Abstract Wilkie proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M})$ is the pentagon lattice ${\mathbf N}_5$ . This theorem implies that every countable nonstandard ${\mathcal M}$ has an elementary cofinal extension ${\mathcal N}$ such that $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$ . It is proved here that whenever ${\mathcal M} \prec {\mathcal N} \models \mathsf {PA}$ and $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$ , then ${\mathcal N}$ must be either an end or a cofinal extension of ${\mathcal M}$ . In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models \mathsf {PA}^*$ such that $\operatorname {\mathrm {Lt}}({\mathcal N}^* / {\mathcal M}^*) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$ .

Abstract For each infinite cardinalκ, the set of algebraic hypergraphs having chromatic number no larger thanκis decidable.

Abstract Previous results about n-grids with acceptable colorings are extended here to n-indexed hyperspaces, which are structures ${\cal A} = \left( {A;{E_0},{E_1}, \ldots ,{E_{n - 1}}} \right)$ , where each ${E_i}$ is an equivalence relation on A.

Abstract D’Aquino, Knight, and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of ɷ1-like models of PA.

Abstract If ${\cal M},{\cal N}$ are countable, arithmetically saturated models of Peano Arithmetic and ${\rm{Aut}}\left( {\cal M} \right) \cong {\rm{Aut}}\left( {\cal N} \right)$ , then the Turing-jumps of ${\rm{Th}}\left( {\cal M} \right)$ and ${\rm{Th}}\left( {\cal N} \right)$ are recursively equivalent.

Abstract Letκbe the cardinality of some saturated model of Peano Arithmetic. There is a set of ${2^{{\aleph _0}}}$ saturated models of PA, each having cardinalityκ, such that wheneverMandNare two distinct models from this set, then Aut( ${\cal M}$ ) ≇ Aut ( $${\cal N}$$ ).

Abstract A structure is an n-grid if each Ei, is an equivalence relation on A and whenever X and Y are equivalence classes of, respectively, distinct Ei, and Ej, then X ∩ Y is finite. A coloring χ: A → n is acceptable if whenever X is an equivalence class of Ei, then {x ∈ X: χ(x) = i} is finite. If B is any set, then the n-cube Bn = (Bn; E0, …, En−1) is considered as an n-grid, where the equivalence classes of Ei are the lines parallel to the i-th coordinate axis. Kuratowski [9], generalizing the n = 3 case proved by Sierpihski [17], proved that ℝn has an acceptable coloring iff 2ℵ0 ≤ ℵn−2. The main result is: if is a semialgebraic (i.e., first-order definable in the field of reals) n-grid, then the following are equivalent: (1) if embeds all finite n-cubes, then 2ℵ0 ≤ ℵn−2: (2) if embeds ℝn, then 2ℵ0 ≤ ℵn−2; (3) has an acceptable coloring.

Abstract Bounded lattices (that is lattices that are both lower bounded and upper bounded) form a large class of lattices that include all distributive lattices, many nondistributive finite lattices such as the pentagon lattice N5. and all lattices in any variety generated by a finite bounded lattice. Extending a theorem of Paris for distributive lattices, we prove that if L is an ℵ0-algebraic bounded lattice, then every countable nonstandard model of Peano Arithmetic has a cofinal elementary extension such that the interstructure lattice Lt( / ) is isomorphic to L.

Abstract For a model of Peano Arithmetic, let Lt( ) be the lattice of its elementary substructures, and let Lt+ ( ) be the equivalenced lattice (Lt( ),≅ ), where ≅ is the equivalence relation of isomorphism on Lt( ). It is known that Lt+( ) is always a reasonable equivalenced lattice. Theorem. Let L be a finite distributive lattice and let (L, E) be reasonable. If 0 is a nonstandard prime model of PA, then 0 has a cofinal extension such that Lt+( ) ≅ (L,E). A general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices.

The theme of this paper is the generalization of theorems about partitions of the sets of points and lines of finite-dimensional Euclidean spaces ℝdto vector spaces over ℝ of arbitrary dimension and, more generally still, to arbitrary vector spaces over other fields so long as these fields are not too big. These theorems have their origins in the following striking theorem of Sierpiński [12] which appeared a half century ago. Sierpiński's Theorem.The Continuum Hypothesis is equivalent to: There is a partition{X, Y, Z}ofℝ3such that if ℓ is a line parallel to the x-axis[respectively:y-axis, z-axis]then X∩ℓ[respectively:Y∩ℓ, Z∩ℓ]is finite. The history of this theorem and some of its subsequent developments are discussed in the very interesting article by Simms [13]. Sierpiński's Theorem was generalized by Kuratowski [9] to partitions of ℝn+2inton+ 2 sets obtaining an equivalence with . The geometric character that Sierpiński's Theorem and its generalization by Kuratowski appear to have is bogus, since the lines parallel to coordinate axes are essentially combinatorial, rather than geometric, objects. The following version of Kuratowski's theorem emphasizes its combinatorial character. Kuratowski's Theorem.Let n < ω and A be any set. Then∣A∣ ≤ ℵnif and only if there is a partition P:An+2→n+ 2such that if i≤n+ 1and ℓ is a line parallel to the i-th coordinate axis, then{x∈ℓ:P(x) =i}is finite.

Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is. For any structure , let Aut( ) be its automorphism group. There are groups which are not isomorphic to any model = (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut(( , <)), must be torsion-free. However, as will be proved in this paper,if(A, <)is a linearly ordered set and G is a subgroup of Aut((A, <)),then there are models ofPAsuch that Aut( ) ≅G. If is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets ofAbe the basic open subgroups. If ′ is an expansion of , then Aut( ′) is a closed subgroup of Aut( ). Conversely, for any closed subgroupG≤ Aut( ) there is an expansion ′ of such that Aut( ′) =G. Thus, if is a model of PA, then Aut( ) is not only a subgroup of Aut((N, <)), but it is even aclosedsubgroup of Aut((N, ′)). There is a characterization, due to Cohn [2] and to Conrad [3], of those groupsGwhich are isomorphic to closed subgroups of automorphism groups of linearly ordered sets.

Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed. Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If λ is a regular cardinal and is aλ-saturated model of PA such that ∣M∣ >λ, then has an elementary extension of the same cardinality which is alsoλ-saturated and which, in addition, is rather classless. The construction in [10] produced a model for which cf( ) =λ+. We asked in Question 5.1 of [10] what other cofinalities could such a model have. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10]. Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed.

The κ-isomorphism property (IPκ) for nonstandard universes was introduced by Henson in [4]. There has been some recent effort aimed at more fully understanding this property. Jin and Shelah in [7] have shown that for κ < ⊐ω, IPκ is equivalent to what we will refer to as the κ-resplendence property. Earlier, in [6], Jin asked if IPκ is equivalent to IPℵ0 plus κ-saturation. He answered this question positively for κ = ℵ1. In this note we extend this answer to all κ. We also extend the result of Jin and Shelah to all κ. (Jin also observed this could be done.) In order to strike a balance between the generalities of model theory and the specifics of nonstandard analysis, we will consider models of Zermelo set theory with the Axiom of Choice; we denote this theory by ZC. The axioms of ZC are just those of ZFC but without the replacement scheme. Thus, among the axioms of ZC are the power set axiom, the infinity axiom, the separation axioms and the axiom of choice. Let(V, E) ⊨ ZC. If a ∈ V, we let *a = {x ∈ V: (V,E) ⊨ x ∈ a}. In particular, i ∈ *ω iff i ∈ V and (V,E) ⊨ (i is a natural number). A subset A ⊆ V is internal if A = *a for some a ∈ V. The standard model of ZC consists of those sets of rank at most ω + ω. In other words, if we let V0 be the set of hereditarily finite sets and for n < ω, then (Vω, ∈) is the standard model of ZC, where Vω = ⋃n<ω. Vn.

Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if T is any consistent extension of CA + AC and cf (λ) > ℵ0, then T has a natural λ-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of . (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11]. A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.

Abstract In a nonstandard universe, the κ -saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ -Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ -sequence in F has a convergent λ -subsequence. We show that if κ < λ are uncountable regular cardinals and β α < λ whenever α < κ and β < λ then there is a κ -saturated nonstandard universe in which the hyperreal numbers have the λ -Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic.

For a structure let φ( ) be the number of nonisomorphic, countably infinite substructures of . The problem considered here, suggested by M. Pouzet, is that of characterizing those countable for which φ( ) ≤ ℵ0. In this paper we will deal exclusively with structures in a finite, binary relational language L. The characterization of those L-structures for which φ( ) ≤ ℵ0 (which turns out to be equivalent to ) is given in Theorem 3. It is the culmination of a three-step process. The first step, resulting in Theorem 1, shows that for a countable stable L-structure , φ( ) ≤ ℵ0 iff is cellular. (See Definition 0.1.) In the second step we consider linearly ordered sets = (A, ≤ ℵ0), and characterize in Theorem 2 the order types of those for which φ( ) ≤ ℵ0. Finally, in Theorem 3, we amalgamate Theorems 1 and 2 to get the classification of all countable L-structures for which φ( ) ≤ ℵ0.

Let L be a countable language which contains only constant and relation symbols but no function symbols. All theories considered here will be L-theories. A theory is coinductive if it can be axiomatized by a set of ∃∀ sentences, and a structure is coinductive if its theory is. The object of this paper is to show that coinductive ℵ0-categorical structures are especially simple. First, they are ω-stable with Morley rank ≤ 1; and second, they have simple algebraic closures, by which is meant that the algebraic closure of the union of two sets is the union of their algebraic closures. While these two properties do not characterize coinductive ℵ0-categorical structures, they do characterize those structures which are cellular. We will say that a countable structure is cellular if there is a finite subset A0 ⊆ A and there are equivalence relations E and F on A∖A0 such that the following hold: (1) There are only finitely many E-classes. (2) If C is an E-class and D an F-class, then ∣C ∩ D ∣ = 1. (3) If a0, a1, …, ak − 1, b0, b1, …, bk − 1 Є A, then 〈a0, a1, …,ak − 1〉 and 〈b0,b1, …, bk − 1〉, satisfy the same quantifier-free formulas provided that: (a) if i < k and either ai Є A0 or bi Є A0, then ai = bi; (b) if i < k, then ai; and bi are E-equivalent; and (c) if i, j < k, then ai; is F-equivalent to aj if bj is F-equivalent to bj. The main result of this paper is the following theorem. Theorem 1. If is coinductive and ℵ0-categorical, then is cellular. The results of this paper were obtained independently of similar results obtained by Lachlan which can be found in [4] and [5]. The proofs here are quite different.

In a model of Peano arithmetic, the isomorphism type of the multiplicative semigroup uniquely determines the isomorphism type of the additive semigroup. In fact, for any prime p of , the function x ↦ px is an isomorphism of the additive semigroup with the multiplicative subsemigroup of powers of p. It was observed by Jensen and Ehrenfeucht [3] that for countable models of PA, the isomorphism type of the additive semigroup (or even the additive group) determines the isomorphism type of the multiplicative semigroup. (See Theorem 3 below.) In this note we will show that the countability restriction cannot be dropped. First, we show (as Theorem 2) that for uncountable models of PA the isomorphism type of the additive group never determines the isomorphism type of the multiplicative semigroup. Our main result is Theorem 5 in which we show that the isomorphism type of the additive semigroup need not determine the isomorphism type of the multiplicative semigroup, thereby improving upon Harnik [2], where Theorem 5 is proved under the assumption of ♢. For completeness, a sketch of the proof of the Jensen-Ehrenfeucht result is included. The history of this paper begins with Nadel's question, asked in 1981, whether the countability assumption can be eliminated in the Jensen-Ehrenfeucht theorem. Soon afterwards, Nadel obtained the strong counterexample of Theorem 2, which applied to the additive group rather than the additive semigroup. A result of Pabion [8] shows that such a strong result is not possible for the additive semigroup.

Abstract No theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional iff its theory has the independence property iff its theory has the multi-order property.

The motivation for the results presented here comes from the following two known theorems which concern countable, recursively saturated models of Peano arithmetic. (1) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ. (See Theorem 10 of [1].) (2) if is a countable, recursively saturated model of PA, then is generated by a set of indiscernibles. (See [4].) It will be shown here that (1) and (2) can be amalgamated into a common generalization. (3) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ and which is generated by a set of indiscernibles. By way of contrast we will also get recursively saturated models of PA which fail to be resplendent and yet are generated by indiscernibles. (4) if is a countable, recursively saturated model of PA, then for each uncountable cardinal κ there is a κ-like recursively saturated generated by a set of indiscernibles. None of (1), (2) or (3) is stated in its most general form. We will make some comments concerning their generalizations. From now on let us fix a finite language L; all structures considered are infinite L-structures unless otherwise indicated.

This paper is a sequel to our earlier paper [2] entitled Saturation and simple extensions of models of Peano arithmetic. Among other things, we will answer some of the questions that were left open there. In §1 we consider the question of whether there are lofty models of PA which have no recursively saturated, simple extensions. We are still unable to answer this question; but we do show in that section that these models are precisely the lofty models which are not recursively saturated and which are κ-like for some regular κ. In §2 we use diagonal methods to produce minimal models of PA in which the standard cut is recursively definable, and other minimal models in which the standard cut is not recursively definable. In §3 we answer two questions from [2] by exhibiting countable models of PA which, in the terminology of this paper, are uniformly ω-lofty but not continuously ω-lofty and others which are continuously ω-lofty but not recursively saturated. We also construct a model (assuming ◇) which is not recursively saturated but every proper, simple cofinal extension of which is ℵ1-saturated. Finally, in §4 we answer another question from [2] by proving that for regular κ ≥ ℵ1; every κ-saturated model of PA has a κ-saturated proper, simple extension which is not κ+-saturated. Our notation and terminology are quite standard. Anything unfamiliar to the reader and not adequately denned here is probably defined in §1 of [2]. All models considered are models of Peano arithmetic.

The purpose of this paper is to study a formal system PA(Q2) of first order Peano arithmetic, PA, augmented by a Ramsey quantifier Q2 which binds two free variables. The intended meaning of Q2xx′φ(x, x′) is that there exists an infinite set X of natural numbers such that φ(a, a′) holds for all a, a′ Є X such that a ≠ a′. Such an X is called a witness set for Q2xx′φ(x, x′). Our results would not be affected by the addition of further Ramsey quantifiers Q3, Q4, …, Here of course the intended meaning of Qkx1 … xkφ(x1,…xk) is that there exists an infinite set X such that φ(a1…, ak) holds for all k-element subsets {a1, … ak} of X. Ramsey quantifiers were first introduced in a general model theoretic setting by Magidor and Malitz [13]. The system PA{Q2), or rather, a system essentially equivalent to it, was first defined and studied by Macintyre [12]. Some of Macintyre's results were obtained independently by Morgenstern [15]. The present paper is essentially self-contained, but all of our results have been directly inspired by those of Macintyre [12]. After some preliminaries in §1, we begin in §2 by giving a new completeness proof for PA(Q2). A by-product of our proof is that for every regular uncountable cardinal k, every consistent extension of PA(Q2) has a k-like model in which all classes are definable. (By a class we mean a subset of the universe of the model, every initial segment of which is finite in the sense of the model.)

Abstract Every ℵ0-categorical partially ordered set of finite width has a finitely axiomatizable theory. Every ℵ0-categorical partially ordered set of finite weak width has a decidable theory. This last statement constitutes a major portion of the complete (with three exceptions) characterization of those finite partially ordered sets for which any ℵ0-categorical partially ordered set not embedding one of them has a decidable theory.

Abstract This paper is primarily concerned with ℵ0-categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ0-categoricity. Among the latter are the following. Corollary 3.3. For every countable ℵ0-categorical there is a linear order of A such that ( , <) is ℵ0-categorical. Corollary 6.7. Every ℵ0-categorical theory of a partially ordered set of finite width has a decidable theory. Theorem 7.7. Every ℵ0-categorical theory of reticles has a decidable theory. There is a section dealing just with decidability of partially ordered sets, the main result of this section being Theorem 8.2. If (P, <) is a finite partially ordered set and KP is the class of partially ordered sets which do not embed (P, <), then Th(KP) is decidable iff KP contains only reticles.

A structure is recursive if the set of quantifier-free sentences in the complete diagram ⊿( ) of is recursive. It has been known for some time that every decidable theory has a recursive model. In fact, every decidable theory has a decidable model (that is a model such that ⊿( ) is recursive). In this paper we find other conditions which imply that a theory have a recursive model. In §1 we study the relation between an ℵ0-categorical theory T having a recursive model and the complexity of the quantificational hierarchy of that theory. We let ∃0 denote the set of quantifier-free sentences, and let ∃n÷1 denote the set of sentences beginning with an existential quantifier and having n alternations of quantifiers. (∀n is defined analogously.) Then we show that if T is an arithmetical ℵ0-categorical theory such that T ⋂ ∃n÷2 is Σn÷10 for each n < ω, then T has a recursive model. We show that this is a best possible result by giving an example of a ⊿n÷20 ℵ0-categorical theory T such that T ⋂ ∃n÷1 is recursive yet T has no recursive model. In §2 we consider the theory of trees. Ershov [1] had proved that every Σ10 theory of trees has a recursive model. We show this to be best possible by giving an example of a ⊿20 theory of trees which has no recursive model.

In this note we give a simple recursive axiomatization for the class of structures of type (ℶω ℵ0). This solves a problem of Vaught which is Problem 13 in the book [1] of Chang and Keisler. The same technique is used to get a recursive axiomatization for the class of κ-like structures where κ is strongly ω-inaccessible. Let us fix throughout some recursive first-order language L, and until further notice let us suppose that included in L is a distinguished unary predicate symbol U. For cardinals κ and λ with κ ≥ λ ≥ ℵ0, we say the structure has type (κ, λ) if card(A)= κ and card . Let K(κ, λ) be the class of all structures of type (κ, λ). For each ordinal α define 2ακby 20κ = κ, and 2ακ= ⋃ {2λ: λ = 2βκ for some β < α} when α > 0. Let Vaught proved the following theorem in [7]. Theorem (Vaught). Suppose a is a sentence such that for each n < ω there are κ, λ with κ > 2λn and a model of σ of type (κ, λ). Then whenever κ ≥ λ ≥ ℵ0, the sentence σ has a model of type (κ, λ).

In this paper we define by means of a partition property a decreasing sequence N = ‹Nα: α is an ordinal› of classes of ordinals. This property is a generalization of the nonexistence of special Aronszajn trees: the successor cardinal κ+ is in N0 iff there does not exist a special Aronszajn κ+-tree. The interest in the classes Nα stems from their applicability in model theory, in particular to that aspect of model theory dealing with ordered and two-cardinal models. A model is κ-like iff < is a linear ordering of A of cardinality κ but such that every proper initial segment has cardinality < κ. is α-ordered iff ≼ is a reflexive, linear ordering of some subset of A with order type α. The sequence N can be characterized by a first-order sentence σ in the following manner: The sentence σ has a κ-like α-ordered model iff κ ∉ Nα. This characterization will allow us to translate various independence statements regarding the sequence N to statements about the independence of transfer properties. We say that the transfer property κ → λ holds iff every first-order sentence which has a κ-like model also has a λ-like model. κ ⇸ λ is the negation of κ → λ.

Let < and ≼ be two distinguished binary relation symbols. A structure is κ-like iff is a linear ordering of A, card(A) = κ, and every proper initial segment of A has cardinality < κ. A structure is α-ordered iff is a (reflexive) linear ordering of type α with field a subset of A. We define when a cardinal κ is α-inaccessible. (In this paper, inaccessible always means weakly inaccessible.) The 0-inaccessible cardinals are just the inaccessible cardinals; if α > 0, then κ is α-inaccessible iff for each β < α, each closed, cofinal subset of κ contains a β-inaccessible. (The (1 + α)-inaccessibles are just the ρα cardinals of Mahlo.) This paper is concerned with the proof of the following theorem. Main Theorem. There is an elementary sentence σ with the property that whenever α is an ordinal and κ an infinite cardinal, then σ has an α-ordered κ-like model iff κ is not α-inaccessible. This theorem gives some additional answers to a question of Mostowski about languages with generalized quantifiers. Fuhrken [1] showed that this question is equivalent to the following one: For which cardinals κ and λ is it true that if an elementary sentence has a κ-like model, then it has a λ-like model? It is actually this question to which the theorem refers. The theorem limits the possible pairs κ, λ of cardinals which answer the question. In fact, if the question is generalized so as to permit sentences from some more extensive language, then the theorem still limits the possible answers. For a more thorough introduction to this problem, the reader is referred to the aforementioned article of Fuhrken as well as Keisler [2] and Vaught [6].

The main result of this paper is the following transfer theorem: If T is an elementary theory which has a κ-like model where κ is hyperinaccessible of type ω, then T has a λ-like model for each λ > card(T). Helling [3] obtained the same conclusion under the stronger hypothesis that κ is weakly compact. Fuhrken conjectured in [1] that the same conclusion would result if κ were merely inaccessible. (He also showed there the connection with a problem about generalized quantifiers.) Thus, our theorem lies properly between Helling's theorem and Fuhrken's conjecture. In [9] it is shown that this theorem is actually the best possible. This theorem and the other results of this paper were announced by the authors in [10]. In §1 we prove as Theorem 1 a slightly stronger form of the above theorem. This theorem is generalized in §2 to Theorem 2 which concerns theories which permit the omitting of types. The methods used in §§1 and 2 are also applicable to problems regarding Hanf numbers as well as to two-cardinal problems.