SOME POSITIVE RESULTS IN THE CONTEXT OF UNIVERSAL MODELS 3

• For every A ⊆ Nα of size κ and type Γ(x) consistent with the complete

theory of (Nα, a)a∈A, Γ(x) is realised in Nα+1.

To do the construction at the successor stage α + 1 we simply apply Lemma 2.2 to

the model Nα. At the end let M =

ακ+

Nα.

2.3

Saturation is not necessary for universality. A weaker notion that still suﬃces

is that of a special model: a model M of size κ is a special model if it is the union

of an elementary chain Mλ : λ λ∗ such that each λ is a cardinal and Mλ is

λ+-saturated. By deﬁnition, saturated models are special. The opposite is not

true. Relationship between saturation, speciality and universality is given by the

following:

Theorem 2.4. Every saturated model is special and every special model is

universal.

Proof. The ﬁrst sentence follows by deﬁnition. Suppose now that M is the

union of a specializing chain Mi : i

i∗

where Mi+1 is κi-saturated for some

cardinals κi increasing to κ, which is the size of M. We may without loss of

generality assume that this chain is continuous. Let N be a model of T of size κ,

enumerated as {xα : α κ}. By induction on α we choose yα ∈ M such that

xα → yα is an elementary embedding. We choose yαs in blocks of κi for i i∗,

that is the induction is on i, so that α κi =⇒ yα ∈ Mi.

Suppose that yα : α κi have been chosen. Choose yα ∈ Mi+1 for α ∈

[κi, κi+1) by induction on α. Suppose that α κi+1 and yβ for β α have been

chosen. We use a modiﬁcation of Cantor’s idea from the proof of the uniqueness of

the rationals: let Γ(x) be the type of xα in (N, xβ)βα. Therefore Γ is a type of

size κi+1 in (Mi+1, yβ)βα. By the saturation of Mi+1 we can ﬁnd yα ∈ Mi+1

which realises this type and the induction continues.

2.4

Theorem 2.5. (Morley and Vaught, [10]) Suppose that κ =

2κ

is uncount-

able2.

Then there is a special model of T of size κ.

Proof. If κ =

λ+

then by the assumption

2λ

= κ and hence by Theorem 2.3

there is a saturated model of T of size κ. Suppose then that κ is a limit cardinal.

Our assumptions allow us to choose an increasing sequence κi : i

i∗

of inﬁnite

cardinals with limit κ, and such that

2κi

= κi+1. Then we build an elementary chain

Mi : i

i∗

by starting with any model M0 of T of size κ0, and applying Lemma

2.2 at successor stages to get a model Mi+1 of size κi+1 which is

κi+-saturated.

Letting M =

ii∗

Mi we obtain a special model as required.

2.5

Conclusion 2.6. Every countable ﬁrst order theory T has a universal model

of size κ for every κ ℵ0 satisfying 2κ = κ.

Results presented above can be found in Chapter V of [1]. We also quote a

selection of theorems which show that some assumptions on the kind of theories

and on cardinal arithmetic are necessary for this conclusion.

Theorem 2.7. (1) (Hausdorﬀ, [5]) There exists a saturated linear order of size

κ ℵ0 iﬀ κ =

κκ.

(2) (Shelah, see [7] for a proof) Suppose that V is a model of GCH in which

κ is a regular cardinal, and let G be V -generic for the Cohen forcing which adds

2The

role of uncountability here is that we need κ to be larger than the size of T .

3 3

• For every A ⊆ Nα of size κ and type Γ(x) consistent with the complete

theory of (Nα, a)a∈A, Γ(x) is realised in Nα+1.

To do the construction at the successor stage α + 1 we simply apply Lemma 2.2 to

the model Nα. At the end let M =

ακ+

Nα.

2.3

Saturation is not necessary for universality. A weaker notion that still suﬃces

is that of a special model: a model M of size κ is a special model if it is the union

of an elementary chain Mλ : λ λ∗ such that each λ is a cardinal and Mλ is

λ+-saturated. By deﬁnition, saturated models are special. The opposite is not

true. Relationship between saturation, speciality and universality is given by the

following:

Theorem 2.4. Every saturated model is special and every special model is

universal.

Proof. The ﬁrst sentence follows by deﬁnition. Suppose now that M is the

union of a specializing chain Mi : i

i∗

where Mi+1 is κi-saturated for some

cardinals κi increasing to κ, which is the size of M. We may without loss of

generality assume that this chain is continuous. Let N be a model of T of size κ,

enumerated as {xα : α κ}. By induction on α we choose yα ∈ M such that

xα → yα is an elementary embedding. We choose yαs in blocks of κi for i i∗,

that is the induction is on i, so that α κi =⇒ yα ∈ Mi.

Suppose that yα : α κi have been chosen. Choose yα ∈ Mi+1 for α ∈

[κi, κi+1) by induction on α. Suppose that α κi+1 and yβ for β α have been

chosen. We use a modiﬁcation of Cantor’s idea from the proof of the uniqueness of

the rationals: let Γ(x) be the type of xα in (N, xβ)βα. Therefore Γ is a type of

size κi+1 in (Mi+1, yβ)βα. By the saturation of Mi+1 we can ﬁnd yα ∈ Mi+1

which realises this type and the induction continues.

2.4

Theorem 2.5. (Morley and Vaught, [10]) Suppose that κ =

2κ

is uncount-

able2.

Then there is a special model of T of size κ.

Proof. If κ =

λ+

then by the assumption

2λ

= κ and hence by Theorem 2.3

there is a saturated model of T of size κ. Suppose then that κ is a limit cardinal.

Our assumptions allow us to choose an increasing sequence κi : i

i∗

of inﬁnite

cardinals with limit κ, and such that

2κi

= κi+1. Then we build an elementary chain

Mi : i

i∗

by starting with any model M0 of T of size κ0, and applying Lemma

2.2 at successor stages to get a model Mi+1 of size κi+1 which is

κi+-saturated.

Letting M =

ii∗

Mi we obtain a special model as required.

2.5

Conclusion 2.6. Every countable ﬁrst order theory T has a universal model

of size κ for every κ ℵ0 satisfying 2κ = κ.

Results presented above can be found in Chapter V of [1]. We also quote a

selection of theorems which show that some assumptions on the kind of theories

and on cardinal arithmetic are necessary for this conclusion.

Theorem 2.7. (1) (Hausdorﬀ, [5]) There exists a saturated linear order of size

κ ℵ0 iﬀ κ =

κκ.

(2) (Shelah, see [7] for a proof) Suppose that V is a model of GCH in which

κ is a regular cardinal, and let G be V -generic for the Cohen forcing which adds

2The

role of uncountability here is that we need κ to be larger than the size of T .

3 3