguments show that for any permutation σ of ω we may find σ̂ ∈ Aut(ℚ, <) such that for all i, j

σ̂[D_{i}] = D_{j}

if and only if σ(i) = j. □

2.6 Definition For X, G, F_{0}, and so on, as in 1.4, ℙ_{0}= Coll(ω, F_{0}), ψ_{0}, ψ_{1} ∈ F_{0}, V_{0}, V_{1} ∈ B_{0}, set

(ψ_{0}, V_{0})R(ψ_{1}, V_{1})

if in 𝕍^{ℙ_{0}} for all x ⊧ φ_{0}

RO({g ∈ G_{0} : g · x ⊧ (ψ_{0})^{ΔV_{0}}}) ∩ RO({g ∈ G_{0} : g · x ⊧ (ψ_{1})^{ΔV_{1}}}) ≠ ∅.

By the density of G_{0} and the continuity of the action with respect to τ_{0}(F_{0}), the issue of whether this intersection is non-empty does not depend on the choice of x ⊧ φ_{0}. For V ∈ B_{0} let B(V) be the set of pairs (φ, W) such that for all ψ ∈ F_{0} and W′ ∈ B_{0}

𝕍^{ℙ_{0}} ⊧ ∀x_{0}((x_{0} ⊧ φ_{0} ∧ φ^{ΔW} ∧ (∃x_{1} ⊧ φ_{0} ∧ φ^{ΔW} ∧ ψ^{ΔW′})) ⇒ x_{0} ⊧ (φ^{ΔW} ∧ ψ^{ΔW′})^{ΔV}).

In other words, B(V) corresponds to the basic open sets witnessing 1.2 for V in the topology τ_{0}(F_{0}). The relation R corresponds to whether relatively open subsets of the orbit overlap.

The next lemma states that if the equivalence class corresponding to φ_{0} requires large forcing to be introduced then the formulas {ψ^{ΔV} : ψ ∈ F_{0}, V ∈ B_{0}} have large R-discrete sets.

2.7 Lemma Let X, G, F_{0}, ℙ, p, σ, φ_{0}, and so on, be as in 1.4. Assume that φ_{0} is nontrivial, in the sense that p ⊩_{ℙ} (σ̇ ⊧ φ_{0}). Let R be as in 2.6. Let κ be a cardinal. Suppose no forcing notion of size less than κ introduces a point in X satisfying φ_{0}.

Then there is no infinite δ < κ such that each B(V) for V ∈ B_{0} has a maximal R-discrete set of size ≤ δ.

Proof. Suppose otherwise and choose large θ > κ so that 𝕍_{θ} ⊧ ZFC^{∗} and choose an elementary substructure

A≺𝕍_{θ}

so that

|A| = δ,
δ + 1 ⊂ A,

and X, G, F_{0}, φ_{0}, and so on, in A. Let N be the transitive collapse of A and

π : N → 𝕍_{θ}

the inverse of the collapsing map. Set ℙ̂ = π^{−1}(ℙ_{0}) (where ℙ_{0}= Coll(ω, F_{0})), φ̂_{0} = π^{−1}(φ_{0}), F̂_{0} = π^{−1}(F_{0}), choose

Ĥ ⊂ ℙ̂,
H ⊂ ℙ_{0}

to be 𝕍-generic, and choose x̂ ∈ N[Ĥ] and x ∈ 𝕍[H] so that

N[Ĥ] ⊧ (x̂ ⊧ φ̂_{0}),
𝕍[H] ⊧ (x ⊧ φ_{0}).

(x and x̂ exist for the reasons mentioned following 1.4.) Note that for each V ∈ B_{0} there will be a maximal R-discrete subset of B(V) included in N. It suffices to show

x̂E_{G}x.

As in the proof of 2.3 find h_{i}, h′_{i} ∈ G_{0}, ψ_{i} ∈ F_{0}, ψ′_{i} ∈ F̂_{0}, V_{i}, V′_{i} ∈ B_{0}, W_{i} ∈ B_{0} and U_{i} ⊂ X basic open so that :

(i) W_{i+1} ⊂ W_{i}, W_{i} = (W_{i})^{−1}, d_{G}(W_{i}) < 2^{−i}, 1_{G} ∈ W_{i}; U_{i+1} ⊂ U_{i}, d_{X}(U_{i}) < 2^{−i};

(ii) h_{2i+1} = h_{2i}; for all g ∈ (W_{2i+1})^{3}h_{2i}
d_{G}(g, h_{2i}) < 2^{−i};
(iii) h′_{2i+2} = h′_{2i+1}; for all g ∈ (W_{2i+2})^{3}h′_{2i+1}
d_{G}(g, h′_{2i+1}) < 2^{−i};
(iv) h_{i+1} ∈ (W_{i})^{3}h_{i} and h′_{i+1} ∈ (W_{i})^{3}h′_{i};
(v) 𝕍[H] ⊧ (h_{i} · x ⊧ (ψ_{i})^{ΔV_{i}});
(vi) N[Ĥ] ⊧ (h′_{i} · x̂ ⊧ (ψ′_{i})^{ΔV′_{i}});
(vii) 𝕍 ⊧ (ψ_{i}, V_{i}) ∈ B(W_{i});

(viii) N ⊧ (ψ′_{i}, V′_{i}) ∈ B(W_{i});
(ix) (π(ψ′_{i}), V′_{i})R(ψ_{i}, V_{i});
(x) h_{i} · x, h′_{i} · x̂ ∈ U_{i}.

Granting all this may be found we finish quickly. By (ii), (iii), and (iv) we get g_{0}= lim h_{i} and g_{1}= lim h′_{i}, whence

g_{0} · x = g_{1} · x̂

by (x) and (i). This would contradict ℙ̂ being too small to introduce a representative of [x]_{G}.

So instead suppose we have built V_{j}, V′_{j}, ψ_{j} and so on for j ≤ 2i and concentrate on trying to show that we may continue the construction up to 2i + 2.

First choose W_{2i+1} ⊂ W_{2i} in accordance with (i) and (ii) and then for (x) and (i) choose U_{2i+1} ⊂ U_{2i} containing h_{2i} · x(=_{df} h_{2i+1} · x) with d_{X}(U_{2i+1}) < 2^{−2i−1}. Then by 1.2 we may choose (V_{2i+1}, ψ_{2i+1}) ∈ B(W_{2i+1}) with

h_{2i} · x ⊧ (ψ_{2i+1})^{ΔV_{2i+1}}.

On the N side we use the assumption on R to find V′_{2i+1} and ψ′_{2i+1} in N so that

N_{ ℙ̂} ⊧ (ψ′_{2i+1}, V′_{2i+1}) ∈ B(W_{2i+1})

and

(π(ψ′_{2i+1}), V′_{2i+1})R(ψ_{2i+1}, V_{2i+1}).

Unwinding the definitions of B(W_{2i}) and B(W_{2i+1}) gives

𝕍^{ℙ_{0}} ⊧ (y ⊧ φ_{0} ∧ π(ψ′_{2i})^{ΔV′_{2i}}) ⇒ y ⊧ ((ψ_{2i})^{ΔV_{2i}} ∧ π(ψ′_{2i})^{ΔV′_{2i}})^{ΔW_{2i}},
𝕍^{ℙ_{0}} ⊧ (y ⊧ φ_{0} ∧ (ψ_{2i})^{ΔV_{2i}}) ⇒ y ⊧ ((ψ_{2i})^{ΔV_{2i}} ∧ (ψ_{2i+1})^{ΔV_{2i+1}})^{ΔW_{2i}},
𝕍^{ℙ_{0}} ⊧ (y ⊧ φ_{0} ∧ (ψ_{2i+1})^{ΔV_{2i+1}}) ⇒ y ⊧ ((ψ_{2i+1})^{ΔV_{2i+1}} ∧ π(ψ′_{2i+1})^{ΔV′_{2i+1}})^{ΔW_{2i+1}}.

After possibly tightening the τ_{0}(F_{0})-open set corresponding to (ψ_{2i+1})^{ΔV_{2i+1}} we can assume

(ψ_{2i+1})^{ΔV_{2i+1}} ⇒ ẋ ∈ U_{2i+1},

and thus we have

𝕍^{ℙ_{0}} ⊧ (y ⊧ φ_{0} ∧ π(ψ′_{2i})^{ΔV′_{2i}}) ⇒ y ⊧ ((π(ψ′_{2i+1}))^{ΔV′_{2i+1}} ∧ ẋ ∈ U_{2i+1})^{Δ(W_{2i})^{3}}.

Thus by elementarity of π we may find h′ ∈ (W_{2i})^{3} ∩ G_{0} so that h′h′_{2i} · x̂ ∈ U_{2i+1} and

N[Ĥ] ⊧ (h′h′_{2i} · x̂ ⊧ (ψ′_{2i+1})^{ΔV′_{2i+1}}).

Then setting h′_{2i+1} = h′h′_{2i} completes the transition from 2i to 2i + 1.

The step from 2i + 1 to 2i + 2 is similar, though easier since it will be a trivial task to find for 𝕍 some (ψ_{2i+2}, V_{2i+2}) ∈ B(W_{2i+2}) with (π(ψ′_{2i+2}), V′_{2i+2})R(ψ_{2i+2}, V_{2i+2}). □

We need a fact from infinitary model theory.

2.8 Theorem Let φ ∈ L_{ω_{1},ω} and suppose

N ⊧ φ

and P is a predicate in the language of N with

|(P)^{N}| ≥ ℶ_{ℵ_{1}}.

Then φ has a model with generating indiscernibles in P.

More precisely there is a model M with language L^{∗} ⊃ L, L^{∗} having a new symbol <, along with new function symbols of the form fφ̂ for φ̂ in the fragment of L_{ω_{1},ω} generated by φ, and distinguished elements (c_{i})_{i∈ℕ}, so that:

(i) (<)^{M} linearly orders (P)^{M};

(ii) each fφ̂ is a Skolem function for φ̂;

(iii) M is the Skolem hull of {c_{i} : i ∈ ℕ} (under the functions of the form fφ̂); (iv) each c_{i} ∈ (P)^{M};

(v) for all ψ in the fragment of L_{ω_{1},ω}^{∗} generated by φ, for all i_{1} < i_{2} < ... < i_{n} and j_{1} < ... < j_{n} in ℕ

M ⊧ ψ(c_{i_{1}}, c_{i_{2}}, ..., c_{i_{n}}) ⇔ ψ(c_{j_{1}}, c_{j_{2}}, ..., c_{j_{n}});

(vi) M ⊧ φ. (See the proof that ℶ_{ℵ_{1}} is the Hanf number of sentences in L_{ω_{1},ω} from [16]. □)

2.9 Theorem Let G be a Polish group for which TVC(G, Σ ∼_{1}^{1}) fails. Then S_{∞} divides G. Proof. Choose some Polish G-space X witnessing the failure of TVC(G, Σ ∼_{1}^{1}). Following 1.5

we may find some (ℙ, p, σ) introducing an equivalence class as in 1.4 that may not be produced by a forcing notion of size less than ℶ_{ℵ_{1}}. Fix φ_{0}, B, B_{0}, F_{0}, G_{0}, and so on, as in 1.4, so that in all generic extensions 𝕍[H] of 𝕍

𝕍[H] ⊧ ∀x ∈ X((p ⊩_{ℙ} xE_{G}σ̇) ⇔ y ⊧ φ_{0}),

and

𝕍 ⊧ p ⊩_{ℙ} (σ̇ ⊧ φ_{0}).

Let 𝕍_{θ} be large enough to contain X, G, φ_{0}, and so on, and satisfy ZFC^{∗}. By 2.7 choose W ∈ B_{0} and P ⊂ 𝕍_{θ} ∩ B(W) of size ℶ_{ℵ_{1}} so that for all

(ψ, V), (ψ′, V′) ∈ P

we have

(ψ, V) ≠ (ψ′, V′) ⇒ ((ψ, V)¬R(ψ′, V′)).

By countability of B_{0} we may assume there is a single V ∈ B_{0} so that every element of P has the form

(ψ, V),

some ψ ∈ F_{0}. Applying 2.8 to

N = (𝕍_{θ}; ∈, P, X, G, G_{0}, φ_{0}, ...)

we may obtain an ω-model M with indiscernibles (ψ_{q}, V)_{q∈ℚ} in P^{M}. Let H ⊂ Coll(ω, (F_{0})^{M}) be M-generic. Choose x ∈ M[H] so that

M[H] ⊧ (x ⊧ φ_{0}).

All this granted we may define G_{1} to be the set of ḡ ∈ G so that for all q ∈ ℚ there exists r ∈ ℚ with

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{q})^{ΔV})})ḡ^{−1} = RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{r})^{ΔV})})

and for q ∈ ℚ there exists r ∈ ℚ with

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{q})^{ΔV})})ḡ = RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{r})^{ΔV})}).

G_{1} is Π ∼_{2}^{0} in G, by 2.2 and since ḡ is in G_{1} if and only if the following four conditions hold: (i) for all q ∈ ℚ there exists r ∈ ℚ with

RO({g ∈ G_{0} : M[H] ⊧ (g ·x ⊧ (ψ_{q})^{ΔV})})ḡ^{−1} ∩RO({g ∈ G_{0} : M[H] ⊧ (g ·x ⊧ (ψ_{r})^{ΔV})}) ≠ ∅,

(ii) for all q, r ∈ ℚ

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{q})^{ΔV})})ḡ^{−1} ∩ RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{r})^{ΔV})}) ≠ ∅

implies

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{q})^{ΔV})})ḡ^{−1} = RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{r})^{ΔV})});

(iii) for all q ∈ ℚ there exists r ∈ ℚ with

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{q})^{ΔV})})ḡ ∩ RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{r})^{ΔV})}) ≠ ∅,

(iv) for all q, r ∈ ℚ

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{q})^{ΔV})})ḡ ∩ RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{r})^{ΔV})}) ≠ ∅

implies

RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ ψ_{q}^{ΔV})})ḡ^{−1} = RO({g ∈ G_{0} : M[H] ⊧ (g · x ⊧ (ψ_{r})^{ΔV})}).

Since G_{1} is a Π ∼_{2}^{0} subgroup of G it must be closed.

For g ∈ G_{1} we may define the permutation π̂(g) of ℚ by the specification that for all q ∈ ℚ

(π̂(g))(q) = r

if and only if r is as in part (i) of the definition of G_{1}.

Now let G_{2} be the set of g ∈ G_{1} such that π̂(g) defines an automorphism of the structure (ℚ, <). G_{2} is a closed subgroup of G_{1} and hence G. Noting that every order preserving permutation of the indiscernibles induces an automorphism of M and using the R-discreteness of the set (P)^{M}, the map

π̂ : G_{2} → Aut(ℚ, <)

is seen to be onto by 2.3. Then by 2.5 S_{∞} divides G. □

References

[1] H. Becker, The topological Vaught’s conjecture and minimal counterexamples, Journal of Symbolic Logic, vol. 59(1994), pp. 757-784.

[2] H. Becker, Polish group actions: dichotomies and generalized elementary embeddings, Journal of the American Mathematical Society, vol. 11 (1998), pp. 397-449.

[3] H. Becker, A.S. Kechris, Borel actions of Polish groups, Bulletin American Mathematical Society, vol. 28(1993), pp. 334-341.

[4] H. Becker, A.S. Kechris, The descriptive set theory of Polish group actions, London Mathematical Society Lecture Notes Series, 232, Cambridge University Press, Cambridge, 1996

[5] S. Buechler, L. Newelski, On the geometry of U-rank 2 types, Logic Colloquium ’90 (Helsinki, 1990), pp. 10-24, Lecture Notes Logic, 2, Springer, Berlin, 1993.

[6] J. Burgess, Effective enumeration of Σ_{1}^{1} equivalence relations, Indiana University Mathematics Journal, vol. 28(1979), pp. 353-364.

[7] E. Effros, Transformation groups and C^{∗}-algebras, Annals of Mathematics, ser 2, vol. 81(1975), pp. 38-55.

[8] S. Gao, Automorphism groups of countable structures, Journal of Symbolic Logic, vol. 63(1998), pp. 891-896.

[9] A. Gregorczyk, A. Mostowski, C. Ryall-Nardzewski, Definability of sets of models of axiomatic theories, Bulletin of the Polish Academy of Sciences, (series Mathematics, Astronomy, Physics), vol. 9(1961), pp. 163-7.

[10] B. Hart, S. Starchenko, M. Valeriote, Vaught’s conjecture for varieties, Transactions of the American Mathematical Society, vol. 342(1994), pp. 173-196.

[11] G. Hjorth, Orbit cardinals: On the effective cardinalities of quotients of the form X/G for X a Polish G-space, Israel Journal of Mathematics, vol. 111(1999), pp. 221-261.

[12] G. Hjorth, Classification and orbit equivalence relations, Mathematical Surveys and Monographs, 75, American Mathematical Society, Providence, RI, 2000.

[13] G. Hjorth, S. Solecki, Vaught’s conjecture and the Glimm-Effros property for Polish transformation groups, Transactions of the American Mathematical Society, vol. 351(1999), pp. 2623-2641.

[14] T. Jech, Set theory, Academic Press, New York, 1978.

[15] A.S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1995.

[16] H.J. Keisler, Model theory for infinitary logic, North-Holland, Amsterdam, 1971.

[17] Y.N. Moschovakis, Descriptive set theory, North-Holland, Amsterdam, 1980.

[18] R. Sami, Polish group actions and the Vaught conjecture, Transactions of the American Mathematical Society, vol. 341(1994), pp. 335-353.

[19] S. Shelah, L. Harrington, M. Makkai, A proof of Vaught’s conjecture for ω-stable theories, Israel Journal of Mathematics, vol. 49(1984), pp. 259-280.

[20] J.R. Steel, On Vaught’s conjecture, Cabal Seminar 76–77, (Proceedings Caltech-UCLA Logic Seminar, 1976–77), pp. 193-208, Lecture Notes in Mathematics, 689, Springer, Berlin, 1978.

[21] J. Stern, Lusin’s restricted continuum problem, Annals of Mathematics, ser 2, vol. 120(1984), pp. 7-37.

[22] R. Vaught, Invariant sets in topology and logic, Fundamenta Mathematicae, vol. 82(1974), pp. 269-94.

6363 MSB

Mathematics

UCLA

CA90095-1555

greg@math.ucla.edu

www.math.ucla.edu/˜ greg