Pseudo-finite model theory

⇤

Jouko Vaananen †

Department of MathematicsUniversity of HelsinkiHelsinki, Finland

jouko.vaananen@helsinki.fi

August 1, 2016

Abstract

We consider the restriction of first–order logic to models, called pseudo–

finite, with the property that every first–order sentence true in the model

is true in a finite model. We argue that this is a good framework for study-

ing first-order logic on finite structures. We prove a Lindstrom Theorem

for extensions of first order logic on pseudo-finite structures.

1 Introduction

First order model theory on infinite structures is a well–developed field of ab-stract mathematics with powerful methods and results. In contrast, finite modeltheory has rather few general methods and also rather few general results. Wepropose an approach where we allow certain infinite models but not all. Someinfinite models have infinite cardinality but no first-order properties which de-pend on this infinity. We distinguish these infinite models from those whichactually satisfy some first-order axioms which do not have finite models at all.In the latter case the infinite cardinality of the model is not an accident but anecessity. We call a model pseudo-finite if every first-order sentence true inthe model is also true in a finite model. We propose the study of first-ordermodel theory restricted to pseudo-finite models as an alternative approach tofinite model theory.

Let us think of the following problem: We have two first-order sentences �and and we want to know if is true in all finite models of �. If � ! isprovable in predicate logic, is true in all models of �, finite or infinite, so wehave solved our problem. But suppose we suspect that � ! is not provablein predicate logic. We may try to demonstrate this by exhibiting a model of �

⇤I am indebted to John Baldwin, Lauri Hella, Tapani Hyttinen, Phokion Kolaitis andKerkko Luosto for useful suggestions.

†Research partially supported by grant 40734 of the Academy of Finland.

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where is false. Such a model-construction may require elaborate techniquesas we know from model theory and axiomatic set theory. We argue belowthat a substantial part of the machinery of infinite model theory is available inpseudo-finite model theory. If we succeed in using these methods to constructa pseudo-finite model of � ^ ¬ , we have solved the problem: The sentence cannot be true in all finite models of �.

Pseudo-finiteness can be defined in several equivalent ways. The Lemmabelow exhibits three. If L is a vocabulary, we use �L to denote the first-ordertheory of all finite L-structures.

Lemma 1 Suppose L is a vocabulary and A is an L–structure. Then the fol-lowing conditions are equivalent

(1) A |= �L.

(2) Every first–order L–sentence which is true in A is true in a finite model.

(3) There is a set {Ai : i 2 I} of finite L–structures and an ultrafilter F on Isuch that

A ⌘Y

i2I

Ai/F .

Proof. (2) ! (3): Let T be the set of first–order L–sentences true in A. LetI be the set of finite subsets of T . For i = {'1, . . . ,'n} 2 I, let Ai be a finiteL–structure such that Ai |= '1^ . . .^'n, and then Xi the set of j 2 I for whichAj |= '1 ^ . . . ^ 'n. The family {Xi : i 2 I} can be extended to an ultrafilterF on I. If now A |= ', then {j 2 I : Aj |= '} contains Xi for i = {'}, henceQi2I

Ai/F |= '. The other directions are trivial. 2

We call models isomorphic toQi2I

Ai/F for some set I, finite Ai and some

ultrafilter F , UP-finite. By the Keisler–Shelah Isomorphism Theorem [11, 15]a model A is pseudo-finite if and only if it has an ultrapower which is UP-finite.This gives a purely algebraic definition of pseudo-finiteness. The random graph(see e.g. [6]) is a well-known example of a pseudo-finite model. More generally,suppose µn is a uniform probability measure on the set Sn(L) of L–structureswith universe {1, . . . , n} and let

µ(') = limn!1

µn({A 2 Sn(L) : A |= '})

for any L–sentence '. Then if A is a model of the almost sure theory of µ

{' : ' first order L–sentence and µ(') = 1},

then A is pseudo-finite. In this case the pseudo-finite structures actually satisfythe stronger condition: true sentences are true in almost all finite models. Thereis a deep result of model theory which can be utilized to get pseudo-finite models:Cherlin, Harrington and Lachlan [3] proved that all models of totally categorical

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(–categorical for each ) theories are pseudo-finite. For a stability theoreticanalysis of pseudo-finite structures, see [10] and [4].

We use common model theoretic notation, as e.g. in [9]. When we talk aboutfinite structures we do not assume that the vocabulary is necessarily finite.

2 Examples

Pseudo-finite models can be and have been used to prove results about finitestructures. Gurevich [7] proved by means of Ehrenfeucht–Fraisse games thateven cardinality is not expressible in first–order logic on ordered structures.One can alternatively use UP-finite models to prove this result. Let An be thelinear order h{1, . . . , n}, <i. It is easy to see that if

A =Y

n<!

A2n/F , B =Y

n<!

A2n+1/F ,

where F is a non–principal ultrafilter on !, then A ⌘ B as both A and B haveorder–type of the form

! +X

i2L

Z+ !⇤,

where L is an @1-saturated dense linear order without endpoints. If even cardi-nality were expressible by a first–order sentence ' on finite ordered structures,then we would have A |= ' and B |= ¬' contradicting A ⌘ B. It is easyto see that a linear order is pseudo-finite if and only if its type is finite or! +

P↵2L Z+ !⇤, where L is a linear order.

Hajek [8] proved that connectedness is not first–order definable on finitegraphs. It was observed by Turan [18] that this can be proved using UP-finitemodels as follows: Let An be the cycle of 2n vertices and Bn the union of twocycles of n vertices. Let A =

Qn<!

An/F and B =Q

n<!Bn/F , where F is a non–

principal ultrafilter on !. It is easy to see that A ⇠= B as both graphs are theunion of continuum many infinite chains of type Z. If there were a first–ordersentence ' expressing connectedness, then A |= ' and B |= ¬' contradictingA ⇠= B.

Fagin [5] proved the stronger result that connectedness is not even monadic⌃1

1 on finite graphs. Also this result can be proved using UP-finite models. Forthis, suppose 9P1 . . . 9Pk' is a monadic ⌃1

1 sentence. Suppose Cn is a cycle oflength 2(k+2)n. Each Cn is connected, so we have predicates P1, . . . , Pk on Cn

making ' true. We can think of these predicates as components of a coloring ofthe vertices of Cn. Let an and bn be vertices of Cn so that the neighborhoodsof an and bn of width n have the same color–pattern. Let C 0

n be the resultof diverting the edge connecting an to its successor to an edge from an to thesuccessor of bn, and the edge connecting bn to its successor to an edge from bnto the successor of an. Let An be the graph Cn and Bn the graph C 0

n, bothendowed with the above coloring. Let A =

Qn<!

An/F and B =Q

n<!Bn/F ,

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where F is a non–principal ultrafilter on !. Then A ⇠= B, whence B |= '. Letn < !, with Bn |= '. We get a contradiction since C 0

n is disconnected.Koponen and Luosto [12] show that neither simplicity nor nilpotency is

first–order definable on finite groups. They accomplish this by using pseudo-finite groups. For simplicity they prove that

Y

p2P

Zp/F ⇠=Y

p2P

(Zp + Zp)/F

where P is the set of primes and F is a non–principal ultrafilter on P . Fornilpotency they use Y

i<!

Z2i/F ⇠=Y

i<!

Z2ip(i)/F .

Starting with [1] algebraists and model theorists have studied pseudo-finitefields.

Familiar model constructions may not yield a pseudo-finite structure. Forexample, if A is pseudo-finite and P is a subset of A, there is no guarantee thatthe expansion (A, P ) is pseudo-finite. However, we may always add constants.Reducts of pseudo-finite structures are pseudo-finite. Ultraproducts of pseudo-finite structures are pseudo-finite, by the Los Lemma. Naturally, any modelelementarily equivalent to a pseudo-finite model is itself pseudo-finite.

UP-finite structures constitute an important special case of pseudo-finiteness.Note that ultraproducts of UP-finite structures are again UP-finite:

Y

i2I

(Y

j2Ji

Ai,j/Gi)/F ⇠=Y

(i,j)2K

Ai,j/H ,

whereK = {(i, j) : i 2 I, j 2 Ij}

andX 2 H () {i : {j 2 Ji : (i, j) 2 X} 2 Gi} 2 F.

The main di↵erence between pseudo-finite and UP-finite structures is that theformer may omit types while the latter are always @1–saturated (if infinite).Also, the former may be countably infinite, while the latter cannot: Shelah [14]showed that the cardinality of an infinite UP-finite

Qi2I

Ai/F is always |I|!. If

we assume the Continuum Hypothesis (CH), the UP-finite models of cardinality@1 become (by @1–saturation) categorical, i.e., CH implies

A ⌘ B , A ⇠= B,

if A and B are UP-finite and of size @1. Shelah [16] constructs a model ofset theory with a non-categorical UP-finite model of cardinality 2!.

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3 Model theory of pseudo-finite structures

Model theory of pseudo-finite structures can be studied in many respects in thesame manner as the model theory of arbitrary structures. The pseudo-finitemodels of a first-order theory T of a vocabulary L are exactly the models of thetheory T [ �L. So we have the following properties: (|=FIN � means � is truein all finite finite models. Equivalently, � is true in all pseudo-finite models, i.e.,�L ` �.)

• Finite Compactness Theorem If T is a first–order theory every finitesubset of which has a finite model, then T has a model (which w.l.o.g. ispseudo-finite).

• Finite Lowenheim–Skolem Theorem If T is a countable first–ordertheory and T has a pseudo-finite model, then T has a countable model(which w.l.o.g. is pseudo-finite).

• Finite Omitting Types Theorem If T is a first-order theory in a count-able vocabulary L and p is an L-type, non-principal relative to T [ �L,then T has a pseudo-finite model which omits p.

• Finite Lyndon Preservation Theorem Suppose T is a first-order the-ory. A sentence is preserved by homomorphic images on pseudo-finitemodels of T if and only if it is equivalent in pseudo-finite models of T toa positive sentence.

• Finite Los–Tarski Theorem Suppose T is a first-order theory. A sen-tence is preserved by pseudo-finite submodels of pseudo-finite models ofT if and only if it is equivalent in pseudo-finite models of T to a universalsentence.

The model theory of pseudo-finite structures is yet to be developed. Somemore advanced results can be immediately derived from the infinite version.An example is the pseudo-finite version of Morley’s Theorem: If a count-able first-order theory has only one pseudo-finite model (up to isomorphism)in some uncountable cardinality, then it has only one pseudo-finite model (upto isomorphism) in every uncountable cardinality. However, for pseudo-finitemodel theory to make sense, it has to be developed systematically. One suchsystematic account is [4].

4 Interpolation

The usual forms of interpolation all fail in finite models. In their usual formu-lation they also fail in pseudo-finite models. However, in each case there is aweaker statement which is true. We start with a crucial example:

Example 2 There are a finite vocabulary L1, a finite vocabulary L2, an L1–sentence �1, an L2–sentence �2, an L1–model A1, and an L2–model A2 such

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that |=FIN �1 ! �2, A1 |= �1, A2 |= ¬�2, A1 � (L0 \ L1) = A2 � (L0 \ L1) ispseudo-finite, and there is no L1 \ L2–sentence ✓ such that |=FIN �1 ! ✓ and|=FIN ✓ ! �2.

Proof. Let L1 = {R} and L2 = {P} where R and P are distinct binary predi-cates. Let �1 say that R is an equivalence relation with all classes of size 2 andlet �2 say ¬(P is an equivalence relation with all classes of size two except one ofsize one). Then |=FIN �1 ! �2. If there were an identity sentence ✓ such that|=FIN '! ✓ and |=FIN ✓ ! ¬ , then ✓ would characterize even cardinality infinite models. The models A1 and A2 as required are easy to construct. 2

We can now immediately see that the usual form of Robinson’s Theorem isfalse: Let T1 be the theory of A1 and T2 the theory of A2. The theories T1

and T2 both have a pseudo-finite model, and all pseudo-finite L1 \ L2-modelsof T1 \ T2 are elementary equivalent. Yet T1 [ T2 has no pseudo-finite model.However, the following obtains easily from the classical Robinson’s Theorem:

Finite Robinson Consistency Theorem Suppose Ti (i 2 {0, 1}) isa first–order Li–theory with a pseudo-finite model. Suppose also thatpseudo-finite L0 \ L1–models of T0 \ T1 are elementary equivalent. Thenthere are pseudo-finite models A0 and A1 such that A0 |= T0, A1 |= T1

and A0 � (L0 \ L1) = A1 � (L0 \ L1).

Example 2 shows that the usual formulation of Craig’s Interpolation Theo-rem does not hold on pseudo-finite models. However, the following formulationfollows from the classical Craig Interpolation Theorem:

Finite Craig Interpolation Theorem Suppose L0 and L1 are vocabu-laries, 'i is an Li–sentence (i 2 {0, 1}) and for all pseudo-finite L0 \ L1–models A and all pseudo-finite expansions Ai of A to an Li–model satisfy

A0 |= '0 ) A1 |= '1.

Then there is an L0 \ L1–sentence ✓ such that

|=FIN '0 ! ✓ and |=FIN ✓ ! '1.

To see more clearly how this follows, we may write it in an equivalent form:Suppose L0 and L1 are vocabularies, 'i is an Li–sentence (i 2 {0, 1}) and

�L0 [ �L1 |= '0 ! '1.

Then there is an L0 \L1–sentence ✓ such that |=FIN '0 ! ✓ and |=FIN ✓ !'1. Example 2 shows that the assumption �L0[L1 |= '0 ! '1 i.e. |=FIN '0 !'1 would not be enough (as is well-known). The situation is the same with BethDefinability Theorem. The above Finite Craig Interpolation Theorem gives:

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Finite Beth Definability Theorem Suppose L is a vocabulary, P anew predicate symbol, and '(P ) a first-order L[ {P}–sentence such thatfor all pseudo-finite L [ {P}–models A and A0 of '(P ) we have

A � L = A0 � L ) PA = PA0.

Then there is an L–formula ✓(~x) such that

'(P ) |=FIN 8~x(✓(~x) $ P (~x)).

Equivalently: If

�L[{P} [ �L[{P 0} [ {'(P ),'(P 0)} ` P = P 0,

then there is an L–formula ✓(~x) such that '(P ) |=FIN 8~x(✓(~x) $ P (~x)). Again,it is easy to see that the weaker assumption

�L[{P,P 0} [ {'(P ),'(P 0)} ` P = P 0

would not be enough, indeed, it is well-known that �(P )^ �(P 0) |=FIN P = P 0

is not enough.Let us call K a pseudo-finite PC–class if there is a vocabulary L and a

first–order ' such that

K = {A � L : A pseudo-finite and A |= '},

and a pseudo-finite ⌃11-class if there is a vocabulary L and a first–order '

such that

K = {A : A pseudo-finite and 9A0(A = A0 � L,A0 |= ')}.

Note the di↵erence between pseudo-finite PC and pseudo-finite ⌃11: the latter

quantifies over more relations than the former. Every non-empty pseudo-finitePC-class contains a finite model, but a pseudo-finite ⌃1

1-class may contain onlyinfinite models. Hence PC and ⌃1

1 are not the same concept on pseudo-finitemodels, but they coincide both on finite models and on all models. On finitemodels they coincide with NP-describability.

A consequence of the Finite Craig Interpolation Theorem is the FiniteSouslin–Kleene Interpolation Theorem for PC: If K is pseudo-finite PCand the complement {A 62 K : A pseudo-finite } is pseudo-finite PC, too, thenK is first–order definable of pseudo-finite models. Thus first–order logic is 4–closed on pseudo-finite structures. To get a similar result for ⌃1

1 we first modifythe Interpolation Theorem:

Alternative Finite Craig Interpolation Theorem Suppose L0 and L1

are vocabularies, 'i is an Li–sentence (i 2 {0, 1}) and for all pseudo-finiteL0 \ L1–models A and all expansions Ai of A to an Li–model

A0 |= '0 ) A1 |= '1.

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Then there is an L0\L1–sentence ✓ such that for all pseudo-finite L0\L1–models A and all expansions Ai of A to an Li–model

A0 |= '0 ! ✓ and A1 |= ✓ ! '1.

The Finite Souslin–Kleene Interpolation for ⌃11 now follows: If K is

pseudo-finite ⌃11 and {A 62 K : A pseudo-finite } is pseudo-finite ⌃1

1, thenK is first–order definable on pseudo-finite structures. On finite models theSouslin-Kleene Theorem for PC (and ⌃1

1) is false, but the analogous question,whether NP\co-NP=PTIME is a famous open problem. On all models theSouslin-Kleene Theorem for PC (and ⌃1

1) holds.

5 A Lindstrom Theorem

On all structures first order logic has a nice characterization, the LindstromTheorem [13]: first-order logic is the maximal logic with the Compactness andthe Lowenheim-Skolem Theorems. Although first order logic seems very robuston finite structures, too, no analogous characterization in terms of model theo-retic properties is known. Indeed, no model theoretic properties comparable toCompactness and the Lowenheim-Skolem Theorems are known for first orderlogic on finite structures. As we have seen above, this is not so in pseudo-finitestructures. In this section we investigate to what extent the Lindstrom Theoremholds on pseudo-finite structures.

For basic definitions concerning abstract logics we refer to [2]. We denote firstorder logic by L!!. The most commonly studied extensions of first-order logic onfinite structures are the fixed point logic FP and its extension L!

1!. Theselogics make perfect sense on pseudo-finite structures, too. However, pseudo-finite structures need not have the finite model property relative to these logics.Let us call a modelA L⇤-pseudo-finite, if every L⇤-sentence true inA is true ina finite structure. We may, and it would make sense, restrict to abstract logicsL⇤ where satisfaction is defined for L⇤-pseudo-finite structures. The pseudo-finite linear order !+Z+!⇤ satisfies a sentence of FP with no finite models, soit is not FP-pseudo-finite. On the other hand, the random graph is a pseudo-finite structure which is even L!

1!-pseudo-finite. An extreme case is the logicL⇤ with the quantifier ”There exists infinitely many”. Here only finite modelsare L⇤-pseudo-finite.

If L⇤ and L⇤⇤ are abstract logics, then L⇤ L⇤⇤ on finite models meansthat for every � 2 L⇤ there is �⇤ 2 L⇤⇤ such that

A |= � () A |= �⇤

for all finite A. If A is a relational structure in a language L and U 2 L, thenA(U) is the relativization of A to the set UA. The abstract logic L⇤ relativizesif for all � 2 L⇤ and all unary predicates U there is �(U) 2 L⇤ such that

A |= �(U) () A(UA) |= �

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for all L⇤-pseudo-finite models A. We assume our abstract logics all relativize.An abstract logic L⇤ satisfies the Finite Compactness Theorem if when-

ever T is a theory in L⇤, every finite subset of which has a finite model, then Thas a model. W.l.o.g. we may take this model to be L⇤-pseudo-finite. L⇤ satis-fies the Finite Lowenheim–Skolem Theorem if whenever T is a countableL⇤-theory and T has an L⇤-pseudo-finite model, then T has a countable model.W.l.o.g. we may take this model to be L⇤-pseudo-finite.

First-order logic satisfies both Finite Compactness Theorem and Finite Lowenheim–Skolem Theorem. Any abstract logic which satisfies the Compactness Theoremon all structures, satisfies also, a fortiori, the Finite Compactness Theorem, andsome infinite models are L⇤-pseudo-finite. Likewise, any abstract logic satisfyingthe Lowenheim–Skolem Theorem (every countable theory with a model has acountable model) satisfies the Finite Lowenheim–Skolem Theorem. This infer-ence shows that the fixed point logic FP satisfies the Finite Lowenheim–SkolemTheorem. The generalized quantifier

QBAuxy�(x, y,~z)

which says that �(x, y,~z) defines on the set of elements satisfying (u) a Booleanalgebra with an even (or infinite) number of atoms, satisfies the CompactnessTheorem on all structures, (assuming GCH) [17]. Hence it satisfies the FiniteCompactness Theorem, too. The logic L!!(QBA) is an example of a logic whichsatisfies the Compactness Theorem and which extends the first-order logic evenon finite structures.

Theorem 3 Suppose L⇤ is an abstract logic such that L!! L⇤ on finite mod-els. If L⇤ satisfies the Finite Compactness Theorem and the Finite Lowenheim-Skolem Theorem, then L⇤ L!! on finite models.

Proof. The proof is an adaption of the original proof of Lindstrom’s Theorem[13]. Suppose ✓ 2 L⇤ is not first order definable on finite models. Because L⇤

satisfies the Finite Compatcness Theorem, we may assume ✓ has finite vocabu-lary ⌧ (see [2, page 79]). Let

�m,kn (x1, ..., xm), n < q(m, k)

be the list of all (up to equivalence) first order formulas of quantifier rank kwith m free variables. Let

�k =^

{�0,kn : n < q(0, k), |=FIN ✓ ! �0,kn }.

Clearly, |=FIN ✓ ! �k. Thus we may choose a finite Ak |= �k ^ ¬✓. Let

k =^

{�0,kn : n < q(0, k),Ak |= �0,kn }.

If |=FIN ✓ ! ¬ k, then |=FIN �k ! ¬ k, contrary to Ak |= �k^ k. Thereforewe can choose a finite Bk |= ✓ ^ k. Thus Ak and Bk satisfy the same firstorder sentences of quantifiers rank k, i.e. Ak ⌘k Bk.

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Let U0 and U1 be new unary predicate symbols. By assumption, the rela-tivisation ✓(U0) of ✓ to U0 and the relativisation (¬✓)(U1) of ¬✓ to U1 are in L⇤.Let P be a new binary predicate and F a new ternary predicate. Let T be theL⇤-theory consisting of the following sentences:

1. ✓(U0)

2. (¬✓)(U1)

3. “P is a linear order”

4. “F codes a P -ranked back-and-forth system1 in the language L betweenU0 and U1.”

5. 9x19x0P (x1, x0)

6. 8xn...8x0[(P (xn, xn�1)^ ...^ P (x1, x0)) ! 9xn+1P (xn+1, xn)] for all n <!.

7. ¬� for each L⇤-sentence in the vocabulary ⌧ .

If T0 ✓ T is finite, we can construct a model C for T0 from Ak and Bk for asu�ciently large k. The model C is finite, hence satisfies also the sentences incondition 7. By the Finite Compactness Theorem there is a model C of thewhole T . Because of the conditions 7, C is L⇤-pseudo-finite. By the FiniteLowenheim.Skolem Theorem, we can choose C to be countable. Let C0 be therestriction of C to UC

0 . Respectively, let C1 be the restriction of C to UC1 .

Note that C0 and C1 are L⇤-pseudo-finite. Now C0 and C1 are countablepartially isomorphic models, hence isomorphic. But C0 |= ✓ and C1 |= ¬✓, acontradiction. 2

A corollary of the theorem is: L⇤ L!! on pseudo-finite models if and onlyif L⇤ satisfies the Finite Compactness Theorem, the Finite Lowenheim-SkolemTheorem, and every pseudo-finite model is L⇤-pseudo-finite.

The above theorem can be interpreted as saying that even though first-orderlogic seems very weak on finite structures, we cannot hope to find logic whichis stronger on finite models and which would admit both compactness and exis-tence of countably infinite models in the proximity of finite models. Somethinghas to be given up. One possibility is to give up countably infinite models. Letus say that an abstract logic L⇤ satisfies the Ultraproduct CompactnessTheorem if whenever T is a theory in L⇤, every finite subset of which hasa finite model, then T has a UP-finite model A. W.l.o.g. A may be takento be L⇤-pseudo-finite. L⇤ satisfies the Ultraproduct Lowenheim–SkolemTheorem if whenever T is an L⇤-theory of cardinality 2@0 and T has an

1This is as in the original proof by Lindstrom [13]. A P -ranked back-and-forth systemin the language L between U0 and U1 is a sequence I = hIi : i 2 P i of non-empty sets ofpartial isomorphimsm relative to the language L. If p 2 Ii and P (j, i), then for every a 2 U0

there is b 2 U1 such that p [ {ha, bi} 2 Ij . Also, for every b 2 U1 there is a 2 U0 such thatp [ {ha, bi} 2 Ij .

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L⇤-pseudo-finite UP-finite model, then T has a UP-finite model of cardinality 2@0 . W.l.o.g. this model can be chosen to be L⇤-pseudo-finite. First-orderlogic satisfies the Ultraproduct Compactness Theorem and the UltraproductLowenheim–Skolem Theorem.

Theorem 4 Assume the Continuum Hypothesis. Suppose L⇤ is an abstractlogic such that FO L⇤ on finite models. If L⇤ satisfies the UltraproductCompactness Theorem and the Ultraproduct Lowenheim-Skolem Theorem, thenL⇤ L!! on finite structures.

Proof. The proof is like that of Theorem 3. At the end we note that C0 andC1 are @1-saturated elementary equivalent models of cardinality 2! = @1,hence isomorphic. 2

We can eliminate the use of Continuum Hypothesis in the above theorem,if we assume instead absoluteness for countably closed forcing, i.e. truthof L⇤ is preserved by countably closed forcing. More exactly, if � 2 L⇤, A is amodel and P is countably closed forcing, then ||�P [A |= �].

Theorem 5 Suppose L⇤ is an abstract logic such that L!! L⇤ on finitestructures and L⇤ is absolute for countably closed forcing. If L⇤ satisfies theUltraproduct Compactness Theorem, then L⇤ L!! on finite models.

Proof. The proof is again like that of Theorem 3. At the end we note that C0

and C1 are @1-saturated and elementary equivalent. If we collapse canonicallytheir cardinalities to @1, they are still @1-saturated and elementary equivalent,hence isomorphic. This leads to a contradiction with the absoluteness assump-tion. 2

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[17] Saharon Shelah. Models with second-order properties. I. Boolean algebraswith no definable automorphisms. Annals of Mathematical Logic, 14 (1978),no. 1, 57–72.

[18] Gyorgy Turan. On the definability of properties of finite graphs, DiscreteMathematics, 49, 1984, 3, 291–302

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