Improving a Bounding Result for
Weakly-Scattered Theories
by
Christina M. Goddard
Bachelor of Science, The University of Queensland, December 1999
Bachelor of Engineering, The University of Queensland, December
1999
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2006
© Christina M. Goddard, MMVI. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
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May 5, 2006
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Improving a Bounding Result for Weakly-Scattered Theories
by
Christina M. Goddard
Submitted to the Department of Mathematics
on May 5, 2006, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
In this thesis, we effectively construct a predecessor function for the type definitions in
the raw hierarchy for any weakly-scattered theory. Using this predecessor function, we
improve a recent bounding result by Sacks for weakly-scattered theories by removing
the assumption of a predecessor function from the k-splitting hypothesis.
We begin by giving an introduction to the infinitary logic C,,, and admissible
sets. We then outline results by Sacks that are important in the construction of the
predecessor function. We introduce scattered and weakly-scattered theories and their
related hierarchies, and explain how they relate to the well-known Scott hierarchy.
Using the raw tree hierarchy, we present Sacks' constructive result called the Effective
Recovery Process.
Using all of these tools, we provide a proof of the existence of a predecessor
function for the type definitions and then use it to improve the bounding result by
Sacks.
Thesis Supervisor: Gerald E. Sacks
Title: Professor of Mathematics
3
4
Acknowledgments
I would like to thank my advisor, Professor Sacks, for his support, advice, and lovely
lunch meetings. Thanks also to my close friend and colleague Alice Chan, who has
been a great help for support and discussion. I would also like to thank my fellow
graduate logic friends and colleagues, especially Nate Ackerman and Cameron Freer,
for their weekly logic seminars, and their companionship. In addition, I would like
to thank my thesis committee, Professors Gerald Sacks, Hartley Rogers, and Akihiro
Kanamori for their expert knowledge and advice.
It has been wonderful being a part of the Boston logic community, and I thank
them all for their constant support and lively dinners over the years. Finally, I'd like
to thank my husband, Michael, and my family back in Australia for their love and
firm belief that I would actually complete this project.
5
6
Contents
1 Introduction
9
1.1 Infinitary Logic .......................
1.2 Admissible Sets ......................
. . . . . . .
. . . . . . .
1.3 Scott Analysis .......................
. . . . . . .
2 Constructing the Raw Tree Hierarchy
2.1 Weakly-Scattered Theories and the Raw Tree Hierarchy.
2.2 Effective Recovery of the Raw Hierarchy .........
2.3 Predecessor Function for the Raw Hierachy ........
2.3.1
Stage 0 . . . ..
. . ..
..
..
2.3.2
Successor Stage ..................
2.3.3
Limit Stage ....................
. . . . . . . . . .
3 A Bound for Weakly-Scattered Theories
3.1 Partial Domains .......................
......... .
......... .
......... .
......... .
......... .
......... .
10
13
21
25
25
28
31
31
32
33
39
39
7
8
Chapter 1
Introduction
Vaught's Conjecture questions the number of models of a complete, countable theory
and is one of the questions that have shaped modern model theory. Since it was
posed by Vaught in 1959, partial results have been achieved by analyzing the Scott
Hierarchy.
In Morley's ground-breaking paper giving a positive result towards Vaught's Conjecture, he introduces the notion of scattered theories. Sacks uses a generalization
of these theories, called weakly-scattered theories, to produce further results. He
introduces a tree hierarchy, called the Raw Hierarchy, which is similar to the Scott
Hierarchy and enumerates the models of a weakly-scattered theory.
We effectively construct a predecessor function for the type definitions in the
Raw Hierarchy for any weakly-scattered theory. Using this predecessor function, we
improve a recent bounding result by Sacks for weakly-scattered theories.
We first introduce the infinitary logic C,,,, and admissible sets. We then provide
an outline of important results by Sacks [7] including the Small Sets Lemma and the
Iterated Bounding Theorem. We describe the Scott Analysis of countable models,
which provides a characterization of countable models in terms of a hierarchy of
countable theories.
In Chapter 2, we introduce scattered and weakly-scattered theories and provide
their related hierarchies. We then outline Sacks' Effective Recovery Theorem given
in [7] and use it to construct the predecessor function.
We conclude by showing how this predecessor function improves Sacks' main
bounding result in [7] by removing the assumption of the existence of such a predecessor function from the effective k-splitting hypothesis.
This thesis is part of Professor Gerald Sacks' program on Vaught's Conjecture.
9
Related work was done by Alice Chan in her thesis [21 that characterizes the ksplitting hypothesis using iterated forcing to obtain Sacks' bounding result in terms
of a Ba hypothesis.
Infinitary Logic
1.1
Let C be a countable first-order language. We define the logic C,w to be built up from
£ as in first-order logic, but we also allow countable conjunctions and disjunctions.
Usually, we will only be concerned with formulas of £w4with finitely many free
variables.
As in first-order logic, we define a set of axioms and rules of inference for a proof
system in
,.
In fact, the axioms and rule are basically the same as those for
first-order logic. Here is a list of the axioms and rules as given by Keisler [3].
Axioms:
1. Every instance of every tautology for first-order propositional logic;
2. (y)
(o-), that is for example, (--VO) - (3-,i);
3. (A I))
for all gE(;
4. (Vxg(x,...))
- g(t,...)
where t is a term substitutable for x in p;
5. x = x; and x = y - y =x;
6.
(x,... ) A x = t -
o(t,...) where t is a term substitutable for x in o.
Rules of Inference:
1. at and ( -. p) implies p;
2. ( - V(x,... )) implies ( -. Vxp(x,... )) if x does not occur free in ;
3. For all p E ), we have that ( - g) implies
- A D.
Only Axiom 3 and Rule 3 do not appear in first-order logic.
The set of theorems of £, , is the least set of formulas of £,
containing all of
the axioms and closed under all of the rules of inference.
One problem with using C,,,, is that it is always uncountable. For many applications, including ours, it is easier to use certain countable subsets of
fragments, that have nice closure properties.
10
,,
called
First, for any set A, let LA = L,,, n A. As in Keisler [3], we define LA to be a
fragment of Lw,
if the following properties hold:
1. A is a nonempty transitive set;
2. If a,b E A,then{ a,b} EA,aUb E A,a x b EA;
3. If a E A and a is the least ordinal that isn't in the transitive closure of a, then
we also have that a C A;
4. If V(x,...) G LA, and t is a term in £,
such that t E A, then
(t,... ) E LA.
Thus a fragment is based on a transitive set that is closed under some very finitary
properties, and is closed under substitution.
Let £ be a countable first-order language, let LA be a countable fragment of
£~~Wand let T C LA be a set of sentences in LA. We now introduce some sufficient
conditions for T to have a model.
We use a Henkin construction to form the model of such a T, and as such let C be
a countable set of new constants. Let M be the first-order language formed by adding
each c
C to L. Form M,
corresponding to M. Let MA be the set of formulas
O(xl,..., Xn, Cl, . .., cm) of Mwl,, obtained from formulas
(P(Xl,...
Xn,
YI, .. ,
Ym)
LA by replacing each free occurrence of yi by ci for 1 < i < m.
We say that T is w-complete if the following properties hold:
1. For any sentence T E LA, either p E T or (-'p) C T. So T is complete in LA in
the usual sense.
2. If the sentence (Viel pi) is in T, then there exists an i E I such that oi E T.
Also, T is finitarily consistent if no contradiction can be derived from T using only the
finitary axioms and inferences of L£,,, that is, all the axioms and rules of inference
mentioned above except for Axiom 3 and Rule 3.
Proposition
1.1. Suppose for all a < 3 <
, we have the following: Ta is a finitarily
consistent, w-complete theory of the fragment 4,; the theory Ta C T0; and La C £~.
Then the union U{Ta a < y} is a finitarily-consistent, w-complete theory in the
fragment U{L
a < y}.
Proposition
1.2 (Model Existence Theorem). Let T be a finitarily-consistent, w-
complete theory in a countable fragment LA. Then T has a model.
11
Proof. Let Y be the least set of formulas such that:
* TCY;
* Y is closed under subformulas;
* If t is a term in MA, c E C, and V(t) E Y, then ¢(c) E Y;
* If ()
E Y, then (_)
E Y;
* Ifc,dEC,then(c=d) EY.
Thus, Y is countable. Let X be the set of all sentences of Y. So X C MA.
Enumerate X = {o,
. .}...and let T = {t o , tl,.. .} be the set of all basic terms of
M.
We construct a sequence To
we have Tn. We construct
T C T2
... as follows. First let To be T. Suppose
T,+l so that:
1. Tn C Tn+1 C MA and Tn+l is finitarily consistent and rw-completein LA.
2. If Tn U {n} is finitarily consistent, then on
3. If Tn U {}n is finitarily consistent and Vn
T,+.
=
Vi,b, then we have that T, U
{3x-Vi ,l}(X) is finitarily consistent. For if not, then
T-3V
i()
i
T, F Vs --
i (Y)
i
T,
a contradiction.
Vi(0
i
Thus Tn U
ji(£)}
{3xFVij
is finitarily
consistent
and 3Vi
i(Y) E LA
So
Thus Tn U {Vi3Yi(Y)} is finitarily consistent (by con(3 Vi bi(Y)) T
traposition) and Vi 3Y0i(Y) E LA. So since Tn is w-complete, there exists an i
such that 3x¢i(Y) E T,. So let 4i(cj E Tn+l.
4. If Tn U {(n} is finitarily consistent and (n = 3xo(x), then since To C LA and
since by stage n, we have only added finitely many formulas in MA, there exists
a c E C not in Tn and not in the formula 3xqo(x). Then Tn U {p(c)} is finitarily
consistent.
So let V(c) C Tn+.
12
5. Let c'
Then T U (t
C be a new constant not added so far.
{Anything added in Steps 2-4} is finitarily consistent, so let (tn = c')
Now we define a model
)U
=
of T. Let T,, = Un< Tn. For c, d E C, let c
T+l
d iff
(c = d) E T,. Using finitary consistency and the above construction, it is easy to see
that - is an equivalence relation on C.
T and cl
I c C}. If p(cl,...,cn)
have the universe {c / Let
dl,..., .
d., then obviously by finitary consistency o(d1 , ..., dn).
Interpret relation, function and constant symbols of M in 2 as follows:
c =tiff(c=t) T,.
C
1. If tis abasictermand c Cthen
2. If R is an n-placed relation symbol and cl, . , n E C then
2C = R(cl,...,
cn) E Tw.
cn) iff R(cl,...,
It is now an easy induction on formula length using finitary consistency that every
[
sentence in T,, holds in 2C, so t 1=T.
1.2
Admissible Sets
Admissible sets were first introduced by Kripke [4] and Platek [6] as a generalization
of metarecursion theory. This recursion process is equivalent to definability in Gbdel's
L. We concentrate on admissible sets in terms of definability.
Since we won't be explicitly using urelements, we follow Keisler's definition of
admissibility from [3] rather than Barwise's definition in [1]. A set A is said to be
admissible if it satisfies the following conditions:
1. A is a nonempty transitive set;
2. If x E A, then the transitive closure tc(x) of x is in A;
3. (Ao-Separation.) If o(x, Yi,... , yn) is a Ao-formula and c, bl, . . ., bn E A, then
{aEcj <A,E>p[a,bl,...,bn]}EA;
4. (-Reflection.)
If (y,
. ..
, yn) is a s-formula, bl ,
<A, E>=(p[bl,...,
13
..
bn]
, bn
CA, and
E
then there is a transitive set a E A such that both
bl,..., b E a and< a, E>= o[bl,... , b].
Remember, a Ao-formula of set theory is built up from atomic formulas and their
negations using the operations
V, A, (x
E y), (3x E y).
A E-formula is built up from these operations and (3x). Similarly a [I-formula is
built up from the four operations and (Vx).
From this definition of an admissible set, we have that admissible sets A also
satisfy the following properties:
1. (Finitary
Properties.)
If a, b E A, then
a x b,aUb, anb,< a,b >,{a,b},a-b
E A.
Also, if f is a function in A, then dom(f), ran(f) E A.
2. (A-Separation.) If a E A, x C a, and x is A on A, then x E A.
3. (E-Replacement.) If f is a function such that f c A x A, f is E on A, a E A,
and aCdom(f),
then f ra
A.
4. (Definition by E-Recursion.) Let G be a binary function such that G
C
G is E on A, and for all a E A such that a C dom(G) we have that G
Then we can define a function F with domain A from G by recursion:
A x A,
a E A.
tc(x)) for all x E A,
F(x) = G(x,F
F is A on A, and F maps A into A.
These properties are proved in both Keisler [3] and Barwise [1].
Weaker forms of first-order completeness and compactness are true for countable
admissible sets. Barwise is responsible for these theorems, and the proofs can be
found in [1].
Theorem 1.3 (Barwise Completeness Theorem). Let LA be a countable admissible
fragment. Then for all p E LA, the following are equivalent:
14
1.
gop;
2. - o;
3. A says that F-V.
Thus the set of valid sentences in LA is E on A.
Theorem 1.4 (Barwise Compactness Theorem). Let LA be a countable admissible
fragment of
,,,,,. Let T be a set of LA-sentences that is El on A. If every To C T
that is an element in A has a model, then T has a model.
We use Barwise Compactness extensively throughout, mostly in the form of the
Small Sets Lemma given below in Lemma 1.5. The Small Sets Lemma is presented in
Sacks [7],and is a generalization of the following result in recursion theory: if a set
S of reals is E l and has cardinality less than 2W,then there is a hyperarithmetic real
H such that s <T H for all s E S and, moreover, an index of H can be computed
uniformly from an index for S.
Let LA be a fragment. We say a theory T is complete for LA if for each sentence
(pin LA, T k ( or T
-
but not both.
Theorem 1.5 (Small Sets Lemma [7]). Let A be a countable admissible set, and let
D(x, y) be a Ao-formula. For p, b E A, define
Sp,b= {x
x C b and D(x,p)}.
Then if Sp,b~ A, the cardinality of Sp,bis 2.
Proof. Let the language L contain the E symbol, a constant ca for all a E A, and a
constant, c distinct from the Ca. We define the AiA set of sentences Z as follows:
1. The sets of sentences
{ca E
b
if a E b for a, b E A} and {ca V
Cb
if a 0 b for a, b e A},
that is the atomic diagram of A.
2. The sentences describing that the constant c is in Sp,b but not in A, that is
(c C_Cb), D(c,
cp), and (c
Ca)
for all a E A.
15
We claim that Z is consistent by contradiction. So suppose Z is inconsistent in LA.
Then by Barwise compactness and completeness, there is some ZO E A such that
Z C Z and ZO is inconsistent. So ZO contains a subset of the atomic diagram, the
sentences (c
C Cb) and
D(c, cp), and {(c 0
Ca)
I a C ao} for some ao E A. But Z
is
inconsistent. So there is a proof in A that
(C C
Cb)and D(c, p)
c ao.
(1.1)
But then Sp,bC ao, that is Sp,b E A, a contradiction.
We now claim that for each sentence p in LA, we have that Z U b is not complete
for LA. Assume that there is some p E A such that ZUp is complete. Then for each
c). By the Z1
ca, there is a deduction Da E A from Z U p of either (a E c) or (a
admissibility of A, there is some deduction D E A that determines which elements of
Cb are in c. But then there is some e E A such that (c = ce) is deducible from Z U A,
a contradiction since c is distinct from ce.
Since ZUp is not complete for all 4 E LA, by Barwise's Theorem 8.1 in [1], there
are 2Wdistinct LA theories for models of Z. The argument extends the Henkin-style
approach given in Proposition 1.2. Each stage Tn is enlarged to cause incompatible
choices
E
for c.
Corollory 1.6. Let A be a countable admissible set. For p, b E A, we have that Sp,b
is countable iff Sp,b E A.
The following Theorem 1.7 is an extremely useful result from Sacks [7] derived
from uniformity hidden in the Small Sets Lemma 1.5.
Theorem 1.7. There is a EZF formula .F(x, y, z) such that for any countableEl admissible set A and any elements p, b, s E A, we have that:
1. If Sp,b is countable, then A
3z.F(p, b, z).
2. For all a E A, we have that A l= F(p, b, a) implies that a = Sp,b.
Proof. We find F(x, y, z) from the proof of the Small Sets Lemma 1.5. The formula
.F(x, y, z) says that there exists a wl such that wl is a subset of the atomic diagram
and there is a deduction of Equation (1.1) from wl such that
z = {s I s E wo and s C y and D(s,x)},
16
where wo is given in Equation
(1.1).
Thus, by the Small Sets Lemma 1.5, Z is inconsistent iff
Sp,b
Sp,b
is countable iff
A. So if Sp,b is countable, then A k 3zF(p, b, z). And the second part also
0
follows immediately.
To create models that have certain properties, we use the Omitting Types Theorem, an extension of the first-order analogue. The presentation given here is from
Keisler [3] and proofs are given in both Keisler [3] and Barwise [1].
Theorem 1.8 (Omitting Types Theorem). Let LA be a countable fragment of L,1 ,,
let T be a set of LA sentences, and for each n < w, let I)n(x, .., xI,) be a set of LA
formulas with at most the free variables xl,.. , x1. Suppose we have the following
assumptions:
1. T has a model;
2. For all n < w and all formulas (x1 , . . ,Xi) in LA,
if TU {(3xl,...,x 1l) (xl, .. ,xj,)} has a model, then there exists a
,E
,nsuch thatT U (3xl,. ..,xl ) A (xi,..., xl)} has a model.
Then T U {An<w (Vxl,...
, xj,)
V 4n} has a model.
The following theorem indicates an important usage of the Omitting Types Theorem 1.8, where we create a model that omits a countable admissible ordinal. The
proof is primarily extracted from the proof of a similar theorem in Keisler [3] p. 5 8 .
Theorem 1.9. Let a > w be a countable admissible ordinal, and let A be the admissible set L(a). Let Z be a El set of sentences in CA as follows:
1. The atomic diagramof L(a). That is, introduceconstants ca for eacha E L(a)
and add the set of sentences
x Ec
V x = ca, for each < a.
-Y<#
2. The axioms for El admissibility, given in Section 1.2.
Then Z has a model that is an end extension of L(a) and omits the ordinal a.
17
Proof. Firstly, Z obviously has a model, any model of Z is an end extension of L(a),
and Z is El on L(a).
Let S be a subset of a that is E on L(a) but not A on L(ca). The construction of
such a set is a simple extension from the classical recursion-theory result and can be
found in Barwise [1] p.156.
Let E(x) be the following set of formulas:
Vy(y G x
y is an ordinal);
=
c# E x, for all c c S;
c,
We now show that TU{-3x
x, for all cp E a-S.
A e(x)} has a model using the Omitting Types Theorem
1.8.
Let +(x) E LA be any formula such that TU {3x+/(x)} has a model. Consider the
set F C LA of formulas ~o(x) such that T 1=-
. Then since T is E on A, we have
that F is too.
Assume, for a contradiction, that F C e. Then S is A on A, since the sets
S
=
{f<C I (c
Ex) ErF} and a
-
S
=
{f < a (c
x) E rF}
are both E on A. These contradict the fact that S is not A on A.
Thus there exists a formula #(x)
F - O such that T U 3x(,(x)
A -O(x)) has
a model. We have now satisfied the assumptions of the Omitting Types Theorem.
Therefore, there is a model 2l that satisfies T U {-Bx A O(x)}, which means that
omits the set S.
Firstly, let a(x) be a E-definition of S in A = L(a). If 21contains an ath ordinal,
then
S-{3
<
I
= ()(c).
Since 2. is admissible, by Ao-separation, there exists an s E 2 such that
21 Vy (y e s 4=
18
a L(a ) (y
)) .
But then
2 I c, E s for
A
c
X S,
s for 3 E a
and
-
S.
But these contradict that 2 omits S, so 2l cannot contain an cath ordinal.
0
We now present an iterated bounding result for admissible sets, given in Sacks [7].
This theorem is very useful in conjunction with the Small Sets Lemma. Let B(x) be
a AO
0 formula with parameter po. We say that B(x) is d-bounded if
Va
[1(a) 4=- L[B, po; a] k 13(a)].
Thus if ad = a n L[/,3,po;a], then B(a) -= B(a).
Note that L[,3,po; a] is the result of iterating first-order definability to the /3th
stage over the transitive closure of the set po and also using the additional atomic
predicate x E a.
For all z, let HYPz be the least El-admissible set containing z, that is HYPZ
=
L(wz, tc(z)).
Theorem 1.10 (Iterated Bounding Theorem [7]). Let B(x) be a A-bounded AOformula with parameter Po. Let F(u, v) be a E1 formula with parameter Pl, and let
P = {Po,Pl}.
Suppose that for all a, the following is true:
If B(a) holds, then there is a unique p,a
such that HYP{p,,,a,} k F(a3
HYP{p,,a~}
, 6 p,3,aO).
Then we have that:
1. There exists a uniform bound 6p,3E HYPp,,}
such that for all a, we have that
if B(a) holds, then p,,,a < p,.
2. We obtain p,3 uniformly, that is p,# is determined by a partialfunction of p
and fl whose restriction to an admissible A has a uniform EA definition.
Proof. Let a be H1 {P}. First we argue that the following set of sentences is
inconsistent. The sentences describe a El-admissible
nonstandard end extension of
1
19
L(a, tc({p, 3, c,})) where there is an element a whose
p,$,a
is in the nonstandard
part.
Let Z be the
1
Pa. set of sentences:
Z1. The axioms for El admissibility.
Z2. The atomic diagram of L(a, tc({p, A, c,})), that is add constants ca to name all
of the elements a E L(a, tc({p, /, c,})), and add sentences that define how each
element is obtained from elements of lower rank.
Z3. Introduce two new constants c and c,
add the sentences to describe c =
cU L[3,po; c], and also that B3(c,) holds.
Z4. Let g(u, v, w) be a AO formula such that F(u, v) is 3wg(u, v, w). For each ca
added in (Z2), add the following sentences:
-5(cp, c, ca), for all 6 < a.
For a contradiction, assume that Z is consistent. Now assume that Z, and hence a,
is countable. Thus, by Theorem 1.9, Z has a model M that is a proper end extension
of L(a, tc({p, A, co})) but omits the ordinal a. Since a is omitted, L(a, tc({p, /, c,}))
is El admissible. Then L(a, tc({p, /3,c})) = HYPpac~,. But from Z, we have that
HYPfp,,c,
[ -F(c3, c 3) for all 6 < a.
This contradicts the supposition in the theorem that
HYPjp,,c)' t
,),
(c~,Cap
since Jpgc E HYP{p,,,cD}. Therefore countable Z is inconsistent.
To remove the assumption that Z is countable, use Cohen's forcing to generically
extend the universe V to a V' where Z is countable. Then by the absoluteness of
provability in £w, we have that Z is inconsistent in V' implies that Z is inconsistent
in V.
We now use a routine Barwise Compactness argument to settle the rest of the
argument. Since Z is inconsistent and EHYP{pA},by the Barwise Compactness Theorem 1.4 and Completeness Theorem 1.3, there is an inconsistent subset W C Z such
that W E HYP{p,#}. Then W contains the following sentences:
20
W1. Various axioms of El admissiblity from (Z1), and the setup of c and c from
(Z3).
W2. A subset Ao of the atomic diagram of L(a, tc({p, 3, c,3 })) from (Z2) such that
A o E HYP{p,#).
W3. For some J& < a, the set of sentences -(ce,
L(61, tc({p, 3,
45, ca,) for
<
and a
})).
Thus, there is a deduction D E HYP{p,} of
(W1) A (W2)
' V{F(cS, c) I < 51}.
Let po be the least p < a such that there is a deduction D E L(p, tc({p, /})) and let
{p,/} be the least such
(1
in any of the Ds in L(po, tc({p,3}).
That is, we minimize
p then (1. Then we have that for any c such that B(c) holds, p,,c < p,3. Thus, we
have proved the first conclusion.
For the second conclusion, the function that gives 3{p,} from p and /3 is uniformly
EA for any admissible A containing p and
because HYP{p,} is a uniformly EAdefinable substructure in A.
1.3
Scott Analysis
In [8], Scott provides invariants that classify countable structures up to isomorphism.
The invariants are single sentences of 4W1,, called Scott Sentences. The Scott Hierarchy, presented below, capitalizes on these Scott Sentences and provides a hierarchy
of theories (and languages) to characterize countable structures.
Let 9. be an arbitrary countable structure with the underlying first-order language
L. We follow Sacks' El recursion in 1[7]
to define the hierarchy and Scott Rank of %.
Let
0o-
*
C = U{L
Ia
< A} for A a limit ordinal;
* T a = the complete theory of t in £C;
*
1
p(i)
= the least fragment L+ such that
is a nonprincipal n-type of
A&{q(x)
I (Y)
T a'
that is realized in 2[, then the formula
E p(Y)} is a member of L+.
21
C L+, and for each n > 0, if
We now list some basic properties of the Scott Hierarchy in Lemma 1.11, given in
Sacks [7]. As such, we need to define a homogeneous model. Following the notation
1 and B are models for £, X C 11, and
in Keisler [3], if LA is a fragment of £,,
f is a function on X into 1]1, then
(2, a)a(x
means that for every formula V (X1 ,..
A
, xn) E LA and al,...,
·, a]
t[al,..
(, f(a))XEX
IA
a, E X, we have that
9 = [(al),..., f(an)].
A model 21 is said to be a LA-homogeneous model if for every set X C_1AI such that
and every function f from X into 1l,
the cardinality of X is less than or equal to 12t1
we have the following: if
(A, a)aEx -=CA (21,f(a))aEx
then for all b
1(1 there exists a f(b) E
such that
(21,a)aEXu{b} -=IA (,
f(a))Exu{b}.
Lemma 1.11 (Properties of the Scott Hierarchy). Let L be a first-order language,
and let 21 be a countable £ structure. Then the following properties hold:
1. If 21
78,
then ^ = L
and T5 = T~~ for all 6.
2. There is some 6 < w 1 such that all the types of T~ realized in 21 are principal.
3. Let d% be the least such 6 such that every distinction in L, , between elements
in 21is made by a formula in L. Then 21 is an atomic model of · ,
4. Let the Scott Rank of 21,sr(2), be the least a such that 21 is the atomic model
of TV. Then if 21! 8, we have that sr(21) = sr(B).
5.21 is a homogeneous model of T,
Proof.
and hence sr(2) < wA + 1.
1. By induction.
2. Fix y and suppose there is a nonprincipal type Py+I in Ta+1 . Let pa be the
restriction of Py+
1 to L£. Then since Py+l is nonprincipal, there is some formula
22
'(x)
E £L+1 such that the formulas
3 [Ap()
and 3
p()]
A/
Apa()
2=
A
(d)
W=
and
()]
A
in A such that
and
. Thus, there are tuples
are both in T
[Ap()
Ap (6 )
A
(b).
So no formulas in L distinguish
and b, otherwise P7 +i would be principal,
but a distinction is made in LI.
Since ( is countable, only countably many
exists.
such distinctions can be made, hence
3. By (2), we have that d exists. Any nonprincipal types in Td, realized in
extend to principal types at d%+ 1, since no more distinctions are made.
4. The Scott Rank of 2. exists from (3), and the result follows from (1).
5. First note that from the inductive definitions of LI and T25, we have that both
are EL(I'
A).
< wL(wi
functions of
Moreover, L
and T8a are developed
in increasing complexity and so, just like in the definition of Gbdel's L, their
graphs are
A L(W'')
41w
)
Let a and b be n tuples in
(n + 1)-type in TI.
IAI,let p(x)
be a n-type in T',
and let q(y, y) be an
Suppose that p(Y) C q(X, y) and that
2 k p(d) A p(b) A 3yq(a, y).
Let c E I21]be an element such that
q(d, c). For ( to be homogeneous,
we need to find a d ( 11 such that 2( k q(b, d). We prove such a d exists by
contradiction. So suppose no such d exists. First let q(Y, y) be the restriction
of q(f, y) to £'. Then q6 (x, y) is the set of all formulas that are true for the
tuple (, c) in T,. Thus,
<Y
. 155
{qa(, y) | 6 < we} is
For each e E
a
E ~( ~ ' ~
I12l,
there
is a
a
<
such that 2t
-,q(b, e). Moreover, there is
function from e to its qs. Thus, by the El admissibility of L(wl, A),
23
there is a d6 <
such that
A
t Vy-qj_ (b, y).
But q (, y) E p(Y) and so k1=
Vy-q8 (, y), a contradiction since c exists.
Thus, a d E 1%Iexists such that
~ q(b, d).
[
24
Chapter 2
Constructing the Raw Tree
Hierarchy
2.1 Weakly-Scattered Theories and the Raw Tree
Hierarchy
Weakly-scattered theories are a generalization of Morley's notion of a scattered theory. Morley introduced scattered theories in his ground-breaking paper [5] to give a
positive result towards Vaught's Conjecture.
We first define scattered theories using Sacks' definition from [7], which is equivalent to Morley's original definition.
Definition 2.1. Let C be a countable first-order language, and let
fragment of £j,.
o be a countable
Fix T C Co, a theory with a model. Let L' be any arbitrary
countable fragment of 4,
extending C, and let T' C
' be any finitarily-consistent,
w-complete theory extending T. We say that T is scattered if the following hold:
1. For all n and all T', the set of all n-types over T', denoted SnT', is countable.
2. For all
', the set {T'
T' C L'} is countable.
We say that T is weakly scattered if only (1) holds.
We now introduce a tree hierarchy to enumerate all the models of a weaklyscattered theory. This notion is introduced in Sacks' [7], and extends a similar tree
hierarchy for scattered theories. Since we're only interested in weakly-scattered theories here, we do not develop the scattered version first. Needless to say, the scattered
25
tree hierarchy is considerably more constructive, and can be developed inside L(W1,T).
However, surprisingly constructive results are obtained for the weakly-scattered case.
Definition
2.2. Let £ be a countable first-order theory, and let Co be a countable
fragment of ,,.
Let T C o be a weakly-scattered theory with a model. Following
Sacks' [7] notation, define
- 1
-
6
if 6 is a successor ordinal,
otherwise.
=
We define the raw hierarchy for T, denoted 7R-(T), as follows.
Level 0. Include every To such that T C_ To and To is a finitarily-consistent,
w-complete theory of 4o. We define Co(To-) to be
o.
Level 6+1. We first define C5+1 (T). Assume that T5 extends a unique predecessor
T3_ on level 6- and that C4(Td_)is countable. If T is an atomic theory, then £÷+1(Tj)
is undefined and T5 has no extensions on level 6 + 1. Otherwise, let £+1 (T5) be the
least fragment of £,,~
extending C(T_) and containing the conjunctions
A{'(
Ep(X)}
E(*)
) I
for each nonprincipal type p(Y) of To. Note that since T is weakly scattered, T is
too, and so
5+1(T&) is countable.
Now for level 6 + 1 of the tree, include every TS+1
that extends T5 and is a finitarily-consistent, w-complete theory of £+1 (Ti).
Limit Level A. We include the theory T on level A if there exists a sequence
< T I 6 < A > such that the following hold:
1. T is a theory on level 6;
2. T
C
T,52 for 61 < 62 <
3. T, = U{T& I
A;
and
<}
Then £X(TX) is U{C(T&_)
6 < A}.
We then define the raw tree rank of a model 21 as
rtr(2) = (least 6) [2 is the atomic model of some To].
It is clear from the definitions that 2l is a countable model of T if and only if there
exists some countable 6 such that 21 is the atomic model of T5.
26
For a given model
of T, we can analyze its path through the raw hierarchy.
Thus, following Sacks' notation [7], we define the raw tree analysis of 2 to be the
following:
*
T(O,) =
o;
* T(O, 21) = the Lo-theory of 21;
*
CT(J+1,)= L5+1(T(6, 21)),as given in Definition 2.2;
* T(6 + 1,2) = the £T(8+l, )-theory of 21;
< A} for A a limit;
*£T(A,2) = U{LT(6,)
* T(A, 2) = U{T(6,2) I a < A}.
We now list two propositions from Sacks [7] that relate the raw tree rank to Scott
rank, given in Section 1.3.
Proposition
2.3. Let 21 be a model of a theory T. Then rtr(2) < sr(2c).
Proof. Clearly by a routine induction on 6, we have that La C £T(,a).
Recall that
£L is defined for Scott analysis in Section 1.3, and LT(6,) is defined above. Thus
T%,(%
) C T(sr(2),9).
Since 21is an atomic model of T(a
it is a homogeneous model of T%(
show that 2 is a homogeneous model of T(sr(),
of
of formulas qo(Y)
We first
2) by induction on the complexity
£T(sr(,.
That is, we show by induction that for all formulas
p(x) (with n free variables) of
£T(sr(a),a)
and all n-tuples
(21,a)-=,Cu)
and 2 t
.
(AX
a and b in 12[, if
b)
(d) then 2 k p(b).
Clearly the only worry is when ¢(g) is of the form 3yp(Y, y). Thus, for some
12, we have that 2Lk= (d,c). Since 2 is T'
homogeneous, there exists a
c G
d E 1211such that
(2 , c)
(2
b1,
d).
Thus, by inductive hypothesis, we have that 2 k '¢(b, d), and so 2 k= (b) as desired.
If p(Y) is an atomic formula for T( ) realized in 2, then since Ts( ) C T(sr(2), 2)
and 21is a homogeneous model of T(sr(21), 21), we have that () determines a unique
27
type for T(sr(2p),
) and thus remains an atom in the larger language. Thus 21 is an
atomic model of T(sr(9A), 21), and so rtr(2t) < sr(2).
Proposition 2.4. Let 21
l
T, and let a < w1 such that L(a, < T, 2t >) s E
admissible. Then
rtr(2t)<a ==- sr(2t)< a.
Proof. We first show by a EL(a,<T,%>)
recursion that for
1
/
< a < wl, we have that
LT(,,) and T(3, 2t) are in L(a, < T, 2t >).
Only the successor stage is nontrivial. Assume that it holds for 3. We use the
Small Sets Lemma 1.5 to show that
£T(+1,.)
E L(a, < T, 21 >). The statement "q
is a nonprincipal type of T(, 2t)" is a A0 statement with parameter p = T(,3, 2Q),
corresponding to the formula D(x,p) in the Small Sets Lemma. Thus the set of
nonprincipal types of T(,3, 2t) is the unique set in L(a, < T, 2L>) that satisfies the E1
formula .F in Theorem 1.7 with p and b both equal to T(,3, 2t). Therefore LT(B+1,t)C
L(a, < T, t >).
Now T(/,3 + 1,21) is the set {
E LT(#+l,t) I 2
o}, SO T(/3 + 1, p1) is El with
parameters from L(a, < T, 21 >). We form the E1 function from each formula
(
to
the least level in L(a, < T, 2 >) in which it's constructed. Because L(a, < T, 2t >)
is E1 admissible, this set is bounded, and thus T(/3 + 1, ) E L(a, < T, 21 >).
We now prove the proposition by contradiction. Suppose that rtr(2%) < a but
sr(21) > a. Then by the above E1 recursion, the set of all distinctions D between
n-tuples of 11 made by formulas in LT(rt(%),) is in L(a, < T, 2 >). But then there
is an unbounded E1 map, similar to the one above, from D into a, which contradicts
the El admissibility of L(a, < T, 2 >).
2.2
El
Effective Recovery of the Raw Hierarchy
Since a weakly-scattered theory could potentially have continuum many extensions
on a given level of the raw hierarchy, it is not generally possible for the raw hierarchy
of a given theory T to exist inside L(a, T) when a < w, and L(a, T) is E1 admissible.
However, it is surprising how much information on the raw hierarchy can be expressed
inside L(a, T).
We now show a result from Sacks' paper [7], which is improved upon in Section
2.3 to give a nice property of the raw hierarchy. We then use these results to prove
28
an improved version of a bounding result by Sacks [7] for weakly-scattered theories
in Chapter 3.
Theorem
2.5 (Effective Recovery Process [7]). Let £ be a countable first-order lan-
guage, and let Lo be a countable fragment of L,,,.
Let T C o be a weakly-scattered
theory. Assume, for convenience, that Lo and L are effectivelyrecoverablefrom T.
Let a be an ordinal such that a _<wl and L(a, T) is El admissible. Let < a
and define A5 to be the set of all theories T5 on level of 1Z1-I(T). Then there
exists a A such that A5 is defined by a -bounded Aoz F formula, denoted rA 5 1, and
rain L(a,T).
Proof. We find ordinals ph such that LC(Tb_) is constructible
T
from T 5 _ using p6 for all
E A8_. Both rA 5 1 and p5 are constructed simultaneously by a EL(aT) recursion
that is uniform in a.
Stage 0. Let £o(To-) be Lo. Then Ao is the set of all finitarily-consistent, wcomplete Lo-theories extending T. Clearly Ao is AZF definable with 3 = 0 and
parameter T. (Note that
o is recoverable from T by assumption.)
Stage 6 + 1. Assume that we have the sequences
<p I y <>
and <rA-I
y<6>
(2.1)
where £,(T-_) is definable over L[py, T; Ty_] and rAyq is a -bounded A z F definition
for An.
Let T5 be an arbitrary theory on level 6, that is T5 E As. We use the sequences
given in Equation (2.1) to effectively reconstruct the unique predecessor T6 E As5such that T5 _ C T8.
We now show how to effectively construct L£+1(T5 ). Let ST 5 denote the set of all
n-types of T5 in C5 (T6_) for n
>
0. Since T is weakly scattered, by the Small Sets
Lemma 1.5, we have that
ST 5 E L(wT6,Ts),
using a method similar to the proof in Proposition 2.4. Let 'yT,be the least -ysuch that
ST
E L(y, T6). Using the uniformity in the Small Sets Lemma, given in Theorem
1.7, we have that yT, is a uniform El function of T6. Thus, by the Iterated Bounding
Theorem 1.10, there exists a 5 < a such that for all Ts E A5 we have:
1. ^T, < 'Y;
29
2. ST
E L(vy, T6 );
3. y is a uniform E1 function of 6 using the parameters from A6 1 and the parameters from the uniform E1 definition of
yT6 .
We now assemble a special set of first-order ZF definitions on level 'y of L(o, T)
that construct all types in all ST6 , given a theory T3. First, let
{PI J E}
(2.2)
be the set of all first-order ZF definitions over L(7y,T) for all y <
with parameter
T6. Let pj(T 6 ) or p'T"represent the set constructed from the definition pT when the
set To is substituted for the parameter T. Thus pj (T)
=
pE · L(y, To).
Let W denote the natural well ordering of the set in Equation 2.2, where each
definition pT is first ordered by the level y <
in which it is constructed, and then
by its Gbdel number e < . Let d6 (T6 ) denote the default type for To and be defined
by the following:
j(T 6 ) = (W-least j) [pj(T6) is an n-type of To for some n];
d6 (T3 ) = pj (T6)(T6)We now tweak the first-order ZF definitions of the types. Let pT be the first-order
ZF definition with parameter T6 defined by:
Tp
p,(T 6 )
d6 (T,)
Let P6 = fp
if pj(T 6 ) is an n-type of T5 for some n;
the default type of T6, otherwise.
j · J}, assembledon level y + 1 in L(a, T). Note that:
1. For all T0 · A5 and all p(g) G ST, there exists a j E J0 such that pT defines
the type p(S) on level iy5+ 1 of L(a, T).
2. For all T0 E A 6 and all j E J,
we have that p
E ST 6 .
3. It is possible that for some T6 E A6 and some j, k E 3a that j :~ k but pT = pT.
Let P6+1be large enough to develop the sequence < p I y < 6 > used to recover T6 from T6 , and also large enough to include the ordinal y to assemble P 6 .
30
We define C+ (T) as follows. Include the definition of £5(Tj_), and for all
P5 add the conjunction of all formulas in p.
E
Then close under the finitary operations
given in Section 1.1 needed to construct a fragment of I~,,.
To determine rA 6 +1 -, note that A6 is the set of all x such that, using the effective
recovery process so far including the sequences < p I y < 6 + 1 > and < rA.- I -y <
6 >, we have that x is a finitarily-consistent, w-complete theory of L6 +1 (T7). Therefore
A5+1 is a pi+-bounded AZF-definable set, and rA 6 +1- c L(a, T).
Limit stage A. Assume again that we have the sequences
<p
y < A >, and < rAs[y
< A>
where £y(T_) is definable over L[p,, T; Ty_] and rAy71is a -bounded AIz F definition
for A7 . To find TA, effectively recover the sequence < T I -y < A > such that
T = U{T 7I ty < A}. Then let C = U{1C(T 7_)
2.3
< A}.
0
Predecessor Function for the Raw Hierachy
The aim of this section is to create a predecessor function for type definitions. In
order to do this, we must alter the type definitions given in the Effective Recovery
Process 2.5, and as such, they are not necessarily defined for all theories on the type's
level anymore. However, these new type definitions will be enough to prove the result
in the next chapter.
We add to the EL(,T) definitions of pa and rA-i in the Effective Recovery Process
2.5 to recursively define the predecessor function f[p, 6] = {f(p, 7Y)I y < 6} for type
definitions p at Stage 6. Thus, the predecessor function is a partial function such that
if p defines an actual type for a given theory on level 6 then f(p, y) defines an actual
type on level 7 <
that is an actual subtype of the type defined by p.
Note that each Stage 6 in our construction of type definitions and the predecessor
function corresponds to one level higher in the effective recovery process in Theorem
2.5. That is, to construct £ 6+1 (T6) and T 6+1, we need to construct ST6 .
2.3.1
Stage 0
Stage 0. Develop Po = {Pjo I jo
Jo} as in Equation (2.2) of the Successor Stage of
the Effective Recovery Process 2.5. So each pjo is the definition of a type in STo for
at least one (and not necessarily all) To E Ao.
31
Then let
f(pjo, O) = pjo for all pjo e Po
0.
2.3.2
Successor Stage
Stage 6 + 1. Assume we have {Pi i < 6 + 1} and {rA. 1 I i <
+ 1}. Using these
sequences, we reconstruct the set of definitions from the previous stage 5:
P6= {Pk I k E K}.
Here KCis the index set of type definitions developed here at level 6, and J is reserved
for the original type definitions developed in Effective Recovery Process 2.5.
Also when defining Lf+1 (T), we want to use the type definitions uniformly to
avoid the language having a domain too. To do so, we alter the type definitions
slightly. Let
cp?
|if
x =x
pj (T5 ) is an n-type of T;
otherwise.
Then
4+1(Ta) = 4(Ta-) U {Arj(Ta) I j e Ka}
and closed under finitary operations.
As in the Effective Recovery Process, we assemble the set Q6+1 = {
+II j E
J6+1 } of first-order definitions at level y+l of L(a,T) using the Small Sets Lemma
and the Iterated Bounding Theorem. Also, Qa+1 has a natural well ordering W+ 1
(as given in the Effective Recovery Process), which is also definable at level Y5+
1.
In what follows, we note that given a theory Ts+1 on level 6+ 1, we can effectively
reconstruct its immediate predecessor Tj using the inductive hypotheses. Now for a
and for each ja+l C $+1,
given fixed k E KAC5
I
ks+
let
E Q6+1 and also assert
+1 Cloasr
be p 6+
kbjbPl
Pjj+,(T6+1) is an m-type of T+1 and
pji+, (Ts+2) D Pk, (Ta), an m-type of Ty.
Let P+l,ks
be the collection all such definitions extending the fixed definition Pkk,
32
and let
P6 +l =- lPXj+,
and ij+ 1 E Js+,}.
| k E KC5
Thus, P6+1 is the union of all such P6+1,k, as these indices range over KC.
We have to check that P&+1is in L(a, T). First, for a given k6 and T3+1, we show
that
P5+l1,k (T+1)
E
L(-y,T; < k6 , T+ 1 >) for some y < a.
But this is true from El-replacement since we can effectively construct
T, T 6+1 , Q6+1, P5
L(wvT'+,T+l)
using the inductive hypotheses.
Let
-y(ks, T+l) = least y [PS+l,k,(k6,T6+l)C L(y,< k6 ,T6 +1 >)].
By Theorem 1.7, we have that y(k,T6 +l) as a function of k6 and T5+1 is uniformly
El. So by the Iterated Bounding Theorem 1.10, there is a uniform bound 'Y+1 of all
these ys, and -y7+1has a uniform El definition from 3 + 1 and its parameters.
Thus, we have that UK P,+l,k, = P5+1
·
L(7y+1 + 1,T), and y+1 < a. Also,
since we've just added repetitions of definitions, we still have that
1. For each T+l E A,+l and p(x) · ST 6 +1 , there is a k E K,+i such that Pk(T+i)
definesp(x) at level y5+1+ 1 of L(a, T), and
2. For all k E K&+
1 , we have that pk(T6+l) · ST&+
1 for at least one T+ 1 · A6 +
1.
Since we've kept track of indices of the immediate predecessors, simply define
f,~~
f °kT'J5+~')-
~Pk
)+
y5
8
f({Tpn
) for
for?y=6
(k,)
for ? < .
Finally let ps+l < a be just large enough to develop the sequence < pi i < 6 >
and the ordinal y6+ 1 + 1 needed to construct P&+1 .
2.3.3
Limit Stage
We introduce a rank, called Schmutz Rank, that is based on CB Rank. We use certain
isolating formulas derived from Schmutz Rank to index the type definitions at the
Ath stage of the construction, for A a limit ordinal.
33
Define Schmutz Rank' for a type p E ST, as follows:
1. SR(p) = 0 if there exists a formula OE £x that isolates p in ST>.
2. SR(p) =
1 if
there exists a formula o E L that isolates p in ST, - {q E
ST, I SR(q) < 1}. Thus, p is the unique type containing o in STx-{q
SR(q) <
1}.
We define the type definitions at level A by recursion. At each stage
, we repeat
the type definitions p and assert that p is a type of Schmutz Rank ,3 and index it by
its isolating formula.
As in the Effective Recovery Process 2.5, we develop the initial set of type definitions {pj(TA) I j E JI.
Stage 0. Define the type definitions of Schmutz Rank 0:
PA,,o= {pP,o I o,o is type defn pj(T,) for some j E J, and also asserts
E C2(T\)and isolates pj(T,) in ST,}.
Then for Pw,o
P\,o, since o E L,(T,),
there is some
> 6. (For
Thus, o isolates p r T for all
a
< A such that
E Ly, we have that
E L£(T).
Ep T
(a-
G T .)
The predecessors of Pp,Oare then
)ET
(
o
(a7
=
if M> 6,
P,1(TY
f(Pj(, TV,)
if
< 6,
where
j(V, Ty) = (least j in the sense of Wy)[p E p (To) ( Pv)].
We now show that the set of rank 0 types is in L[a, T; T,] uniformly for use in the
next step.
Let Rx,o denote the set of rank 0 types of Tx on level A, that is,
RA,o= {x I x C_T and 3o E CVE
4[O
E x 4 ( -- V) E Tx]
L
and x is complete, finitarily consistent, and w-complete}.
By the Small Sets Lemma 1.5, we have that Rx,o E L(wVT,TA). But then by Iterated
'The name was coined by Prof. Sacks.
34
Bounding 1.10, there exists a y,0 < wA such that RA,oE L(y,o, To) for all T E A>.
Alter yA,0so that it's large enough to develop all of Stage 0 (but still less than a).
Stage / >
. Assume the recursion has produced the sequence of ordinals
6 < 3}, and construct the set of type definitions for T, of rank <
{,
, that
is
RA,< = U R ,
E L(a, T)
We first develop an intermediate set of type definitions of rank 3, denoted Qx,,
where
Q,3 = p,
I p,
is a type defn pj (T,) for some j E J, and also asserts
W E C(Tx) and p,
Vq E STA[o E q -
E ST) - RA,< and
(q - p v q E R,<
)]}.
In other words, pe, is the definition of the unique type isolated by T in ST - RA,<O.
By the Small Sets Lemma and Iterated Bounding, Q,
E L(A, 1 , Tx) for some y,1a<
wT for all T, E A,.
We now look at a given type definition pyr (abbreviated as p for now) and
its isolating formula. We claim that there exists a bound
q
< A such that all types
ST, containing o that have rank less than/3 have an isolating formula below
stage 6.
We find it using Barwise Compactness. Let Z be the set of axioms with parameters
p, p, and /3all in L(wTA,Tx):
1. q E ST>; q
p;
E q;
2. b does not isolate q in ST - RA,<y, for all O C 5, all
< A, and all
y </ 3;
3. the structure of L(w1 x , T) (so we have an end extension).
Then Z is El in L(wTA,TA) thanks to the ordinals we've constructed along the way
in the recursion.
If Z is consistent, then p would have two extensions in ST
p is the unique such, so Z is inconsistent.
of rank > 3. But
By Barwise Compactness, there is a
zo
L(wT, TN) and z C Z that is inconsistent. Let 6 < wTA bound the ordinals
mentioned in zo of axiom type (2). Then the axioms (3) and (1) in zo imply there
exists an isolating formula in £5 for any such q. Let p be the E1 function (uniform
in TA) that takes (p, 'p,/) to 6.
35
We now have enough information to define the set of type definitions of rank /3:
Pix,/ ={pp,
E Qx,, and also asserts o is
for a =
A(p~,a
r T 6) E £C+(T5 )
1
(p,,, o,/3)}.
This definition is effective since we can reconstruct all the T ( < A) from Tx and/~
is bounded E1.
We claim that such a
(given in Px,) always isolates pa, in ST>. If not, then
it splits to say p (= pa,/) and q E ST,.
We then have that q is an extension of
V and different to p, so it must have Schmutz Rank < . Thus q is itself isolated
p r T6
(in ST - RA,,, a < ) by a formula O E La by construction. But (-'p)
otherwise p = q, and } = A(p' T)
q would mean that O is inconsistent with q. A
contradiction.
As in Stage 0, the predecessors of p,3 E Px,, are then
f (PW,,ls, =
'r(yPj(,,T?) if > ,
Pj (;o,
T'
(P3(,., 7) if < ,
where
j(o, T) = (least j in the sense of Wy)[;o E pj( T ,) ( Pa)].
To complete Stage
, we just need to include the set of types of rank 3 for the
following stages in the recursion. Let
R, = {x x C T x and x ¢ Rx,<3 and
3O E
xVq E STX[p E q -
(q = x V q E R,<3),
and x is complete, finitarily consistent, and w-complete}.
Since the parameters 4, ST), and RA,<a are all in L(wTA,T A), then by the Small Sets
Lemma, R,3 E L(wT, Tx). And then by Iterated Bounding, there is a y < wT\ such
that R,
E L(-y, T ax) for
all Tx E AX. Expand
,
to include 7y,if necessary.
We now have to show that this recursion is bounded within L(a, T) so that we can
construct Stage A + 1, and continue the Effective Recovery Process. First note that
the definition of Schmutz Rank is E1, and the domain of the rank is ST, E L(a, Tx).
36
Therefore, by El-replacement, the range is also bounded in L(a, Tx). Let
YTA =
(least y)[Vq E STA(SR(q)
< )].
So by Iterated Bounding, there is a y that bounds all the yTA,and has a uniform
E1 definition. We only need recurse through the Schmutz Ranks less than Ay\.This
recursion is El at worst with bounded input, and so let p < a be the least such
ordinal such that L[px, Lo; TA] constructs it all.
37
38
Chapter 3
A Bound for Weakly-Scattered
Theories
Partial Domains
3.1
We now present the bounding theorem from Sacks [7] without the assumption of his
predecessor property. With the definitions altered correctly so that type definitions
can have partial domains and include the predecessor function from Section 2.3, the
proof goes through in nearly exactly the same way. We include the whole proof from
Sacks [7] here, however, for the sake of completeness.
Let L be a countable first-order language, and let Co be a countable fragment of
£,,
.
Let T C Co be a weakly-scattered theory with a model. Finally, let L(ca,T)
be El-admissble.
In the Effective Recovery Process, develop the improved set of type definitions P6
at level y6 of L(a, T), as given in the Predecessor Function Section 2.3. So
Pa-{p
Ij
6}
for the improved index set J.
We define a A
(a
T)
predicate that determines the domain of a type definition,
that is, whether a definition is an actual type for a given theory on a given level of
the hierarchy.
dom(p~, T,
) iff p (T) C ST5 and T5 is a theory on level J.
39
This definition is clearly AL(aT)
from the Effective Recovery Process and the Prede'-1
cessor Function Section 2.3.
We now define a
of 7?t(T).
iL(aT)
set of sentences Ba whose models have a node on level a
The set Ba consists of the following sentences:
1. T C To and To is an w-complete, finitarily-consistent theory of 4o.
2. T C TS+l and T+l is an w-complete, finitarily-consistent theory of C5+1(T6)
for all
< a.
3. T = U{T I
< A} for all limit ordinals A < a.
4. For all 6 < ca,we have that T5 has a nonprincipal n-type for some n.
The definition of Ba is the same as Sacks' version in [7] because our language is
not dependent on the type domains. Note that Ba is AL(aT) because we construct
£5(T 3 _) via the ordinal p5, as defined in the EL(aT) recursion in Section 2.3.
Using the improved type definitions and index Pa and J
that p is on level
in Section 2.3, we say
if
V -pi'
iEJ8
The crucial definition to be changed is splitting types.
We define a split <
p, r, r' >72+1 at level 6 to be the sentence
dom(p, T,
)
dom(r, T+ 1, 6 + 1) A
dom(r', T+,
+ 1) A r
r' A r' and r extend p.
We then define < p, r, r' >X+1 to be a k-split if < p, r, r' >7r+ splits at level 6 and
p(T3 ) has arity k. Let K be a set of k-splits. Then we say that
K is unbounded if Vf, < a 36 > B [K has a k-split on level 61.
Let p
be some type definition at stage 6, and let f be a EL(a' T) predecessor function,
as given in Section 2.3. We say that pi is K-unbounded if the set of all y such that
B<p,r,r' > [<p,r,r'>EKApisonlevelyAf(p, 6)=p ]
a T then
is unbounded
in
. Note
Note that
that if
then since
is E,L(a,T) and
is unbounded
in a~.
if K
K is
is AL(,T),
since ff is
adpisop is on
40
level y" is a bounded sentence in L(a, T), we have that K-unboundedness is a
11L(aT)
property.
In order to continue, we need to define what we mean by proofs with these definitions and constants for theories. When we say B, ]-F
, we mean for a given
set of theories T that satisfy Ba, the formula F(Ty), constructed from F and our
particular Ty, is true via a finitary proof.
We say that the effective k-splitting hypothesis hold for T at a if there exists an
set K of k-splits such that Ba, and K are consistent if Bo, is.
unbounded
AL(aT)
Theorem
3.1. Let L(a, T) be a countable E 2 -admissible set. Let T be a weakly-
scattered theory such that for each /3 < a, we have that T has a model of Scott rank
at least /3. If there exists a k such that the effectivek-splitting hypothesisholdsfor T
at a, then T has a countable model 2 such that both
WI= a andsr(2) = a + 1.
Proof. We first prove that B, is consistent. Using Barwise's Compactness Theorem
1.4, T has a model QAsuch that L(a, < T, 2i >) is E1 admissible and sr(2) > a.
Therefore by Proposition 2.4, we have that rtr(2t) > a, and so B, is consistent.
Since we're assuming the effective k-splitting hypothesis holds, let K be an unbounded
AL(aT)
set of k splits. We construct a model of B, U K in a Henkin-style
argument so that:
1. T , has a nonprincipal type q, on level a; and
2. L[a, T; T,, qa] is E1 admissible. Note that in this structure, we have atomic
predicates for the sets {Ti j /3 < a} and
q#i /3 < a} but not T and q,
themselves.
Once we've created this model, we use a type-omitting argument to show that T has
realizing q,, such that wa = a.
a model
For our construction, let Sn be the set of sentences chosen by the end of stage n.
By the end of the entire construction, we decide all sentences of rank less than a in
the
AL(aT)
language of L[a, T; Ta, qa], denoted I,,T.
Stage 0. We construct the set of sentences So. First add B, and K to So. Then
add to So the following sentences:
1. the sentences of £a,T that construct the elements of L[a, T; Ta, q,] from those
of lower rank;
41
j E J[pj = q A dom(pj, T, /3)],for all /3 < a; that is, q(Tp) is a real type
2.
for T# on level /3;
3. the type qy extends q,3, for all /3 <
<a;
p(TO), for all p that are K-bounded type definitions and for all /3< a.
4. q
We mentioned above that K-unboundedness is a I L
( T)
property, and thus S is
EL(a,T)
2
We now show that So is consistent. Since B,, U K is consistent and definable over
L[a, T; T,], let 9A be a model of Ba UK that also specifies the structure of L[a, T; T].
Fix
T
< a. We show that 9 can be interpreted as a model of the subset of So
that only mentions q for /3 < r. And thus by compactness, So has a model.
Note that for all y < a, there is a K-unbounded type on level y. For if not, then
for each type definition pj on level ? there is a least /3j such that for all > /3j, there
is no such extension of pj on level in K. But then /3j is a EL(' T) function of j and
thus bounded by /3 < a, a contradiction.
We now choose a type definition p, on level r that is K-unbounded. Let
U, = s 3t3t'[< s,t,t' >E K] and f(s, r) =p,}, and
U = s s
U, and f (s,7y)= r}, for
< -r.
So U, is the set of all extensions of p, in K, and U is the set of all extensions of p,
in K that also extend r on level y.
Also note that for all y < , there is a U,-bounded type definition r on level Y.
For if not, then U is bounded for every r on level y. But clearly
U = U{U7 I r on level y}.
Hence by a similar E2 -admissibility argument to above, U is bounded, and thus p,
is bounded, a contradiction.
For each y < -, choose a U,-unbounded r on level y. We claim that
B,, U K F r is extended by p.
Let s
U2~ and assume B, U K. Then s extends p, = f(s, r) and s extends
r = f(s, y). So Pr extends ry. Moreover, continuing this argument, we have that for
42
'YI< 72 < ,
Bo U K F r
is extended by ry2 .
We have now shown that 93 can be interpreted as a model of the sentences of So
that only mention q for 7 < T by letting q be r 7 . And therefore by compactness,
So is consistent.
Stage n + 1. Assume that Sn is consistent and Er.(O'T)definable. There are two
nontrivial cases for adding sentences to Sn+xCase 1. If V is V{ji
i E 27}and SnU{op} is consistent, then let Sn+l be SnU{bji}
for some i E I such that Sn,,U {Pi} is consistent.
Case 2. If D(x, y) is a Ao
0 formula in La,T, we establish that Ao bounding holds
for D(x, y), and thus El replacement holds too. This shows that D(x, y) does not
violate El admissiblity for L[a, T; T,,, qo].
Let p < a be an ordinal such that D(x, y) denotes a possibly many-valued Ao
function d(x) from p into a. We wish to show that the range of d is bounded.
For each
< a, let H3 = {-'D(, 7y)
< a}. Then if Sn U He, is consistent for
some ' < a, then let Sn+1 = S U He,. When this case occurs, D(x, y) is not a
candidate for bounding. If this does not happen, then for each 6 < a we have that
Sn F V{D(6, y) I
< a}.
Thus, by Barwise Compactness 1.4 (applied to sentences of the form Ha), there is an
ordinal c(a) such that
S,, F v{D(,y)
17
I <
c(a)}.
From the uniform nature of Barwise Compactness, c(a) is a EL(,rT) function of 6.
Then let c = sup{c(6)
< p}. Then we have that c < a and d(6) < c for all a < p,
and thus D(x, y) satisfies AO
0 bounding. Let Sn+, = Sn.
Let S = U{Sn n < w}. Then S is a consistent set of sentences that specifies the
structure of L[a, T; T, q] by Case 1, and specifies that o is a nonprincipal type in
To by the argument in So, and also determines that L[a, T; To,,q,,] is El-admissible
by Case 2.
We now use an omitting types argument to find the model B for the theorem.
Let Z be the following EL(o,T)set of sentences:
1. The set of sentences S defined above.
2. The axioms of El admissibility.
43
3. The ordinal d > f for all
< a and d does not occur in (1).
4. The theory Td is a theory on level d of the RRH(T), and B is a countable atomic
model of Td. Also, we have that T,_ C Td.
5. For each atom p(Y) E Ta, add the sentence saying that o(Y) is an atom in Td.
First note that Z is consistent by the above construction. Therefore by the OmittingTypes Theorem 1.8, Z has a model 9A that is a proper end extension of L[a, T; Ta, qa,]
but omits a.
Then wl < a, otherwise a is recursive in 8, and so a E
contradiction.
Also, by assumption,
3 1 To for , < a, and so by Proposition
1, a
2.3, we
have that sr(!) > a. But since sr(B) is at most wa + 1, we have that wO= a.
For a contradiction, assume that sr(s)
=
a. Then by definition, B is the atomic
model of Ta. Since a is a limit ordinal, let the rank of an atom V(s) in Ta be the
least B < a such that T(Y) is an atom for TO. Let f be the function that maps each
E Z to the rank of the atom of Ta that defines its type in Ta. From the sentences
in (4) in Z and effective recovery, f is definable from Td and so f E . But then
lub(ran(f)) = a E C, a contradiction. Thus sr(9) = a + 1 as desired.
44
0
Bibliography
[1] Barwise, J., "Admissible Sets and Structures, An Approach to Definability Theory," Springer-Verlag,
1975.
[2] Chan, A., "Models of High Rank for Weakly Scattered Theories," MIT Thesis,
2006.
[3] Keisler, H.J., "Model Theory for Infinitary Logic," North-Holland Publishing
Company, 1971.
[4] Kripke, S., "Transfinite recursion on admissible ordinals, I, II (abstracts)," Journal of Symbolic Logic 29, 161-162 (164).
[5] Morley, M., "The number of countable models," Journal of Symbolic Logic 35
(1970), 1-30.
[6] Platek, R., "Foundations of recursion theory," Doctoral Dissertation and Supplement, Stanford CA, Stanford University, 1966.
[7] Sacks, G.E., "Bounds on Weak Scattering," Preprint.
[8] Scott, D., "Logic with denumerably long formulas and finite strings of quantifiers," 1965 Theory of Models, North-Holland Publishing Company, 329-341.
45