SOME POSITIVE RESULTS IN THE CONTEXT OF UNIVERSAL MODELS 11

that the theory is amenable. As an exercise, the reader may check that natural def-

initions of approximation families for this class will fail to be workable. However,

in [12] Shelah proved that it is consistent that there is a universal linear order of

size ℵ1 in a model where CH fails 5. In the terminology of the rest of this paper,

this shows that the theory of dense linear orders is amenable at ℵ1. Note that the

method of the proof uses oracle-proper forcing, which is a technique limited to ℵ1.

Again by high non-amenability we conclude that the analogue of [12] cannot be

obtained at cardinals larger than ℵ1, and at the same time that amenability is not

implied by amenability at ℵ1.

3.4. Amenability at ℵ1. Our presentation so far leaves open the question if

every ‘reasonable’ theory T is amenable at ℵ1. Namely, let D(T ) denote the set of

complete types over the empty set in ﬁnitely many variables. It is known that if

this set is uncountable then it has to have the cardinality of the continuum, and

it is easy to see that every type in D(T ) must be realised in the universal model.

Therefore if D(T ) is uncountable there cannot be a universal model of T in ℵ1 if

CH fails. It remains to ask what happens if D(T ) is countable, namely if it is

possible that every T with D(T ) countable is amenable at ℵ1. A negative answer

to this is given in §1 of [7], where there is an example of a theory with countable

D(T ) which has a universal model at ℵ1 iﬀ CH holds.

Conclusion. In conclusion, we have presented the methods that are currently

available for making the universality number at ℵ1 small while failing CH. These

methods come with amalgamation-type requirements on the theory in question and

we have discussed the prototypical examples of theories that satisfy or not these

amalgamation properties. We have discussed the division amenability/high non-

amenability deﬁned in terms of the ability of a theory to have a small universality

number in circumstances where the relevant instances of GCH are violated. We end

by mentioning that there is a programme of characterising this division in syntactic

terms i.e. in terms that do not discuss models of a theory but properties of its types

and formulae. The present state of this programme is described in the Introduction

to [4].

References

[1] C. Chang and H.J. Keisler, Model Theory (3. edition), Studies in Logic, vol. 73, North-

Holland, (Amsterdam-New York-Oxford-Tokyo), 650 pp.,(1990).

[2] M. Dˇ zamonja, Club guessing and the universal models, Notre Dame Journal for Formal Logic,

vol. 46, No. 3, pp. 283-300, (2005).

[3] M. Dˇ zamonja and S. Shelah, On the existence of universals, Archive for Mathematical Logic,

vol. 43, pp. 901-936, (2004).

[4] M. Dˇ zamonja and S. Shelah, On properties of theories which preclude the existence of uni-

versal models, Annals of Pure and Applied Logic, vol. 139, no. 1-3, pp. 280-302, (2006).

[5] F. Hausdorﬀ, Grundz¨ uge einer Theorie der geordneten Mengenlehre, Mathematische An-

nalen, vol. 65, pp. 435-505. (In German) (1908).

[6] M. Kojman, Representing embeddability as set inclusion, Journal of the London Mathematical

Society (2nd series), vol 58, no. 185, Part 2, pp. 257-270 (1998).

[7] M. Kojman and S. Shelah, Non-existence of Universal Orders in Many Cardinals, Journal

of Symbolic Logic, vol. 57, pp. 875-891, (1992).

5

Unfortunately, the published proof of this very important result is very sketchy and the

author of this article would not claim that she understands it completely. Shelah has kindly

explained some of the major missing and inaccurate points in a recent conversation and promised

to make a written version available.

11 11

that the theory is amenable. As an exercise, the reader may check that natural def-

initions of approximation families for this class will fail to be workable. However,

in [12] Shelah proved that it is consistent that there is a universal linear order of

size ℵ1 in a model where CH fails 5. In the terminology of the rest of this paper,

this shows that the theory of dense linear orders is amenable at ℵ1. Note that the

method of the proof uses oracle-proper forcing, which is a technique limited to ℵ1.

Again by high non-amenability we conclude that the analogue of [12] cannot be

obtained at cardinals larger than ℵ1, and at the same time that amenability is not

implied by amenability at ℵ1.

3.4. Amenability at ℵ1. Our presentation so far leaves open the question if

every ‘reasonable’ theory T is amenable at ℵ1. Namely, let D(T ) denote the set of

complete types over the empty set in ﬁnitely many variables. It is known that if

this set is uncountable then it has to have the cardinality of the continuum, and

it is easy to see that every type in D(T ) must be realised in the universal model.

Therefore if D(T ) is uncountable there cannot be a universal model of T in ℵ1 if

CH fails. It remains to ask what happens if D(T ) is countable, namely if it is

possible that every T with D(T ) countable is amenable at ℵ1. A negative answer

to this is given in §1 of [7], where there is an example of a theory with countable

D(T ) which has a universal model at ℵ1 iﬀ CH holds.

Conclusion. In conclusion, we have presented the methods that are currently

available for making the universality number at ℵ1 small while failing CH. These

methods come with amalgamation-type requirements on the theory in question and

we have discussed the prototypical examples of theories that satisfy or not these

amalgamation properties. We have discussed the division amenability/high non-

amenability deﬁned in terms of the ability of a theory to have a small universality

number in circumstances where the relevant instances of GCH are violated. We end

by mentioning that there is a programme of characterising this division in syntactic

terms i.e. in terms that do not discuss models of a theory but properties of its types

and formulae. The present state of this programme is described in the Introduction

to [4].

References

[1] C. Chang and H.J. Keisler, Model Theory (3. edition), Studies in Logic, vol. 73, North-

Holland, (Amsterdam-New York-Oxford-Tokyo), 650 pp.,(1990).

[2] M. Dˇ zamonja, Club guessing and the universal models, Notre Dame Journal for Formal Logic,

vol. 46, No. 3, pp. 283-300, (2005).

[3] M. Dˇ zamonja and S. Shelah, On the existence of universals, Archive for Mathematical Logic,

vol. 43, pp. 901-936, (2004).

[4] M. Dˇ zamonja and S. Shelah, On properties of theories which preclude the existence of uni-

versal models, Annals of Pure and Applied Logic, vol. 139, no. 1-3, pp. 280-302, (2006).

[5] F. Hausdorﬀ, Grundz¨ uge einer Theorie der geordneten Mengenlehre, Mathematische An-

nalen, vol. 65, pp. 435-505. (In German) (1908).

[6] M. Kojman, Representing embeddability as set inclusion, Journal of the London Mathematical

Society (2nd series), vol 58, no. 185, Part 2, pp. 257-270 (1998).

[7] M. Kojman and S. Shelah, Non-existence of Universal Orders in Many Cardinals, Journal

of Symbolic Logic, vol. 57, pp. 875-891, (1992).

5

Unfortunately, the published proof of this very important result is very sketchy and the

author of this article would not claim that she understands it completely. Shelah has kindly

explained some of the major missing and inaccurate points in a recent conversation and promised

to make a written version available.

11 11