It has been considered desirable by many set theorists to find maximality properties which state that the universe has in some sense “many sets”. The properties isolated thus far have tended to be consistent with each other (as far as we know). For example it is a widely held view that the existence of a supercompact cardinal is consistent with the axiom of determinacy holding in L(R). This consistency has been held to be evidence for the truth of these properties. It is with this in mind that the first author suggested the following:

Maximality Principle If P is a partial ordering and G ⊆ P is a V-generic ultrafilter then either

• a) there is a real number r ∈ V [G] with r ∉ V, or

• b) there is an ordinal α such that α is a cardinal in V but not in V[G].

This maximality principle applied to garden variety partial orderings has startling results for the structure of V.

For example, if for some , then P = 〈{p: p ⊆ κ, ∣p∣ < κ}, ⊆〉 neither adds a real nor collapses a cardinal. Thus from the maximality principle we can deduce that the G. C. H. fails everywhere and there are no inaccessible cardinals. (Hence this principle contradicts large cardinals.) Similarly one can show that there are no Suslin trees on any cardinal κ. These consequences help justify the title “maximality principle”.

Since the maximality principle implies that the G. C. H. fails at strong singular limit cardinals it has consistency strength at least that of “many large cardinals”. (See [M].) On the other hand it is not known to be consistent, relative to any assumptions.

The uniform halting problem (UH) can be stated as follows.

Give a decision procedure which for any given Turing machine (TM) will decide whether or not it has an immortal instantaneous description (ID).

An ID is called immortal if it has no terminal successor. As it is generally the case in the literature (see e.g. Minsky [3, p. 118]) we assume that in an ID the tape must be blank except for some finite numbers of squares. If we remove this restriction the UH becomes the immortality problem (IP). The UH should not be confused with the initialised uniform halting problem (whether or not a TM has an immortal ID when started in a specified state) which can easily be shown to be undecidable (see e.g. Minsky [3, p. 151]).

If the bounded proper forcing axiom BPFA holds and ω1 = ω1L, then there is a lightface Σ31 well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of “David's trick.” We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ41 for many “consistently locally certified” relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ2 stable ordinals of L.

There are two origins for first-order theories. One type of theory arises by generalizing the common features of a number of different structures, e.g. the theory of groups, and formulating a set of axioms to encode these common features. Here the set of axioms is well understood, frequently it is finite or at least recursive, but there usually is no clear understanding of all the logical consequences of these axioms. The second type of theory arises by considering the set, T = Th(A), of all sentences true in a fixed structure A,3 e.g. the theory of arithmetic (N, +, 0) or the theory of the field of complex numbers (alias: the theory of algebraically closed fields of characteristic zero). The second case gives little more insight as to the truth in A (i.e. membership in T) of a given sentence ∅. But it does guarantee that for a given sentence ∅, either ∅ or ¬∅ is in T, that is, that T is a complete theory. When does a theory T of the first type, i.e. with well-understood axioms, posses this completeness property? An obvious sufficient condition is that T be secretly of the second type, that it have only one model, or, in jargon, T is categorical. Unfortunately (or fortunately depending on your point of view) for any theory with an infinite model, the Löwenheim-Skolem theorem shows this to be impossible: The theory has a model in every infinite power. In the mid-50's Łoś and Vaught discovered that if a theory T with no finite models is categorical in some infinite power α (all models with cardinality α are isomorphic) then T is complete. We will be dealing below with countable complete theories and will assume, unless stated to the contrary, that each theory has no finite models.

In this paper we continue, from [2], the development of provably recursive analysis, that is, the study of real numbers defined by programs which can be proven to be correct in some fixed axiom system S. In particular we develop the provable analogue of an effective operator on the set of recursive real numbers, namely, a provably correct operator on the set of provably recursive real numbers. In Theorems 1 and 2 we exhibit a provably correct operator on which is discontinuous at 0; we thus disprove the analogue of the Ceitin-Moschovakis theorem of recursive analysis, which states that every effective operator on is (effectively) continuous. Our final theorems show, however, that no provably correct operator on can be proven (in S) to be discontinuous.

In this paper we study (self-)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on = {0, 1}* are exactly the polynomial time computable functions. Our treatment of PTO is proof-theoretic and very much in the spirit of reductive proof theory.

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 logic of relational and partial variable sets, seen as a generalization of set-valued presheaves, allowing transition functions to be arbitrary relations or arbitrary partial functions. We find that such a logic is the usual intuitionistic and co-intuitionistic first order logic without Beck and Frobenius conditions relative to quantifiers along arbitrary terms. The important case of partial variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independence results are obtained for different kinds of Beck and Frobenius conditions.

Where a is a Turing degree and ξ is an ordinal < (ℵ1)L1, the result of performing ξ jumps on a, a(ξ), is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through (ℵ1)L1 of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.

One of the main theorems of the paper states the following. Let R-K-M be finite extensions of a rational one variable function field R over a finite field of constants. Let S be a finite set of valuations of K. Then the ring of elements of K having no poles outside S has a Diophantine definition over its integral closure in M.

The main subject of the article is the finite submodel property for ℵ0-categorical structures, in particular under the additional assumptions that the structure is simple, 1-based and has trivial dependence. Here, a structure has the finite submodel property if every sentence which is true in the structure is true in a finite substructure of it. It will be useful to consider a couple of other finiteness properties, related to the finite submodel property, which are variants of the usual concept of saturation.

For the rest of the introduction we will assume that M is an ℵ0-categorical (infinite) structure with a countable language. We also assume that there is an upper bound to the arity of the function symbols in M:s language and that, for every 0 < n < ℵ0 and R ⊆ Mn which is definable in M without parameters, there exists a relation symbol, in the language of M, which is interpreted as R; these assumptions are not necessary for most results to be presented, but it simplifies the statement of a result which I mention in this introduction.

First we will consider ‘canonically embedded’ substructures of Meq. Here, a structure N is canonically embedded in Meq if N's universe is a subset of Meq which is definable without parameters and, for every 0 < n < ℵ0 and R ⊆ Nn which is ε-definable in Meq there is a relation symbol in the language of N which is interpreted as R; we also assume that the language of N has no other relation (or function or constant) symbols.

We obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like ∃≥3/4y which says that “for at least 3/4 of all y”. These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are:

1. We deal with the quantifier ∃≥ry, where y is a tuple of variables.

2. We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope.

3. Instead of the unbiased measure where each model with universe n has the same probability, we work with any measure generated by independent atomic probabilities PR for each predicate symbol R.

4. We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs).

5. We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers.

6. We allow each PR, as well as each r in the probability quantifier (∃≥ry), to depend on the size of the universe.

We generalize a well-known theorem binding the elementary equivalence relation on the level of PAC fields and the isomorphism type of their absolute Galois groups. Our results concern two cases: saturated PAC structures and nonsaturated PAC structures.

We modify the definable ultrapower construction of Kanovei and Shelah (2004) to develop a ZF-definable extension of the continuum with transfer provable using countable choice only, with an additional mild hypothesis on well-ordering implying properness. Under the same assumptions, we also prove the existence of a definable, proper elementary extension of the standard superstructure over the reals.

Let S be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of S over a field K of characteristic 0. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over $RC{A_0}$ (or even over $RCA_0^{\rm{*}}$) to the statement that ${\omega ^\omega }$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

We consider separably closed fields of characteristic p > 0 and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the p-component functions.

We show that if A and are automorphic via Φ then the structures (A) and () are Δ30-isomorphic via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.

In Team Semantics, a dependency notion is strongly first order if every sentence of the logic obtained by adding the corresponding atoms to First-Order Logic is equivalent to some first-order sentence. In this work it is shown that all nontrivial dependency atoms that are strongly first order, downwards closed, and relativizable (in the sense that the relativizations of the corresponding atoms with respect to some unary predicate are expressible in terms of them) are definable in terms of constancy atoms.

Additionally, it is shown that any strongly first-order dependency is safe for any family of downwards closed dependencies, in the sense that every sentence of the logic obtained by adding to First-Order Logic both the strongly first-order dependency and the downwards closed dependencies is equivalent to some sentence of the logic obtained by adding only the downwards closed dependencies.

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.