The Countable Admissible Ordinal Equivalence Relation

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by  William Chan

THE COUNTABLE ADMISSIBLE ORDINAL EQUIVALENCE RELATION
WILLIAMCHAN
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n by countable structures. It will be shown here that Fω1 must be classified by structures of high Scott
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8 functionf :X →Y whichisareductionofEtoF,exceptpossiblyonco1untablemanyclassesofE. Usinga
2 resultofZapletal,theexistenceofameasurablecardinalimpliesEω1 ≤a∆11 Fω1. However,itwillbeshown
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O isomorphismrelationinducedbyacounterexample toVaught’s conjecture cannotbe∆11 reducibletoFω1.
L Thisshowstheconsistency ofanegativeanswertoaquestionofSy-DavidFriedman.
.
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a
m
[
1. Introduction
1
v If x∈ω2, ωx denotes the supremum of the order types of x-recursive well-orderings on ω. Moreover, ωx
1 1
4
is also the minimum ordinal height of admissible sets containing x as an element. The latter definition will
2
be more relevant for this paper.
9
7 The eponymous countable admissible ordinal equivalence relation, denoted by Fω1, is defined on ω2 by:
0
x F y ⇔ωx =ωy
1. ω1 1 1
0 It is an Σ1 equivalence relation with all classes ∆1. Moreover, F is a thin equivalance relation, i.e., it
6 1 1 ω1
has no perfect set of inequivalence elements. Some further properties of F as an equivalence relation will
1 ω1
be established in this paper.
:
v Some basic results in admissibility theory and infinitary logic that will be useful throughout the paper
Xi will be reviewed in Section 2. This section will cover briefly topics such as KP, admissible sets, Scott ranks,
andthe Scottanalysis. Inthis section,aspects ofBarwise’stheoryofinfinitary logicincountableadmissible
r
a fragments,whichwillbethemaintoolinmanyarguments,willbereviewed. Asaexampleofanapplication,
a proof of a theorem of Sacks (Theorem 2.16), which establishes that every countable admissible ordinal
is of the form ωx for some x ∈ ω2, will be given. This proof serves as a template for other arguments.
1
Sacks theorem also explains why it is appropriate to call F the “countable admissible ordinal equivalence
ω1
relation”.
TherehavebeensomeearlyworkonwhetherF satisfiescertainpropertiesofequivalencerelationsrelated
ω1
togeneralizationofVaught’sconjecture. Forexample,Markerin[14]hasshownthatF isnotinducedbya
ω1
continuousactionofaPolishgrouponthe Polishspaceω2. Beckerin[3],page782,strenghenedthistoshow
that: theequivalencerelationF isnotanorbitequivalencerelationofa∆1 groupactionofaPolishgroup.
ω1 1
A natural question following these results would be whether F is ∆1 reducible to equivalence relations
ω1 1
induced by continuous or ∆1 actions of Polish groups. If such reductions do exist, another question could
1
January28,2016
ResearchpartiallysupportedbyNSFgrantsDMS-1464475andEMSW21-RTGDMS-1044448
1
be what properties must these reductions have.
In Section 3, Fω1 will be shown to be ∆11 reducible to a continuous action of S∞, i.e., it is classifiable by
countablestructures. Anexplicit∆1classificationofF bycountablestructuresinthelanguagewithasingle
1 ω1
binary relation symbol, due to Montalb´an, will be provided. The classification of F will use an effective
ω1
constructionof the Harrisonlinearordering. This classification,denotedf, hasthe additionalpropertythat
for all x∈ω2, SR(f(x))= ωx+1. This example was provided by Montalb´an through communication with
1
Marks and the author.
The explicit classification, f, mentioned above has images that are structures of high Scott rank. In
Section4, itwillbe shownthatthis is anecessaryfeatureofallclassificationofF by countablestructures.
ω1
The lightface version of the main result of this section is the following:
Theorem 4.2 Let L be a recursive language. Let S(L) denote the set of reals that code L-structures on
ω. If f : ω2 → S(L) is a ∆11 function such that x Fω1 y if and only if f(x) ∼=L f(y), then for all x,
SR(f(x))≥ωx.
1
The more general form considers reductions that are ∆1(z) and involves a condition on the admissible
1
spectrum of z. Intuitively, Theorem 4.2 (in its lightface form as stated above) asserts that any potential
classification of F must have high Scott rank in the sense that the image of any real under the reduction
ω1
is a structure of high Scott rank. High Scott rank means that SR(f(x)) is either ωx or ωx+1.
1 1
Section5isconcernedwithaweakformofreductionofequivalencerelations,inventedbyZapletal,known
asalmost∆1 reduction. IfE andF aretwoΣ1 equivalencerelationsonPolishspaceX andY,respectively,
1 1
then E is almost ∆11 reducible to F (in symbols: E ≤a∆1 F) if and only there is a ∆11 function f :X →Y
1
and a countable set A such that if x,y ∈/ A, then x E y if and only if f(x) E f(y).
An almost Borel reduction is simply a reduction that may fail on countably many classes. Often Σ1
1
equivalence relation may have a few unwieldly classes. The almost Borel reduction is especially useful since
it can be used to ignore these classes. One example of such an Σ1 equivalence relation is E which is the
1 ω1
isomorphism relation of well-orderings with a single class of non-well-orderings. It is defined on ω2 by:
x E y ⇔(x,y ∈/ WO)∨(ot(x)=ot(y))
ω1
E is a thin Σ1 equivalence with one Σ1 class and all the other classes are ∆1.
ω1 1 1 1
Zapletal isolated an invariant of equivalence relations called the pinned cardinal. This invariant involves
pinnednamesonforcings: anideathatappearsimplicitlyorexplicitlyintheworksofSilver,Burgess,Hjorth,
and Zapletal in the study of thin Σ1 equivalence relations. Zapletal showed that there is a deep connection
1
between E , almost ∆1 reducibilities, and pinned cardinals under large cardinal assumptions:
ω1 1
Theorem 5.7 ([20] Theorem 4.1.3)If there exists a measurable cardinal and E is a Σ1 equivalence relation
1
with infinite pinned cardinal, then Eω1 ≤a∆11 E.
Given that this result involves large cardinals, a natural question would be to explore the consistency
results surrounding Zapletal’s theorem. For example, a naturalqustion is whether ZFC can prove the above
result of Zapletal. More specifically, is this result true in Go¨del constructible universe L? This investigation
leads to F in the following way: It will be shown that F has infinite pinned cardinal. Hence, with a
ω1 ω1
measurable cardinal, Eω1 ≤a∆11 Fω1 via the result of Zapletal. (The author can show that 0♯ can prove the
statement that Eω1 ≤a∆11 Fω1. A proof of this will appear in a future paper on pinned cardinals.)
The main result of this section is
Theorem 5.11 The statement Eω1 ≤a∆11 Fω1 is not true in L (and set generic extensions of L).
This result is proved by using infinitary logic in admissible fragments to show that if f is a ∆1(z) function
1
which witnesses Eω1 ≤a∆11 Fω1, then z has anadmissibility spectrum which is full of gaps relative to the set
of all admissible ordinals. No constructible real (or even set generic over L real) can have such property.
2
The final sectionaddressesa questionof Sy-DavidFriedmanusing the techniques of the previoussection.
Essentially, the question is:
Question 6.3IsitpossiblethattheisomorphismrelationofacounterexampletoVaught’sconjectureis∆1
1
bireducible to F ?
ω1
The main result of this final section is:
Theorem 6.9 In L (and set generic extensions of L), no isomorphism relation of a counterexample to
Vaught’s conjecture can be ∆1 reducible to F .
1 ω1
This yields a negative answer to Friedman’s question in L and set generic extensions of L.
TheauthorwouldliketoacknowledgeandthankSy-DavidFriedman,SuGao,AlexanderKechris,Andrew
Marks,andAntonioMontalb´anforveryhelpfuldiscussionsandcommentsaboutwhatappearsinthispaper.
2. Admissibility and Infinitary Logic
The reader should refer to [2] for definitions and further details about admissibility.
Let ∈˙ denote a binary relation symbol. Let L be a language such that ∈˙ ∈ L. KPL denotes Kripke-
Platek Set Theory in the language L with ∈˙ serving as the distinguished membership symbol. The L
subscript will usually be concealed. KP+INF is KP augmented with the axiom of infinity.
Definition 2.1. Let L be a language containing ∈˙. A L-structure A=(A,∈˙A,...) is an admissible set if
and only if A|=KP, A is a transitive set, and ∈˙A =∈↾A.
If A is an admissible set, then o(A)=A∩ON.
Anordinalαisanadmissible ordinal ifandonlyifthereisanadmissiblesetAsuchthatα=o(A). More
generally, if x∈ω2, an ordinal α is x-admissible if and only if there is an admissible set A such that x∈A
and α=o(A).
The admissibility spectrum of x is Λ(x)={α:α is an x-admissible ordinal}.
If x∈ω2, O(x)=min(Λ(x)).
Definition 2.2. For x∈ω2, let HYP(x) denotes the ⊆-smallest admissible set containing x.
Definition 2.3. If x∈ω2, let ωx =O(x).
1
Proposition 2.4. The function (α,x) →L (x), where α∈ON and x is a set, is a Σ function in KP. In
α 1
fact, it is ∆ .
1
Proof. See [2], Chapter II, Section 5 - 7. Also note that the function is defined on a ∆ set. (cid:3)
1
Proposition 2.5. If A is an admissible set with x∈A and α=o(A), then L (x) is an admissible set. In
α
fact, L (x) is the ⊆-smallest admissible set A such that x∈A and o(A)=α.
α
In particular, if α is an x-admissible ordinal, then L (x) is an admissible set.
α
Proof. See [2], Theorem II.5.7. (cid:3)
Proposition 2.6. If x⊆ω, then HYP(x)=LO(x)(x)=Lωx(x).
1
Proof. See [2], Theorem II.5.9. (cid:3)
Definition 2.7. Let x ∈ ω2. Suppose HYP(x) = (Lωx(x),∈). Let HYPx = ω2∩Lωx(x). HYPx is the set
1 1
of all x-hyperarithmetic reals.
Inparticular, x-hyperarithmetic realsare exactlythose reals thatappear in alladmissible sets containing
x.
Next, the relevant aspects of first order infinitary logic and admissible fragments will be reviewed. The
detailed formalization can be found in [2], Chapter III.
3
Definition 2.8. LetL denoteafirstorderlanguage(asetofconstant,relation,andfunctionsymbols). Fix
a ∆ class {v :α∈ON}, which will representvariables. L denotes the collection of finitary L-formulas
1 α ωω
using variables from {vi : i < ω}. L∞ω denotes the collection of all infinitary formulas with finitely many
free variables.
Proposition 2.9. In KP+INF, Lωω is a set. In KP, L∞ω is a ∆1 class.
Proof. See [2], Proposition III.1.4 and page 81. (cid:3)
Proposition 2.10. (KP) “M |=L ϕ(x¯)” as a relation on the language L, L-structure M, infinitary
L-formula ϕ, and tuple x¯ of M is equivalent to a ∆ predicate.
1
Proof. See [2], pages 82-82. (cid:3)
Definition 2.11. Let L be a language. Let A be an admissible set such that L is ∆ definable in A. The
1
admissible fragment of L∞ω given by A, denoted LA, is defined as
LA ={ϕ∈A:ϕ∈L∞ω}={ϕ∈A:A|=ϕ∈L∞ω}
The last equivalence follows from ∆ absoluteness.
1
Definition 2.12. LetL bealanguageconsistingofabinaryrelation∈˙. LetMbeaL-structuresuchthat
(M,∈˙M)satisfiesextensionality. Define WF(M)asthesubstructureconsistingofthewell-foundedelements
of M. WF(M) is called the well-founded part of M.
M is called solid if and only if WF(M) is transitive.
Remark 2.13. The notion of solid comes from Jensen’s [8]. Every structure has an isomorphic solid model
that is obtained by Mostowski collapsing the well-founded part.
The notion of solidness is mostly a convenience: In our usage, ω ⊆ M. Therefore, Mostowski collapsing
will not change reals. Transitivity is desired due to the definition of admissibility and in order to apply
familiar absoluteness results. Rather than having to repeatedly Mostowski collapse WF(M) and mention
reals are not moved, one will just assume the well-founded part is transitive by demanding M is solid.
Lemma 2.14. (Truncation Lemma) If M|=KP, then WF(M)|=KP. In particular, if M is a solid model,
then WF(M) is an admissible set.
Proof. See [2], II.8.4. (cid:3)
The following is the central technique used in the paper:
Theorem 2.15. (Solid Model ExistenceTheorem) Let A be a countable admissible set. Let L be a language
which is ∆ definable over A and contains a binary relation symbol ∈˙ and constant symbols a¯ for each
1
a ∈ A. Let T be a consistent L-theory in the countable admissible fragment LA, be Σ1 definable over A,
and contains the following:
(i) KP
(ii) For each a∈A, the sentence (∀v)(v∈˙a¯⇒ v =z¯).
Then there exists a solid L-structure B such thatWBz∈|=a T and ON∩B =ON∩A.
Proof. See [8], Section 4, Lemma 11. (cid:3)
Theorem 2.16. (Sacks’ Theorem) If α > ω is an admissible ordinal, then there exists some x ∈ ω2 such
that α=ωx.
1
Let z ∈ω2. If α∈Λ(z)∩ω1, then there exists y ∈ω2 with ω1y =α and z ≤T y.
Proof. See [17], Corollary 3.16. The following proof is similar to [8], Section 4, Lemma 10. The second
statement will be proved below:
Since α ∈ Λ(z), let A be an admissible set such that z ∈ A and o(A) = α. (For example, A = L (z) by
α
Proposition 2.5.)
Let L be a language consisting of the following:
(I) A binary relation symbol ∈˙.
(II) Constant symbols a¯ for each a∈A.
4
(III) One other distinguished constant symbol c˙.
The elements of L can be appropriately coded as elements of A so that L is ∆ definable over A.
1
Let T be a theory in the countable admissible fragment LA consisting of the following:
(i) KP
(ii) For each a∈A, (∀v)(v∈˙a¯⇒ v =z¯).
Wz∈a
(iii) c˙ ⊆ω¯.
(iv) For each ordinal σ ∈α, “σ¯ is not admissible relative to c˙”. More formally, “L (c˙)6|=KP+INF”.
σ¯
(v) z¯≤ c˙.
T
T can be coded as a class in A in such a way that it is Σ in A. T is consistent: Find any u ∈ ω2 which
1
codes an ordinal greater than α. Let c = u⊕z. Consider the following L-structure M: The universe M
is Hℵ . For each a ∈ A, a¯M = a. (Since A is countable and transitive, A ∈ Hℵ .) ∈˙M =∈↾ Hℵ . c˙M = c.
1 1 1
M clearly satisfy (i), (ii), (iii), and (v). For (iv), suppose there is an σ < α such that L (c) |= KP. Since
σ
c∈L (c)andL (c)|=KP, u∈L (c)becausec=u⊕z. Sincethe MostowskicollapsemapisaΣ definable
σ σ σ 1
function in KP, if reals code binary relations in the usual way, then KP proves the existence of ot(u). Thus
ot(u) ∈ L (c). However, ot(u) > α > σ. Contradiction. It has been shown that M also satisfy (iv). T is
σ
consistent.
TheSolidModelExistenceTheorem(Theorem2.15)impliesthereisasolidL-structureB |=T suchthat
ON∩B = ON∩A = α. Let y = c˙B. The claim is that ωy = α. By Lemma 2.14, WF(B) is an admissible
1
set containing y and z. o(WF(B)) = ON∩WF(B)= ON∩B = ON∩A = α. Thus ωy ≤ α. Now suppose
1
that ω1y < α. In V, Lωy(y) |= KP. Since the function (α,x) 7→ Lα(x) is ∆1 (by Proposition 2.4) and the
1
satisfactionrelationis ∆ (by Proposition2.10), by ∆ absolutenessbetween the transitive sets WF(B)and
1 1
V, one has WF(B) |= Lωy(y) |= KP. Again by absoluteness of ∆1 formulas between the transitive (in the
sense of B) sets WF(B) a1nd B, B |= Lω1x |= KP. Letting σ = ω1x < α, B |= Lσ¯(c˙) |= KP. This contradicts
B |=T. A similar absoluteness argument shows that z ≤ y. (cid:3)
T
Remark 2.17. This proofofSacks theoremis the basic template for severalother argumentsthroughoutthe
paper. This proof will be frequently referred.
Next,variousaspects ofthe Scottanalysiswillbe reviewed. Since there aresomeminorvariationsamong
the definitions of Scott rank, Scott sentences, canonical Scott sentences, etc., these will be provided below.
See [15], page 57-60 or [16] for more information.
Definition 2.18. Let L be a language. Define the binary relation (M,a) ∼ (N,b) where α ∈ ON,
α
a∈<ωM, and b∈<ωN as follows:
(i) (M,a)∼ (N,b) if and only if for all atomic L-formulas ϕ, M|=ϕ(a) if and only if N |=ϕ(b).
0
(ii) If α is a limit ordinal, then (M,a)∼ (N,b) if and only if for all β <α, (M,a)∼ (N,b).
α β
(iii) If α = β +1, then (M,a) ∼ (N,b) if and only if for all c ∈ M, there exists a d ∈ N such that
α
(M,a,c)∼ (N,b,d) and for all d∈N, there exists a c∈M such that (M,a,c)∼ (N,b,d).
β β
LetMbeaL-structureanda∈kM forsomek∈ω. Forα∈ON,theL∞ω formulaΦMa,α(v)(invariables
v such that |v|=k) is defined as follows:
(I)LetX be the setofallatomicandnegationatomicL-formulaswithfreevariablesv suchthat|v|=k.
Let ΦM(v)= X
a,0 V
(II) If α is a limit ordinal, let X ={ΦM(v):β <α}. Let ΦM (v)= X.
a,β a,α V
(III) If α = β +1, then let X = {(∃w)ΦM (v,w) : b ∈ M} and Y = {ΦM (v,w) : b ∈ M}. Then let
ab,β ab,β
ΦM (v)= X∧(∀w) Y.
a,α
V W
For M, a L-structure, and a∈kM (for some k), define ρ(M,a¯) to be the least α∈ON such that for all
b∈kM, (M,a)∼ (M,b) if and and only if for all β, (M,a)∼ (M,b).
α β
Define SR(M)=sup{ρ(M,a)+1:a∈<ωM}. Define R(M)=sup{ρ(M,a):a∈<ωM}.
Let α=R(M). Let
X ={(∀v¯)(ΦM (v)⇒ΦM (v)):a∈<ωM}
a,α a,α+1
CSS(M)=ΦM ∧ X
∅,α ^
5
CSS(M) is the canonical Scott sentence of M. SR(M) is the Scott rank of M.
The following are well-known results. Usually, a careful inspection of the proof indicates what can be
done in KP+INF or ZFC.
Proposition 2.19. The relation ∼ is equivalent to ∆ formula over KP+INF.
1
Proof. It can be defined by Σ-recursion. (cid:3)
Proposition 2.20. Let A be an admissible set such that A|= INF. Let L ∈ A be a language. Let M ∈ A
range over L-structure, a range over elements of <ωM, and α range over ON∩A. Then the function
f(M,a,α)=ΦM (v) is ∆ definable in A.
a,α 1
In particular, if R(M)∈A, then CSS(M)∈A.
Proof. It can be defined by Σ-recursion. (cid:3)
Proposition 2.21. (KP+INF) Let L be a language. Let M and N be L-structures, a∈kM, b∈kN for
some k ∈ω, and α∈ON. Then (M,a)∼ (N,b) if and only if N |=ΦM (b).
α a,α
Proof. This is proved by induction. See [15], Lemma 2.4.13. (cid:3)
Proposition 2.22. (KP+INF) If (M,a)≡L∞ω (N,b), then for all α, (M,a)∼α (N,b).
Proof. M|=ϕ (a). So N |=ϕ (b). By Proposition 2.21, (M,a)∼ (N,b). (cid:3)
a,α a,α α
Definition 2.23. LetL bealanguage. LetϕbeaformulaofL∞ω. Thequantifier rank ofϕdenotedqr(ϕ)
is defined as follows:
(i) qr(ϕ)=0 if ϕ is an atomic formula.
(ii) qr(¬ϕ)=qr(ϕ)
(iii) qr( X)=qr( X)=sup{qr(ψ):ψ ∈X}.
V W
(iv) qr(∃vϕ)=qr(∀vϕ)=qr(ϕ)+1.
Proposition 2.24. The relation “qr(ϕ)=α” is ∆ definable in KP+INF.
1
Proof. It can be defined by Σ-recursion. (cid:3)
Proposition 2.25. (KP+INF) Let L be a language. M,N be L-structures. a∈kM and b∈kN for some
k ∈ ω. Then for all α ∈ ON, (M,a) ∼ (N,b) if and only if for all ϕ with qr(ϕ) ≤ α, M |= ϕ(a) if and
α
only if N |=ϕ(b).
Proof. This is proved by induction. (cid:3)
Proposition 2.26. (ZF) Let L be some language. Let M and N be L-structures. Suppose A is an
admissible set with L,M,N ∈A. Then A|=M≡L N if and only if M≡L N.
∞ω ∞ω
Proof. See [16], Theorem 1.3. (cid:3)
Remark 2.27. Acommonphenomenonisthatcertainpropertiesarereflectedbetweenappropriateadmissible
sets and the true universe. A useful observation is that if such a property holds from the point of view of
an admissible set then it is true in the universe. The above proposition asserts that infinitary elementary
equivalence is such a property.
Another familiar example is the effective boundedness theorem. Suppose ϕ:WO→ω is a Π1 rank. Let
1 1
B ⊆ WO be Σ1. Let A be a countable admissible set containing the parameters used to define B. Inside
1
of A, ϕ(B) is bounded by o(A). A priori, the true bound on ϕ(B) may be higher as the true universe has
more countable ordinals and more members of B. However, the effective boundedness theorem asserts that
in fact, in the true universe, ϕ(B) is bounded by o(A).
The following proposition with an included proof shows countable admissible sets can also be used to
produce true bounds on the Scott rank.
Proposition2.28. LetL beacountablelanguage andMbeacountableL-structure. Onemayidentify M
as a real by associating it with an isomorphic structureon ω. If A is an admissible set with L,M∈A, then
R(M)≤ON∩A. R(M)≤O(M). SR(M)≤O(M)+1. Inparticular, R(M)≤ωM andSR(M)≤ωM+1.
1 1
6
Proof. See [16], Corollary 1.
It suffices to show that R(M) ≤ O(M). Suppose not. Then there exists a and b such that for all
α < O(M), (M,a) ∼ (M,b) but for β = O(M), (M,a) 6∼ (M,b). Let A be an admissible set with
α β
M ∈A and o(A) =O(M). By ∆1-absoluteness and Proposition 2.25, A |= (M,a)≡L∞ω (M,b). Thus by
Proposition2.26,(M,a)≡L∞ω (M,b). However,(M,a)6∼β (M,b)implies M|=ΦMa,β(a)andM6|=ΦMa,β(b)
by Proposition 2.21. This shows (M,a)6≡L (M,b). Contradiction.
∞ω
(cid:3)
Definition 2.29. Let L be a language. Let M be a L-structure. ϕ is a Scott sentence if and only if for
all L-structure N and M, N |=ϕ and M|=ϕ implies M≡L N.
∞ω
Theorem 2.30. (ZFC) Let L be a language. Let M be a countable L-structure. Then there exists a
L∞ω-sentence ϕ such that for all countable L-structure N, N ∼=L M if and only if N |= ϕ. In fact, ϕ is
CSS(M).
(KP+INF) If ϕ is a Scott sentence for a countable structure M, then for all countable N, N |=ϕ if and
only if N ∼=L M.
Proof. Observe the first statement asserts that there exists a sentence such that whenever a countable
structure satisfies this sentence, there exists an isomorphism between it and M. The existence of this
sentence requires working beyond KP+INF. The second statement asserts that KP+INF can prove that
if a Scott sentence happens to exist, then for any countable structure satisfying this sentence, there is an
isomorphism between it and M.
This is the Scott’s isomorphism theorem. See [15], Theorem 2.4.15 for a proof. The results in KP+INF
follows essentiallythe same proofwith the assistanceof some of the abovepropositions provedin KP+INF.
(cid:3)
Definition 2.31. Let L be a countable language. Let S(L) denote the set of all L-structures on ω.
Definition 2.32. Let ∈˙ be a binary relation symbol. Let S∗ denote the subset of S({∈˙}) consisting of
ω-models of KP+INF.
Proposition 2.33. Let {φ : e ∈ ω} be a recursive enumeration of {∈˙} -formulas. The relation on
e ωω
x∈S({∈˙}) and e∈ω asserting “x|=φ ” is ∆1.
e 1
Also S∗ is ∆1.
1
Proof. See [12], page 14-16 for relevant definitions and proofs. (cid:3)
Remark 2.34. One can check that there is a ∆1 function such that given A ∈ S∗ and n ∈ ω, the function
1
givestheelementofAwhichAthinksisn. Usingthis,onecandetermineina∆1 waywhetherA∈S∗ thinks
1
some x ∈ ω2 exists. In the following, if A ∈ S∗ and x ∈ ω2, the sentence “x ∈ A” should be understood as
this informally described ∆1 relation.
1
Proposition 2.35. Let L be a recursive language. Let ϕ∈HYP(x)∩L∞ω. Then Mod(ϕ)={s∈S(L):
s|=L ϕ} is ∆11(x).
Proof. Note that s∈Mod(ϕ) if and only if
(∃A)(A∈S∗∧x∈A∧s∈A∧A|=s|=L ϕ)
if and only if
(∀A)(A∈S∗∧x∈A∧s∈A)⇒A|=s|=L ϕ)
These equivalences are established using the absoluteness of satisfaction. This shows that Mod(ϕ) is ∆1(x).
1
(cid:3)
Remark 2.36. Later, the paper will be concerned with relating countable admissible sets and isomorphism
of countable structures. The second statement of Theorem 2.30 captures the essence of these types of
arguments: Isomorphism of countable structures is reflected between the true universe and admissible sets
which witness the countability ofthe relevantstructures and possessesa Scottsentence for these structures.
TheoriginalargumentsforsomeresultsofthispaperusedmoredirectlythesecondstatementofTheorem
2.30. The argument presented below is simpler using the Scott isomorphism theorem and Proposition 2.35
but may conceal this essential idea.
7
Now to introduce the main equivalence relation of this paper:
Definition 2.37. Let F be the equivalence relationdefined on ω2 by x F y if and only if ωx =ωy. F
ω1 ω1 1 1 ω1
is a Σ1 equivalence relation with all classes ∆1.
1 1
The first claim from the above definition is well known and follows easily from the characterizationof ωx
1
as the supremum of the x-recursive ordinals. The next proposition implies each class is ∆1. There will be
1
much to say later about the complexity of each F -equivalence class.
ω1
Proposition 2.38. Let α be a countable admissible ordinal and z ∈ ω2 be such that α < ωz. Then the set
1
{y ∈ω2:ωy =α} is ∆1(z).
1 1
Proof. If u and v are reals coding linear orderings on ω, then u(cid:22)v means there exists an order preserving
injective functionf fromthe linearorderingcodedby utothe linearorderingcodedbyv. (cid:22)isaΣ1 relation
1
in the variables u and v.
Since α < ωz, there exists some e ∈ ω such that {e}z is the characteristic function of a well-ordering
1
isomorphic to α. Let B ={y ∈ω2:α=ωy}. Then
1
y ∈B ⇔(∀n) ({n}y ∈WO⇒{n}y (cid:22){e}z) ∧
(cid:16) (cid:17)
(∀k)(∃j)({j}y (cid:22){e}z∧{e}z ↾k(cid:22){j}y)
B is Σ1(z). Also
1
y ∈/ B ⇔(∃j)(∀n)({n}y ∈WO⇒({n}y (cid:22){e}z ↾j)∨(∃n)({n}y ∈LO∧{n}y (cid:22){e}z∧{e}z (cid:22){n}y)
B is Π1(z). Hence B is ∆1(z). (cid:3)
1 1
3. Classifiable by Countable Structures
Definition 3.1. Let x ∈ ω2. A linear ordering R on ω is an x-recursive x-pseudo-wellordering if and only
if R is an x-recursive linear ordering on ω which is not a wellorderingbut Lωx(x)|=R is a wellordering,i.e.
1
R has no x-hyperarithmetic descending sequences.
Proposition 3.2. (Harrison, Kleene) For all x∈ω2, there exists an x-recursive x-pseudo-wellordering.
Proof. See [9] or [19], III.2.1. A generalized form of this construction will be used below.
ThiscanalsobeprovedusingTheorem2.15andinfinitarylogicinadmissiblefragments. Intheapplication
of Theorem 2.15, Barwise compactness is used to show the consistency of the appropriate theory in the
countable admissible fragment. See Nadel’s proof given in [1] VIII, Section 5.7 for more details. (cid:3)
The following characterizes the order type of x-recursive x-pseudo-wellorderings:
Theorem 3.3. (Harrison) Let R be a x-recursive x-pseudo-wellordering, then ot(R)=ωx(1+η)+ρ where
1
η =ot(Q) and ρ<ωx.
1
Proof. See [7] or [19], Lemma III.2.2. (cid:3)
Proposition 3.4. Recall if y ∈ω2, then HYPy =Lωy(y)∩ω2, the set of y-hyperarithmetic reals.
1
The relation x∈HYPy is a Π1 relation in the variable x and y.
1
Proof. The claim is that:
x∈HYPy ⇔(∀A)((A∈S∗∧y ∈A)⇒(x∈A))
SeeRemark2.34aboutwhat“y ∈A”shouldpreciselymean. ThelatterpartoftheequivalenceisΠ1. Hence
1
the result follows from the claim.
To prove the claim:
(⇒) Suppose A ∈ S∗. Let n ∈ ω be the representative of y in A. Since A |= KP, by Lemma 2.14
(Truncation Lemma), WF(A) |=KP. Let π be the Mostowski collapse of WF(A) onto an admissible set B.
y ∈ B since y = π(n). Since x ∈ HYPy, x is in every admissible set containing y. x ∈ B. Then π−1(x)
represents x in A.
(⇐)Recall HYP(y)is the smallestadmissiblesetcontaining xandω. The domainofHYP(y)is Lωy(y).
It is countable. Let π : Lωy(y) →ω be any bijection. The bijection gives an element A ∈S∗ isomorph1ic to
1
8
HYP(y). π(y) represents y in A. There exists some n∈ω suchthat n representsx in A, by the hypothesis.
Then x∈Lωy(y) since x=π−1(n). x∈HYPy. (cid:3)
1
The following propositions uses the ideas from [19] III.1 and III.2.
Proposition 3.5. There exists a recursive tree U on 2×ω such that for all x∈ω2, Ux has a path but has
no x-hyperarithmetic paths.
Proof. By Proposition3.4, there is a recursive tree V on 2×2×ω such that x∈/ HYPy if and only if V(x,y)
is ill-founded. Define the relation Φ on ω2×ωω by
Φ(y,f)⇔(∀n)((f (n)=0∨f (n)=1)∧V(f ↾n,y ↾n,f ↾n))
0 0 0 1
where f (n)=f(hi,ni), for i=0,1. Φ is Π0. Let U be a recursive tree on 2×ω such that
i 1
Φ(y,f)⇔(∀n)((y ↾n,f ↾n)∈U)
For any y, if Uy has a path f, then Φ(y,f). Therefore, f1 ∈[V(f0,y)]. f0 ∈/ HYPy. So Uy can not have a
y-hyperarithmetic path f, since otherwise f ∈ HYPy, which yields a contradiction. Uy has a path: Let x
0
be any real which is not in HYPy. [V(x,y)] is non-empty. Let g ∈ [V(x,y)]. Let f be such that f = x and
0
f =g. Then Φ(y,f). f ∈[Uy]. (cid:3)
1
Definition 3.6. The Kleene-Brouwerordering < is defined on <ωω as follows: s< t if and only if
KB KB
(i) t(cid:22)s and |t|<|s|
or
(ii) If there exists an n∈ω such that for all k <n, s(k)=t(k) and s(n)<t(n).
Proposition 3.7. Let T be a tree on ω. T is wellfounded if and only if <KB↾T is wellfounded. Moreover,
if there is an x-hyperarithmetic infinite descending sequence in <KB↾T, then there is an x-hyperarithmetic
path through T.
Proof. If f ∈[T], then {f ↾n:n∈ω} is an infinite descending sequence in < ↾T.
KB
Let S ={s ∈<ω2:n∈ω} be an x-hyperarithmetic descending sequence in < ↾T. Define f ∈ωω by
n KB
f(n)=i⇔(∃p)(∀q ≥p)(s (n)=i)
q
f ∈[T] and f is Σ0(S). f is also x-hyperarithmetic. (cid:3)
2
NowtoproduceaclassificationofF bycountablestructures. Theideawillbetosendxtoanx-Harrison
ω1
linearordering. Using Proposition3.5andapplying the Kleene-Brouwerordering,onecanobtaina function
gsuchthatg(x)isanx-recursivex-pseudo-wellordering. Nowsupposeωx =ωy. Letαdenotethisadmissible
1 1
ordinal. By Theorem 3.3, ot(g(x))=α(1+η)+ρ and ot(g(y))=α(1+η)+ρ , where ρ <α and ρ <α.
x y x y
However,it could happen that ρ 6=ρ . One way to modify g to get a classificationof F would be to “cut
x y ω1
off” the recursivetail ofg(x). To do this, one uses a trick, suggestedMontalban,to cut offthe recursivetail
of the order type by taking a product of ω copies of g(x). The details follow:
Proposition 3.8. Fix x∈ω2. Let ρ<ωx and η =ot(Q). Then (ωx(1+η)+ρ)ω =ωx(1+η).
1 1 1
Proof. Let P be any x-recursive x-pseudo-wellorderings of order type ωx(1 +η)+ρ. Let P ×ω be the
1
x-recursivestructureisomorphictoω copiesofP followingeachother. P×ω isstillanx-recursivex-pseudo-
wellordering. It has no x-recursive tail. By Theorem 3.3, ot(P ×ω)=(ωx(1+η)+ρ)ω =ωx(1+η). (cid:3)
1 1
Proposition 3.9. There exists an e ∈ ω such that for all x ∈ ω2, {e}x is isomorphic to (<KB↾ Ux)·ω,
where U comes from Proposition 3.5.
Proof. This is basic recursion theory using the previous results. (cid:3)
Theorem 3.10. (Montalba´n) The equivalence relation F is classifiable by countable structures. In fact,
ω1
there is an e∈ω such that f(x)={e}x is the desired classification.
9
Proof. LetL ={R˙}, whereR˙ is a binaryrelationsymbol. F willbe classifiedby countableL-structures.
ω1
Ux isanx-hyperarithmetictreewithpathsbutnox-hyperarithmeticpath. Hence< ↾Ux isanx-recursive
KB
linear ordering with infinite descending sequences but no x-hyperarithmetic infinite descending sequences.
So < ↾ Ux is an x-recursive x-pseudo-wellordering. It has order type ωx(1+η)+ρ for some ρ < ωx.
KB 1 1
Therefore, (< ↾ Ux)·ω has order type ωx(1+η), i.e., it is an x-Harrison linear ordering. Hence x F y
if and only ω1xK=B ω1x if and only ω1x(1+η)1=ω1y(1+η) if and only (≤KB↾Ux)·ω ∼=L (≤KB↾ Uy)·ω ifωa1nd
only if {e}x ∼=L {e}y. This gives a classification of Fω1. (cid:3)
4. Finer Aspects of Classification by Countable Structures
The previous section provided an explicit classification f :ω2→S(L) which was ∆1 and for all x∈ω2,
1
SR(f(x))=ωx+1. This section will show that any classification of F by countable structures must have
1 ω1
a similar property.
ThenextresultwillcalculatethecomplexityofeachF classaccordingtoeffectivedescriptivesettheory.
ω1
Theorem 4.1. For any x∈ω2, [x] is not Π1(z).
Fω1 1
Proof. Suppose [x] is Π1(x). Let B = ω2−[x] . B is then Σ1(x). Let U be a tree on 2×ω recursive
Fω1 Fω1 1
in x such that
y ∈B ⇔[Uy]6=∅
Observe U ∈Lωx(x).
Let L be the1language consisting of the following:
(I) A binary relation symbol ∈˙.
(II) Constant symbol a¯ for each a∈Lωx(x).
1
(III) Two other distinguished constant symbols c˙ and d˙.
L can be considered a ∆1 definable subset of Lωx(x).
L may be regarded as a ∆1 subset of Lωx(x). 1
Let T be a theory in the countable adm1issible fragment L consisting of the following:
(i) KP Lω1x(x)
(ii) For each a∈Lω1x(x), (∀v)(v∈˙a¯⇒Wz∈av =z¯).
(iii) c˙ ⊆ω¯ and d˙:ω¯ →ω¯.
(iv) For each ordinal σ ∈ωx, “σ¯ is not admissible relative to c˙”.
1
(v) d˙∈[U¯c˙].
T can be considered a Σ1 on Lωx(x) theory.
T is consistent: Find any y ∈1ω2 such that ωy > ωx. Then y ∈ B. There exists some z ∈ ωω such that
1
z ∈ [Uy]. Consider the L-structure M defined as follows: M = Hℵ1. ∈˙M =∈↾ Hℵ1. For each a ∈ Lω1x(x),
let a¯M =a. Let c˙M =y and d˙M =z. M|=T.
By Theorem 2.15, T has a solid model N such that ON∩N = ON∩Lω1x(x) = ω1x. Let u = c˙N and
v =d˙M. As in Theorem 2.16, ωu =ωx.
1 1
N |=v ∈[Uu]. By ∆ absoluteness,WF(N)|=v ∈[Uu]. Since N is solid,WF(N) is transitive as viewed
1
in V. So by ∆ absoluteness, V |=v ∈[Uu]. [Uu]6=∅. u∈B. ωu 6=ωx. Contradiction. (cid:3)
1 1 1
Suppose f is a classification of F by countable structures in some recursive language. The Scott rank
ω1
of the image of f must be high:
Theorem 4.2. Let L be a recursive language. If f :ω2→S(L) is a ∆1(z) function such that x F y if
1 ω1
and only if f(x)∼=L f(y), then for all x such that ω1x ∈Λ(z), SR(f(x))≥ω1x.
Proof. Supposethereexistsanx∈ω2withωx ∈Λ(z)andSR(f(x))<ωx. Letα=ωx. ByProposition2.16,
1 1 1
there exists a y with z ≤T y and ω1y = α. Since ω1y = α = ω1x, x Fω1 y. This implies that f(x) ∼=L f(y).
Hence SR(f(y)) = SR(f(x)) < ω1x = α = ω1y. z ≤T y implies that z ∈ Lωy(y), and in particular, z is
1
in every admissible set containing y. Since f is ∆1(z), f(y) is ∆1(z,y) = ∆1(y) since z ≤ y. f(y) is
1 1 1 T
hyperarithmetic in y. f(y) is in every admissible set that has y as a member. Since SR(f(y)) < ωy and
1
f(y) is in every admissible set containing y, CSS(f(y)) is in every admissible set containing y. In particular
CSS(f(y))∈Lωy(y).
1
10