Bulletin of the Section of Logic
Volume 39:3/4 (2010), pp. 107–122
Mohamed Khaled
Tarek Sayed Ahmed
VAUGHT’S THEOREM HOLDS FOR L2 BUT FAILS FOR
Ln WHEN n > 2
Abstract
Vaught’s theorem says that if T is a countable atomic first order theory, then T
has an atomic model. Let Ln denote the finite variable fragment of first order
logic with n variables. We show that a strong form of Vaught’s theorem holds
for L2 while it fails for Ln when n > 2. An analogous result is proved for the
finite variable fragments without equality.
Let L be a first order countable language, and T be a (complete) theory
in L. A formula φ(x0 , . . . xn−1 ) is complete in T if for all ψ(x0 , . . . xn−1 ),
we have
T |= φ → ψ or T |= φ → ¬ψ.
A formula θ(x0 , . . . xn−1 ) is completable in T if there is a complete formula
φ(x0 , . . . xn−1 ) with T |= φ → θ. A theory T is atomic if every L formula
which is consistent with T is completable in T . A L-structure M is said to
be atomic, if every n-tuple a0 , . . . an−1 ∈ M satisfies a complete formula in
T h(M). We give a few (well known) examples, cf. [5]:
1. Let T be the theory of real closed ordered fields (in the language
of fields). The ordered field of real algebraic numbers is the unique
atomic model of T . The ordered field of real numbers is not atomic.
2. The standard model N of number theory in the language {+, ·, 0, S, <}
is atomic. Here S stand for the successor function (S(n) = n + 1, for
n ∈ N).
3. Every model of pure identity theory is atomic. This gives an example
of uncountable atomic models.
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The following is a classic Theorem of Vaught in model theory:
Theorem 34 . Let T be a countable complete theory. Then T has a
countable atomic model if and only if T is atomic.
The hard implication is of course, if T is atomic, then T has a countable
atomic model. It is proved by applying the omitting types Theorem, cf.
[5] Theorem 2.3.2., Shelah in [17] extended Vaught’s theorem to theories
of cardinality ≤ ℵ1 . We note that countable atomic models are prime.
Therefore every countable atomic theory has a prime model. M is prime
if its elementary embeddable in every model of T h(M). For example N
elementary embeds into any (non-standard) model of T h(N) and the field
of algebraic real numbers elementary embeds into any model of the theory
of real closed field. We note that every dense linear order without endpoints
is atomic, that is (R, <) and (Q, <) are atomic. (Q, <) is prime but (R, <) is
not. Finally we should mention that there are atomic theories of cardinality
≥ ℵ1 having no prime models [13].
The condition on finiteness of models often provides us (in a certain
sense) with more complex theories, but the condition of finiteness for variable sets usually make theories simpler (for example, algorithmically) nevertheless some desirable properties can be lost. Our next result is of such
kind. The above definitions make perfect sense when we restrict our languages to the first n variables. Letting Ln denote first order logic with n
variables, we have:
Theorem 1 .
(i) Vaught’s Theorem holds for L2 even in the uncountable case.
(ii) Vaught’s theorem for the countable case does not hold for Ln when
2 < n < ω.
Vuaght’s Theorem holds for L1 as proved in [12]. We prove our result
using polyadic algebras. The abstract version of such algebras are defined
in [16]. Indeed in [16] polyadic algebras are viewed as expansions of cylindric algebras of dimension with the unary operations pij i, j < n which
happen to be Boolean endomorphisms of the boolean part of the algebra.
The generic examples of polyadic algebras are algebras of formulas and set
algebras. These are closely related, for models of first order theories give
rise in a natural way to set algebras. (There is a converse, too, see below.)
Vaught’s Theorem holds for L2 but fails for Ln when n > 2
109
Let L be a first order language and T be an L-theory. Let F mL denote
the set of L formulas. φ ∈ F mL is restricted if all its relational atomic
subformulas have the form Ri (v0 , v1 , . . .), i..e the variables occur only in
their natural order. Let F mL
r be the set of restricted formulas. It is not
hard to see that, in the presence of ω variables, every first order formula
is logically equivalent to a restricted one. We can, therefore, assume that
T consists solely of restricted sentences (formulas without free variables.)
Let
FmL = hF mL , ∨, ¬, ∃vk , vk = vl ik,l<ω
and
L
FmL
r = hF mr , ∨, ¬, ∃vk , vk = vl ik,l<ω
Then the Tarski-Lindenbaum quotient algebras FmL /T and FmL
r /T are
locally finite cylindric algebras of dimension ω [7][4.3]. Substitutions are
definable in such algebras [6] [1.11.9-1.11-11], reflecting the metalogical
operation of simultaneous substitution of variables for free variables such
that the substitution is free. Now what happens if we restrict our attention
to finitely many variables? Let Ln denote the restriction of L to the first
n variables, and T be an Ln -theory. Substitutions are no longer definable
for the quotient algebra of restricted formulas. But adding the pij ’s and
interpreting them as interchanging the variables vi and vj (both free and
bound), in formulas, substitutions are recovered. (FmLn /T is a PEAn
n
while FmL
r /T is a CAn that might not be closed under the operations
of substitutions.) Polyadic algebras are therefore more appropriate for
dealing with finite variable fragments of first order logic, for they capture
all formulas and not just restricted ones. Now let M be a model of T .
We occasionally write FmT for the more cumbersome FmLn /T when Ln is
clear from context. Let
φM = {s ∈ n M : M |= φ[s]}.
That is φM is the set of all assignments satisfying φ in M. Let
B = {φM : φ ∈ F mLn }.
Then, it can be easily checked that B is the universe of a PEAn and the
map φ/T 7→ φM is a homomorphism with domain FmT . Indeed, we have
(φ ∧ ψ)M = φM ∩ ψ M ,
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Mohamed Khaled and Tarek Sayed Ahmed
(¬φ)M = n M \ φM ,
ck (φ)M = (∃vk φ)M ,
slk (φM ) = (∃vl (vk = vl ) ∧ φ)M ,
pkl (φM ) = {s ∈ n M : s ◦ [i, j] ∈ φM } = (φ[vk , vl ])M .
and
dkl = (xk = xl )M .
Here φ[vk , vl ] denotes the operation of interchanging the variables vk and
vl . (Note that satisfiability is reflected algebraically by homomorphisms.)
A model M of T gives a representation of the algebra FmT . More
generally, a representation of A ∈ PEAn on a set X is an injective boolean
homomorphism h : A → ℘(n X) (the power set of n X) such that
S
• h(1) = i∈I n Xi where the Xi ’s are disjoint. Here 1 is the greatest
element of the boolean reduct of A and I is an arbitrary non-empty
set.
• For all i, j < n, x̄ ∈ h(dij ) iff xi = xj
• For all i < n, a ∈ A and x̄ ∈ h(ci a) iff x̄iy ∈ h(a) for some y ∈ X. Here
x̄iy is the sequence that agrees with x̄ except for i where its value is
y.
• For all i, j < n, a ∈ A and x̄ ∈ h(sij a) iff x̄ ◦ [i|j] ∈ h(a)
• For all i, j < n, a ∈ A and x̄ ∈ h(pij a) iff x̄ ◦ [i, j] ∈ h(a).
A is representable if it has a representation. When A is simple (has
no proper congruences) then h(A) has unit a cartesian square, i.e a set
of the form n X. Such algebras are called polyadic equality set algebras
of dimension n. They can be singled out from the representable algebras
by the intrinsic property of being simple. The class of all such algebras is
denoted by PEAsn . A typical set algebra is that corresponding to a model
M, that is the algebra with universe {φM : φ ∈ L} defined above. We
denote this algebra by PEAsM
n with the superscript indicating the role of
M. For algebras A and B with the same similarity type, let Is(A, B) stand
for the class of isomorphisms of A to B. Now how are algebras of formulas,
models and set algebras related? To wrap up, what has been said, we
restrict our attention to the case when algebras in question are simple. (In
this case theories considered are complete.) Now for any A ∈ PEAsn and
h ∈ Is(FmT , A), there is a model M for T such that A = PEAsM
n and
Vaught’s Theorem holds for L2 but fails for Ln when n > 2
111
h(φ/ ≡T ) = φM . Conversely for any model M of L and A = PEAsM
n ,
there is an h ∈ Is(FmT , A) such that h(φ/ ≡T ) = φM.
Now assume that A is simple and representable on X via h. Let s ∈ n X.
Let h−1 (s) = {x ∈ A : s ∈ h(x)}. It is easy to see that h−1 (s) is a boolean
ultrafilter of A. h is an atomic representation if h−1 (s) is a principal
ultrafilter for every s ∈ n X. That is s is in the range of an atom of A.
It is easy to see that if A has an atomic representation then it is atomic.
Now what about the converse? Then, a moment of reflection reveals that,
we are asking whether every complete atomic theory has an atomic model?
We answer this as follows:
Theorem 2 .
(i) Every representable polyadic equality algebra of dimension 2 has an
atomic representation.
(ii) For 2 < n < ω, there exists a simple atomic countable representable
PEAn such that its Df reduct has no atomic representation.
The Df reduct of a polyadic algebra A is the reduct obtained by deleting diagonal elements and substitutions and is denoted by Rddf A. Its
cylindric reduct is obtained by deleting only substitutions and is denoted
by Rdca A. The latter is a CAn i.e a cylindric algeba of dimension n.
Theorem 1 follows immediately from Theorem 2.
Proof of Theorem 1 (i). We prove (i). (ii) is deferred to the next section
since it is more involved. Let A be a simple representable atomic polyadic
algebra of dimension 2. We claim that A has an atomic representation. We
closely follow lemmas 3.2.59 and Theorem 5.4.32 in [7]. By theorem 3.2.65
in [7] A has no defective atoms, since it is representable. Define Dat, small,
big, Aab as in lemma 3.2.59.
S For a ∈ Dat, let Xa = {(a, i) : i < µ} where
µ = |AtA| + ω. Let U = a∈Dat Xa . A mapping φ from AtA to ℘(2 U ) is
defined by defining φ Aab in [7] lemma 5.4.3. Then for any x ∈ A, let
f (x) =
[
{φ(a) : a ≤ x, a ∈ Dat}.
Then clearly f is an atomic representation. The preservation of all operations other than the diagonal elements can be done as in theorem 5.4.32.
The preservation of diagonal elements can be done by adapting the corresponding part in lemma 3.2.59.
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Proof of the negative part
We construct our desired algebra using ceratin graphs. In [8] a set of
coloured graphs G is defined. The reader is referred to [8] for their definition
which we briefly recall.
Definition 3 . A coloured graph is an undirected graph Γ such that every
edge of Γ is coloured by a unique edge colour (below), and some (n − 1) tuples have unique colours, too. The edge colours are :
greens: gi (i = 1, . . . , n − 2) and g0i (i < ω);
whites: wi (i = 0, . . . , n − 2);
reds: rim (i < m < ω.)
The colours for (n − 1) - tuples are :
yellows: yS (S ⊆ ω, S = ω or S finite).
We will sometimes write Γ(x, y) for the colour of an edge (x, y), and
Γ(a1 , . . . , an−1 ) for the colour of an (n−1)-tuple a1 , . . . , an−1 in the coloured
graph Γ.
Let Γ, 4 be coloured graphs, and ψ : Γ → 4 be a map. ψ is said to be
coloured graph embedding, or simply an embedding, if it is one to one and
preserves all edges, and all colours, where defined, in both directions.
Let i < ω and let Γ be a coloured graph consisting of n nodes, x0 , . . . ,
xn−2 , y, such that (xj , y) is an edge of Γ for each j < n − 1. We call Γ an
i − cone if for each j < n − 1, the edge (xj , y) is coloured gj if j > 0, and
g0i if j = 0, and no other edge of Γ (if any) are coloured green. The apex of
the cone is y, its base {x0 , . . . , xn−2 }. The tint of the cone is i. These are
well-defined, as any Γ can be viewed as a cone in at most one way. Notice
that a cone induces a linear ordering on its base, namely, x0 , . . . , xn−2 .
Definition 4 . The class G consists of all coloured
graphs Γ with the following properties.
(1) Γ is a complete graph (all possible edges are present)
(2) Γ contains no triangles of the following types:
• (g, g 0 , g ∗ ) any green coloures g, g 0 , g ∗
• (gi , gi , wi ) any i = 1, . . . , n − 2
• (g0j , g0k , w0 ) any j, k < ω
0
• (rim , rjm , rkm∗ ) unless m = m0 = m∗ and |{i, j, k}| = 3.
Vaught’s Theorem holds for L2 but fails for Ln when n > 2
113
(3) If a1 , . . . , an−2 ∈ Γ are distinct, and no edge (ai , aj )(i < j < n − 1) is
coloured green, then the sequence (a1 , . . . , an−2 ) is coloured a unique
shade of yellow. No other (n − 1) - tuples are coloured yellow.
(4) If D = {d0 , . . . , dn−2 , δ} ⊆ Γ and Γ D (the coloured graph induced
on D) is an i-cone with apex δ, inducing the ordering d0 , . . . , dn−2 on
its base, and the tuple (d0 , . . . , dn−2 ) is coloured yS , then i ∈ S.
Clearly, G is closed under isomorphism and under induced subgraphs.
G depends on n.
In the next definition we show how these graphs can be used to construct a polyadic atom structure.
Definition 5 .
Consider the class Kn of surjective maps a : n(=
{0, . . . , n − 1}) → Γa , any Γa ∈ G. Many of these maps, though formally distinct, will differ only because the nodes in the image graphs will
not be the same. So we define an equivalence relation ∼ on Kn
a ∼ b ⇐⇒ a(i) = a(j) ⇔ b(i) = b(j)
and
Γa (a(i), a(j)) = Γa (b(i), b(j)),
if defined, and
Γa (a(k0 ), . . . a(kn−2 )) = Γb (b(k0 ) . . . b(kn−2 )),
if defined, for all i, j ∈ n and (n − 1)-tuples k̄ of elements of n. In other
words, a and b define isomorphic coloured graphs. This is an equivalence
relation on Kn . We define an atom structure Cn0 with domain
Cn0 = {[a] : a ∈ Kn }.
For every i, j ∈ n and [a], [b] ∈ Cn0 we define Eij ⊆ Cn0 , Ti ⊆ 2 Cn0 and
Pij ⊆ 2 Cn0 as follows:
[a] ∈ Eij ⇐⇒ a(i) = a(j)
[a]Ti [b] ⇐⇒ a[n\{i}] ∼ b[n\{i}].
That is [a]Ti [b] if and only if for some c ∈ [a], b(j) = c(j) for all j 6= i.
Finally
[a]Pij [b] ⇐⇒ b ◦ [i, j] ∼ a
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Mohamed Khaled and Tarek Sayed Ahmed
These definitions are sound (do not depend on the representatives). Now
consider the atom structure
Cn0 = (Cn0 , Ti , Pij , Eij )i,j<n .
Let Dn be the complex algebra over Cn0 . That is Dn = ℘(Cn0 ). The boolean
operations are the usual set theoretic intersections and taking complements;
the extra non boolean operations are defined for X ∈ ℘(Cn0 ) as follows
ci X = {[b] ∈ Cn0 : ∃[a] ∈ X [a]Ti [b]},
pij X = {[b] ∈ Cn0 : ∃[a] ∈ X [a]Pij [b]},
dij = Eij.
Let Cn to be the subalgebra of Dn generated by the atoms, i.e. by the set
{[a] : a ∈ Kn }.
Lemma 6 . The algebra Cn is generated by n − 1 dimensional elements
Q
Proof. It suffices to show that {[a]} = {ci {[a]} : i < n} for any a ∈ Kn .
Assume that a : n → Γ with Γ ∈ G. Clearly ≤ holds. For the other
direction assume that b : n → ∆ and [a] 6= [b]. We show that b cannot be
an element of the right hand side. Since a and b are not equivalent, we can
assume that
1. (∃i, j < n)∆(b(i), b(j)) 6= Γ(a(i), a(j)) or
2. (∃i1 , . . . in−1 < n)∆(b(i1 ), . . . b(in−1 )) 6= Γ(a(i1 ), . . . a(in−1 )).
In the first case, let k be distinct from i and j. Then [b] ∈
/ ck {[a]}. In the
second case, choose k ∈
/ {i1 , . . . in−1 } and proceed the same way.
We shall need the notion of atomic networks which are basically finite
approximations to complete representations.
Definition 7 . Let D be an atomic arbitrary n-dimensional polyadic-type
algebra. Let AtD denotes the set of its atoms. An atomic network N is a
set of nodes 4 and a total function N : n 4 → At(D) such that
- N (δ̄) ≤ dij iff δi = δj (for any δ̄ ∈ n 4 and any i, j < n).
- N (δ̄di ) ≤ ci N (δ̄) (for any i < n, δ̄ ∈ n 4, d ∈ 4).
-pij N (δ) = N (δ ◦ [i, j].)
A complete representation is a representation that preserves infinitary
meets and joins whenever defined. In [8] it is proved that an algebra has a
Vaught’s Theorem holds for L2 but fails for Ln when n > 2
115
complete representation if and only if it has an atomic one. We now prove
the main result of this paper:
Theorem 8 . Cn is elementary equivalent to a completely representable
P EAn , hence is representable, but its Df reduct has no complete representation. In particular, a simple component of Cn satisfies (ii) in theorem
2.
Proof. A “graph game” is defined between two players ∃ (female) and
∀ (male) in [8], cf. lemma 30. It is shown that for all k < ω, ∃ has a
winning strategy in the graph game of length k [8] proposition 33 while
∀ has a winning strategy for the graph game of length ω [8] proposition
32. Another game on networks Gk (D), D is an atomic polyadic algebra,
has k rounds, k ≤ ω, and is defined as follows. In the zero’th round, ∀
picks any atom a of D. ∃ has to respond with a finite atomic network N0
¯ = a for some n-tuple of nodes d¯ ∈ n0 N . Without loss,
such that N0 (d)
|N0 | ≤ n. In any further round, let the last network played be N . ∀ picks
an index i < n, a “face” F = (f0 , f1 , . . . , fn−2 ) ∈ N n−1 , and an atom
b ≤ ci N (f0 , . . . fi−1 , x, fi . . . , fn−2 ) (the choice of x ∈ N is arbitrary, as
the second part of the definition of an atomic network together with the
fact that ci (ci x) = ci x for all x ∈ D ensures that the right-hand side does
not depend on it). ∃ must respond, if possible, with a network N ⊆ N +
with at most one more node, such that there is a node d ∈ N + with
N + (f0 , . . . , fi−1 , d, fi . . . , fn−2 ) = b. If she can do this in every round, she
has won the play. It is proved in [8], lemma 31, that ∃ has a winning
strategy in the graph games of all finite lengths if and only if she has a
winning strategy in the games Gk (Cn ) and that ∃ has a winning strategy
in the graph game of length ω if and only if she has a winning strategy
in the game Gω (Cn ). Cn is clearly countable. Now for all k < ω, ∃ has
a winning strategy σk in Gk (Cn ). Let B be a non-principal ultrapower
of Cn . Then ∃ has a winning strategy σ in Gω (B), essentially she uses
σn in the nth component of the ultraproduct so that at each round of
Gω (B) ∃ is still winning in co-finitely many components, this suffices to
show that she has still not lost. Now we can use an elementary chain
argument to construct countable elementary subalgebras of B containing
Cn . Cn = A0 ≤ A1 . . . ≤ B. For this let Ai+1 be a countable elementary
subalgebra of B containing Ai and all elements of B that σ selects
S in play of
Gω (B) in which ∀ only chooses elements from Ai . Let A0 = i∈ω Ai . This
116
Mohamed Khaled and Tarek Sayed Ahmed
is a countable elementary subalgebra of B and ∃ has a winning strategy in
Gω (A0 ). We prove that A0 has a complete representation, this will prove
that Cn is representable (since the class of representable algebras is a variety
and Cn ≡ A0 .) Consider a play N0 ⊆ N1 ⊆ . . . of Gω (A0 ) in which ∃ uses
her strategy, so that all the Nt are atomic networks, and ∀ eventually picks
up every face (f0 , . . . fn−2 ) every i < n and every atom b. That is ∀ plays
every possible legal move in some stage of the play. He can do this because
there are countably many nodes that appear in the play and countably
many atoms in A0 . If ∃ uses her
S winning strategy to this particular game,
then the limit network M = t<ω (Nt ) will satisfy the following condition
(*)
For every face (f0 , . . . fn−2 ) ∈ n−1 M for all i < n for every atom
b of A if b ≤ ci M (f0 , . . . , x, fn−2 ), then there exists a node l such that
b = M (f0 , . . . l, . . . fn−2 ). For r ∈ A0 , let
¯ ≤ r)}.
h(r) = {d¯ ∈ n M : ∃t < ω(d¯ ∈ n Nt and Nt (d)
¯ such that
For every d¯ ∈ n M there is a t < ω and an atom a = Nt (d)
¯
¯
d ∈ h(Nt (d)), so that h is an atomic representation. We may assume that
A0 is simple. If not, then we can replace it by its simple components and
prove that every such component is completely representable. Then we glue
these components. In more detail, consider the elements {cn a : a ∈ AtA0 }.
Then every simple component Sa of A0 can be obtained by relativizing to
cn a for an atom a. So for each atom a, Sa has a complete representation
ha . The domain of the representation will then be the disjoint union of the
domains of ha , and now represent the whole of A0 by
[
g(α) =
{ha (α · cn a)}.
a∈AtA
We first prove that h is a homomorphism from the diagonal free reduct
of A0 to ℘(n M ) (The diagonal free reduct of the full polyadic set algebra
with unit n M .) We check the boolean operations. We have d¯ ∈ h(r + s)
¯ ≤ r + s. Because Nt (d)
¯ is an
iff ∃t < ω, such that d¯ ∈ n Nt and Nt (d)
n
¯
¯
¯ ≤ s.
atom this is equivalent to ∃t < ω, d ∈ Nt and Nt (d) ≤ r or Nt (d)
¯
Equivalently s̄ ∈ h(r) ∪ h(s). Complement is just as easy: d ∈ h(−r) iff
¯ ≤ −r. d¯
d∈
/ h(r). Indeed assume that d¯ ∈ h(−r). Then ∃t1 < ω, Nt1 (d)
¯
cannot be in h(r) for else we get t2 < ω such that Nt2 (d) ≤ r. By taking
¯ = 0 which is impossible because
t = max{t1 , t2 } it follows that Nt (d)
¯
Nt (d) is an atom. The reverse inclusion is the same. Now h preserves
Vaught’s Theorem holds for L2 but fails for Ln when n > 2
117
cylindrifications by (*). Indeed, the following follows from (*): there exists
¯ ≤ ci r iff there exists t2 ∈ ω such that Nt (d¯i ) ≤ r
t1 < ω such that Nt1 (d)
u
2
for some u. Now we check substitutions. Let d¯ ∈ h(pij r). Then there exists
¯ ≤ pij r. Hence pij Nt (d)
¯ = Nt [d¯ ◦ [i, j]) ≤ r The
t < ω such that Nt (d)
other inclusion is similar. To preserve diagonals we have to factor out by
an equivalence relation which turn out to be a congruence relation. Define
a binary relation ∼ on M as follows:
x ∼ y ⇐⇒ ∃z̄ ∈ n M, ∃k, l < n(zk = x ∧ zl = y ∧ z̄ ∈ h(dkl )).
The following holds: If x̄, ȳ ∈ n M , i < n and x̄ ≡i ȳ and xi ∼ yi , then
x̄ ∈ h(r) iff ȳ ∈ h(r) for all r ∈ A0 . To see this, let t < ω be so large
such that x̄, ȳ ∈ n Nt , there is z̄ ∈ n Nt and k, l < n with zk = xi , zl = yi
and Nt (z̄) ≤ dkl , and ∀ has already played the moves d, x̄, r and d, ȳ, r
before round t of the game. Therefore Nt (x̄) ≤ r or Nt (x̄) ≤ −r and
similarly for ȳ. From the definition of network and the existence of z, we
have Nt (x̄).Nt (ȳ) 6= 0. Therefore we must have Nt (x̄) ≤ r iff Nt (ȳ) ≤ r.
Since this holds for large t, we obtain x̄ ∈ h(r) iff ȳ ∈ h(r). It follows that
∼ is an equivalence relation on M . It is clearly reflexive and symmetric.
For transitivity, suppose x ∼ y ∼ z. We may suppose that x 6= y. Take
w ∈ n M witnessing x ∼ y: say wk = x and wl = y, and w̄ ∈ h(dkl ). Then
k 6= l. Define v̄ ∈ n M by v̄ ≡l w̄, vl = z. We have vk = wk = x. Therefore
v̄ ∈ h(dkl ), too, so x ∼ z. Similarly, it follows that ∼ is an h congruence
in the sense that if x̄, ȳ ∈ n M and xi ∼ yi for all i < n, then x̄ ∈ h(r) iff
ȳ ∈ h(r) for all r ∈ A”. To see this, for j ≤ n define z̄ j ∈ n M by zij = yi if
i < j and zij = xi if j ≤ i < d. Then
x̄ = z̄ 0 ≡0 z̄ 1 ≡1 z̄ 2 ≡2 ... ≡n−1 z̄ d = ȳ
For all i ≤ n, since z̄ i ≡i z̄ i+1 and zii = xi ∼ yi = zii+1 , we have z̄ i ∈ h(r)
iff z̄ i+1 ∈ h(r) for all r ∈ A0 . Thus by taking i = 0, 1, . . . n − 1 in turn we
obtain x̄ ∈ h(r) iff ȳ ∈ h(r) for all r ∈ A0 as required. Let N = M/ ∼
be the set of equivalences classes. For d¯ ∈ n M with d¯ = (d0 , . . . dn−1 ), let
¯ ∼ = (d0 / ∼, . . . dn−1 / ∼). For r ∈ A0 , let
d/
¯ ∼: d¯ ∈ h(r)}.
g(r) = {d/
Then g : A0 → ℘(n N ) is as required. Now we show that Rddf Cn has
no complete representation. Assume, seeking a contradiction, that it has a
complete representation g. Then we claim that Cn has a complete representation h. This follows from the following: if A ∈ CAn and f : A → B is a
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complete representation of Rddf A into a (diagonal free) set algebra B, then
there exists C ∈ Csudfn with base U and g : B → C such that g ◦ f : A → C
[U ]
is complete representation with dij ⊆ g ◦ f (dij ). Indeed, we canSassume
that B has
Qa pairwise disjoint base system (Vi : i < n). Let U = i<n Vi .
Let dn = i<j<n dij . Then cn∼{k} dn = 1, hence cn∼{k} f (dn ) = 1, it follows
that for all u ∈ Vk there is an x ∈ f (dn ) such that xk = u. Hence for each
k < n there is a function gk mapping Vk into f (dn ) such that (gk u)k = u
for all u ∈ Vk . Now define for each k < n a function tk mapping U onto Vk
by setting tk v = (gj v)k where j is such that v ∈ Vj . Then for x ∈ A, let
g(x) = {u ∈ n U : (tk (uk ) : k < n) ∈ x}.
Then C = Img and g are as required. Now assume that Rddf Cn isQcompletely representable, via the isomorphism h, as a set algebra D ⊆ P ( Ui :
i < n). We can assume that U0 = . . . = Un−1 = U and dU
ij ⊆ h(dij ). Define
R on U as follows: let i, j < n be distinct, then
R = {(u, v) ∈ U × U : s(i) = u and s(j) = v for some s ∈ h(dij )}
Then R is independent of the choice of i and j and is an equivalence relation
on U [7] lemma 5.1.49. Let
E = {x ∈ C : (∀s, t ∈ n U )[(∀i < n)((s(i), t(i)) ∈ R =⇒ (s ∈ h(x) ←→
t ∈ h(x))]}.
Then {x ∈ C : ∆x 6= n} ⊆ E and E is a subuniverse of C [7] lemma 5.1.50.
Thus E = C. Then we can factor U by R so that C can be completely
embedded into (℘n (U/R), ci )i<n via the isomorphism f given by
f (x) = {(s(i)/R : i < n) ∈ n (U/R) : s ∈ h(x)}.
Moreover, as easily checked, diagonals are preserved. Since Cn is generated
as a boolean algebra by {x ∈ Cn : ∆x 6= n}, the rest follows from [7] 5.4.26.
We have proved that Cn has a complete representation h. Then ∃ can use
h as a guide and win Gω (Cn ), which contradicts the above. Let the base
of h be U . Any finite subset N ⊆ U defines an atomic network by defining
¯ to be the unique atom a ∈ Cn such that d¯ ∈ h(a). Such an atom
N (d)
exists since the representation is atomic. It is easy to see that, so defined,
N is a network. ∃ ensures that each network is played this way. For t = 0
let ∀ choose an atom a. ∃ chooses d¯ ∈ n U with d¯ ∈ h(a). She defines
the atomic network by stating that its notes are d0 , d1 . . . dn . Given the
Vaught’s Theorem holds for L2 but fails for Ln when n > 2
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inductive hypothesis Nt ⊆ U at round t, ∀ chooses (f0 , . . . fn−2 ) ∈ n−1 Nt ,
i < n and atom b ≤ ci Nt (f0 , x, . . . fn−2 ). Then (f0 , x, . . . fn−2 ) ∈ h(ci b).
Hence there exists z ∈ U such that (f0 , z, . . . fn−2 ) ∈ h(b). ∃ selects such a
z and forms Nt+1 by stating that its nodes are those of Nt together with
z. This completes the proof.
Concluding remarks
• Let L6=
n denote the fragment first order logic without equality restricted to the first n variables. From the above, together with [7]
5.1.56, Vaught’s theorem holds for L6=
n if and only if n ≤ 2. Using
Theorem 8, together with [2], one can also prove that the omiting
types theorem fails for the finite variable fragment of first order logic
without equality as long as the number of variables > 2. See also
[11].
• The omitting types theorem is proved to hold for infinitary extensions
of first order logic in [14] and [15].
• A word on non - classical interpretations of Ln as multi-modal logic
seems timely. The most prominent citizens of first order logic are the
quantifiers, whose meaning is defined as follows:
M |= ∃vi φ[s] ⇐⇒ ∃d ∈ M : M |= φ[sdi ].
Here sdi is the sequence which agrees with s except at i where its
value is d. There is an obvious modal view on this definition. Let
A = ω M be the set of all assignments, and define a set of equivalence
relations ≡i on A × A by s ≡i t iff s(j) = t(j) for all j 6= i. Moving
the assignment to the front, we get a familiar “modal” pattern:
M, s |= ∃vi φ ⇐⇒ (∃t ∈ A)(t ≡i s ∧ M, t |= φ.)
According to the Tarskian tradition, A is the set of all evaluations
of the variables into M. Here we treat elements of t ∈ A as states
or possible worlds in a certain Kripke frame (A, ≡i , . . .). The various
logical connectives, like φ 7→ ∃xφ will re-appear here as modalities
whose accessibility relations (the relations ≡i ) on A will determine
their meanings. So we can view first order logic as one particular
instance of the modal logic of assignments. Relativized semantics can
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Mohamed Khaled and Tarek Sayed Ahmed
be obtained by looking at subsets of A. It is harmless and sometimes
even desirable to look at only some of the states and discard the rest.
• Algebraic logic has recently become of great interest in modal logic:
both fields are often studying the same phenomena, by different
names. Algebraic logicians use the term atom structures, while modal
logicians use the term Kripke frame. Let n be finite. Consider a frame
F = (W, Ti , Pij , Eij )i,j<n of polyadic type. (That is both Ti and Pij
are binary relations and Eij is unary). Then we can form the complex
algebra of F which we denote abusing notation slightly by ℘(F). ℘(F)
is of the same similarity type as PEAn . If K is a class of frames,
then K + = {A : A ∼
= ℘(F) for some F ∈ K}. Let K be a class of
frames, let F ∈ K and let T erm(X) be the set of terms generated
from a countable set of variables X (in the language of PEAn ). Let
v be a function from X to ℘(F). We call M = (F, v) a Kripke model
over F. A truth relation |= can be defined by recursion as follows:
For s ∈ F we define
M, s |= x ⇐⇒ s ∈ v(x).
The booleans are as expected and the extra non boolean operations,
the cylindrifications, say, are defined as follows:
M, s |= ci τ ⇐⇒ (∃t ∈ F )sTi t and M, t |= τ.
The definition for pij is entirely analogous. A concrete frame is of the
form (V, ≡i , Pij , Dij ) where V ⊆ α U for some set U and for s, t ∈ V
we have
s ≡i t ⇐⇒ s(j) = t(j) ∀j 6= i
sPij t ⇐⇒ s ◦ [i, j] = t
Dij = {s ∈ V : si = sj }.
Kcube is the class of frames whose domain is of the form α U and K is
the class of arbitrary concrete frames. There is a whole landscape of
classes of frames between K and Kcube obtained by imposing extra
conditions on V the domain of frames in question. Let L be such a
class where V is not necessarily a cartesian square. The logic corresponding to SP L+ has syntax like first order logic, and it can be
viewed as multi-modal propositional logic enriched with constants.
Vaught’s Theorem holds for L2 but fails for Ln when n > 2
121
In particular, we can look at quantifiers ∃vi as if they are modal operators ♦i whose meaning is given by the relation ≡i . The semantics
in a frame with domain V are relativized to the states in V . For
example
M, s |= ♦i φ iff (∃t ∈ V )(t ≡i s ∧ M, t |= φ.)
Such logics, from which the guarded fragment arose, behave nicer
than n variable fragments of ordinary first order logic in many respects (particularly concerning decidability and completeness).
• Negative results (for finite variable fragments of first order logic) mentioned above do not occur for the guarded fragment of first order logic
introduced in [3]. The guarded fragment (GF ) was introduced as a
fragment of first order logic which combines a great expressive power
with nice modal behavior. It consists of relational first order formulas
whose quantifiers are relativized by atoms in a certain way. GF has
been established as a particularly well-behaved fragment of first order
logic in many respects. The main point of the GF (and its variants
e.g. the packed fragments) is that (inside the GF ) we are safe of the
above negative results for Ln . So it seems to us that it is likely that
(some form of) Vaught’s Theorem holds for GF , but further research
is needed.
References
[1] H. Andréka, J. D. Monk, I. Németi, (editors) Algebraic Logic, NorthHolland, Amsterdam, 1991.
[2] H. Andréka, I. Németi, T. Sayed Ahmed, Omitting types for finite variable fragments and complete representations for algebras. To appear in the
Journal of Symbolic Logic.
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[5] C. Chang, J. Keisler, Model Theory, North Holland, 1994.
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[12] T. Sayed Ahmed, Martin’s axiom, omitting types and complete representations in algebraic logic, Studia Logica, vol. 72 (2002), pp. 1–25.
[13] T. Sayed Ahmed, An atomic theory with no prime models, Australasian
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[14] T. Sayed Ahmed, Omitting types for infinitary extensions of first order logic
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[15] T. Sayed Ahmed and B. Samir, Omitting types for first order logic with infinitary predicates, Mathematical Logic Quaterly 53/6 (2007), pp. 564–
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[16] I. Sain, R. J. Thompson, Strictly finite schema axiomatizations of QuasiPolyadic Algebras, H. Andréka H., J.D Monk and I. Németi, editors Algebraic Logic, (Proc. Conference Budapest 1988), vol 54 of Colloq,
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Department of Mathematics, Faculty of Science
Cairo University, Giza, Egypt