A very important example of almost strongly minimal theories are the algebraically closed fields. A. Macintyre has shown [3] that every ω1-categorical field is algebraically closed. Therefore every ω1-categorical field is almost strongly minimal. It will be shown that not every ω1-categorical ring is almost strongly minimal.

Let R0 be the factor ring C[y/(y2), where C[y] is the ring of polynomials in the indeterminate y over the field of complex numbers and (y2) the ideal generated by y2 in C[y].

It is straightforward to prove that R0 has the following properties:

1. R0 is a commutative ring with identity.

2. R0 is of characteristic 0.

3. For every polynomial p(x) = ∑ a1x1 ∈ R0[x] with of ai2 ≠ 0 for some i > 0 there is an a ∈ R0 such that p(a) · p(a) = 0.

4. For all x, y ∈ R0 such that x2 = 0 and y ≠ 0 there exists a z ∈ R0 with y · z = x.

5. There is an x ≠ 0 such that x2 = 0.

These properties can be ∀∃-axiomatised in a countable first order logic (see [4]). Let T be the set of these sentences. With Theorem 7 we get that T is model-complete.

If R is a model of T then I shall denote {a ∈ R ∣ a2 = 0}.

The present report is a, somewhat lengthy, addendum to [4], where the elimination of cuts from derivations in sequent calculus for classical logic was studied ‘from the point of view of linear logic’. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives.

The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions.

We give a detailed description of the simulation of tq-reductions by means of reductions of the interpretation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative.

We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e.. by only one specific combination, and structural rules.

This paper concerns automorphisms of the computably enumerable sets. We prove two results relating semilow sets and prompt degrees via automorphisms, one of which is complementary to a recent result of Downey and Harrington. We also show that the property of effective simplicity is not invariant under automorphism, and that in fact every promptly simple set is automorphic to an effectively simple set. A major technique used in these proofs is a modification of the Harrington-Soare version of the method of Harrington-Soare and Cholak for constructing Δ30 automorphisms; this modification takes advantage of a recent result of Soare on the extension of “restricted” automorphisms to full automorphisms.

We prove the consistency result from the title. By forcing we construct a model of g = ℵ1, b = cf(Sym(ω)) = ℵ2.

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).

In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).

Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

Here, we present a category ${\mathbf {pEff}}$ which can be considered a predicative variant of Hyland's Effective Topos ${{\mathbf {Eff} }}$ for the following reasons. First, its construction is carried in Feferman’s predicative theory of non-iterative fixpoints ${{\widehat {ID_1}}}$ . Second, ${\mathbf {pEff}}$ is a list-arithmetic locally cartesian closed pretopos with a full subcategory ${{\mathbf {pEff}_{set}}}$ of small objects having the same categorical structure which is preserved by the embedding in ${\mathbf {pEff}}$ ; furthermore subobjects in ${{\mathbf {pEff}_{set}}}$ are classified by a non-small object in ${\mathbf {pEff}}$ . Third ${\mathbf {pEff}}$ happens to coincide with the exact completion of the lex category defined as a predicative rendering in ${{\widehat {ID_1}}}$ of the subcategory of ${{\mathbf {Eff} }}$ of recursive functions and it validates the Formal Church’s thesis. Hence pEff turns out to be itself a predicative rendering of a full subcategory of ${{\mathbf {Eff} }}$ .

In the early eighties, answering a question of A. Macintyre, J. H. Schmerl ([13]) proved that every countable recursively saturated structure, equipped with a function β encoding the finite functions, is the β-closure of an infinite indiscernible sequence. This result implies that every countably saturated structure, in a countable but not necessarily recursive language, is an Ehrenfeucht-Mostowski model, by which we mean that the structure expands, in a countable language, to the Skolem hull of an infinite indiscernible sequence (in the new language).

More recently, D. Lascar ([5]) showed that the saturated model of cardinality ℵ1 of an ω-stable theory is also an Ehrenfeucht-Mostowski model.

These results naturally raise the following problem: which (countable) complete theories have an uncountably saturated Ehrenfeucht-Mostowski model. We study a generalization of this question. Namely, we call ACI-model a structure which can be expanded, in a countable language L′, to the algebraic closure (in L′) of an infinite indiscernible sequence (in L′). And we try to characterize the λ-saturated structures which are ACI-models.

The main results are the following. First it is enough to restrict ourselves to ℵ1-saturated structures: if T has an ℵ1-saturated ACI-model then, for every infinite λ, T has a λ-saturated ACI-model. We obtain a complete answer in the case of stable theories: if T is stable then the three following properties are equivalent: (a) T is ω-stable, (b) T has an ℵ1-saturated ACI-model, (c) every saturated model of T is an Ehrenfeucht-Mostowski model. The unstable case is more complicated, however we show that if T has an ℵ1-saturated ACI-model then T doesn't have the independence property.

We continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].