We will mainly be concerned with a result which refutes a stronger variant of a conjecture of Macpherson about finitely axiomatizable ω-categorical theories. Then we prove a result which implies that the ω-categorical stable pseudoplanes of Hrushovski do not have the finite submodel property.

Let's call a consistent first-order sentence without finite models an axiom of infinity. Can we somehow describe the axioms of infinity? Two standard examples are:

ϕ1: A first-order sentence which expresses that a binary relation < on a nonempty universe is transitive and irreflexive and that for every x there is y such that x < y.

ϕ2: A first-order sentence which expresses that there is a unique x such that, (0) for every y, s(y) ≠ x (where s is a unary function symbol),

and, for every x, if x does not satisfy (0) then there is a unique y such that s(y) = x.

Every complete theory T such that ϕ1 ϵ T has the strict order property (as defined in [10]), since the formula x < y will have the strict order property for T. Let's say that if Ψ is an axiom of infinity and every complete theory T with Ψ ϵ T has the strict order property, then Ψ has the strict order property.

Every complete theory T such that ϕ2 ϵ T is not ω-categorical. This is the case because a complete theory T without finite models is ω-categorical if and only if, for every 0 < n < ω, there are only finitely many formulas in the variables x1,…,xn, up to equivalence, in any model of T.

In a previous work with Antonio Bucciarelli, we introduced indexed linear logic as a tool for studying and enlarging the denotational semantics of linear logic. In particular, we showed how to define new denotational models of linear logic using symmetric product phase models (truth-value models) of indexed linear logic. We present here a strict extension of indexed linear logic for which symmetric product phase spaces provide a complete semantics. We study the connection between this new system and indexed linear logic.

We construct, assuming that there is no inner model with a Woodin cardinal but without any large cardinal assumption, a model Kc which is iterable for set length iterations, which is universal with respect to all weasels with which it can be compared, and (assuming GCH) is universal with respect to set sized premice.

We describe a general method how to construct from a prepositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described “implicitly” by polynomial size circuits.

As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory and hence, in particular, polynomially simulates the quantified prepositional calculus G and the -consequences of proved with one use of exponentiation. Furthermore, the soundness of iEF is not provable in . An iteration of the construction yields a proof system corresponding to T2 + Exp and, in principle, to much stronger theories.

1. We show that if p is a real type which is internal in a set Σ of partial types in a simple theory, then there is a type p′ interbounded with p, which is finitely generated over Σ, and possesses a fundamental system of solutions relative to Σ.

2. If p is a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σω, then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts on p while fixing Σ generically In case p is Σ-internal and T is stable, this is the binding group of p over Σ.

In this paper, we study the bi-isolation phenomena in the d.c.e. degrees and prove that there are c.e. degrees c1 < c2 and a d.c.e. degree d ∈ (c1, c2) such that (c1, d) and (d, c2) contain no c.e. degrees. Thus, the c.e. degrees between c1 and c2 are all incomparable with d. We also show that there are d.c.e. degrees d1 < d2 such that (d1, d2) contains a unique c.e. degree.

A formalized version of Kleene realizability for intuitionistic first-order arithmetic HA was axiomatically characterized by Troelstra (see [2, 3.2]) as follows: for an arbitrary HA-sentence φ, HA ⊢ ∃x(x realizes φ) if and only if HA + ECT0 ⊢ φ.

Many notions of realizability have been characterized in this fashion: see [2] or [3] for details. For some notions, for example extensional realizability, it is necessary to pass to an extension of HA: realizability in HA is characterized by deducibility from certain axioms in an extension of HA.

The present note is concerned with modified realizability, seen as interpretation for HA. From semantical considerations (see [4]) it follows that this interpretation can be constructed as a combination of three ingredients:

i) Kleene realizability;

ii) Kripke forcing over a 2-element linear order P;

iii) The Friedman translation [1].

This will be shown in section 2. The Friedman translation (in the way we use it) introduces a new propositional constant V; hence we move to an extension HA(V) of HA. We must then define Kleene realizability and forcing for the extended language. Now let (φ)v be the result of the Friedman translation applied to φ. We obtain, in HA, that the sentence saying that φ is modified-realizable, is equivalent to the sentence which says that the statement “(φ)v is Kleene-realizable” is forced in P (see section 2).

We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties.

If we have a complete decidable atomic theory with all types of the theory computable, we show that for every degree d with 0 < d < 0', there is a prime model with elementary diagram of degree d. Indeed, this is a corollary of the fact that if T is a complete decidable theory and L is a computable set of c.e. partial types of T, then for any degree d > 0, T has a d-decidable model omitting the nonprincipal types listed by L.

Employing the theory of Birkhoff polarities as a model of model theory yields an inductively defined dual structure which is a formalization of semantics and which allows for simple proofs of some new results for model theory.

Herzog and Rothmaler gave the following purely topological characterization of stable theories. (See the exercises 11.3.4 – 11.3.7 in [2]).

A complete theory T is stable iff for any model M and any extension M ⊂ B the restriction map S(B) → S(M) has a continuous section.

In fact, if T is stable, taking the unique non-forking extension defines a continuous section of S(B) → S(A) for all subsets A of B, provided A is algebraically closed in Teq. Herzog and Rothmaler asked, if, for stable T, there is a continuous section for any subset A of B. Or, equivalently, if for any A, S(acleq(A)) → S(A) has a continuous section.

This is an interesting problem, also for unstable T. Is it true that for any T and any set of parameters A the restriction map S(acl(A)) → S(A) has a continuous section? We answer the question by the following two theorems.

Theorem 1. Let A be a subset of a model of T. Assume that the Boolean algebra of acl(A)-definable formulas is generated by

• some countable set of formulas,

• all A–definable formulas,

• all formulas which are atomic over acl(A).

Then S(acl(A)) → S(A) has a continuous section.

The conditions of the theorems are satisfied if, for example, L and A are countable, or, if there are only countably many non-isolated types over acl(A).

Theorem 2. There is a theory of Morley rank 2 and Morley degree 1 such that S(acl(∅)) → S(∅) has no continuous section.

In this paper we introduce a new property of families of functions on the Baire space, called pseudo-dominating, and apply the properties of these families to the study of cardinal characteristics of the continuum. We show that the minimum cardinality of a pseudo-dominating family is min {τ, ∂}. We derive two corollaries from the proof: τ ≥ min{∂, u} and min{∂, τ} = min{∂, τσ}. We show that if a dominating family is partitioned into fewer that s pieces, then one of the pieces is pseudo-dominating. We finally show that u < g implies that every unbounded family of functions is pseudo-dominating, and that the Filter Dichotomy principle is equivalent to every unbounded family of functions being finitely pseudo-dominating.

A relation on a linearly ordered structure is called semi-bounded if it is definable in an expansion of the structure by bounded relations. We study ultimate behavior of semi-bounded relations in an ordered module M over an ordered commutative ring R such that M/rM is finite for all nonzero r ϵ R. We consider M as a structure in the language of ordered R-modules augmented by relation symbols for the submodules rM, and prove several quantifier elimination results for semi-bounded relations and functions in M. We show that these quantifier elimination results essentially characterize the ordered modules M with finite indices of the submodules rM. It is proven that (1) any semi-bounded k-ary relation on M is equal, outside a finite union of k-strips, to a k-ary relation quantifier-free definable in M, (2) any semibounded function from Mk to M is equal, outside a finite union of k-strips, to a piecewise linear function, and (3) any semi-bounded in M endomorphism of the additive group of M is of the form x ↦ σx, for some σ from the field of fractions of R.

For a stationary set S ⊆ ω1, and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E over ω1 ∖ S there exists a gap with no subgap that is E-Hausdorff.

A new type of chain condition, called polarized chain condition, is introduced. We prove that the iteration with finite support of polarized c.c.c. posets is again a polarized c.c.c. poset.

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity of reals. We define the class of “Schnorr trivial” reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members.

We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f ∈ [0,1] computes a non-computable subset of ℕ there is a non-total degree between Turing degrees a <Tb iff b is a PA degree relative to a; ⊆ 2ℕ is a Scott set iff it is the collection of f-computable subsets of ℕ for some f ∈ [0,1] of non-total degree; and there are computably incomparable f, g ∈ [0,1] which compute exactly the same subsets of ℕ. Proofs draw from classical analysis and constructive analysis as well as from computability theory.

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [34], [33], we give syntactical conditions necessary and sufficient for a relation to be intrinsically on a structure. We consider some examples of computable structures and intrinsically relations R. We also consider a general family of examples of intrinsically relations arising in computable structures of maximum Scott rank.

For three of the examples, the maximal well-ordered initial segment in a Harrison ordering, the superatomic part of a Harrison Boolean algebra, and the height-possessing part of a Harrison p-group, we show that the Turing degrees of images of the relation in computable copies of the structure are the same as the Turing degrees of paths through Kleene's . With this as motivation, we investigate the possible degrees of these paths. We show that there is a path in which ∅′ is not computable. In fact, there is one in which no noncomputable hyperarithmetical set is computable. There are paths that are Turing incomparable, or Turing incomparable over a given hyperarithmetical set. There is a pair of paths whose degrees form a minimal pair. However, there is no path of minimal degree.

Let n be a positive integer. By a βn-model we mean an ω-model which is elementary with respect to formulas. We prove the following βn-model version of Gödel's Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a βn-model of S, then there exists a βn-model of S + “there is no countable βn-model of S”. We also prove a βn-model version of Löb's Theorem. As a corollary, we obtain a βn-model which is not a βn+1-model.