We prove Los conjecture = Morley theorem in ZF. with the same characterization, i.e., of first order countable theories categorical in ℵα for some (eqiuvalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality ℵα is either ≥ ∣α∣ for every α or it has a small upper bound (independent of α close to ⊐2).

We specialise Aronszajn trees by an ωω-bounding forcing that adds reals. We work with creature forcings on uncountable spaces.

As an application of these notions of forcing, we answer a question of Moore, Hrušák and Džamonja whether implies the existence of a Souslin tree in a negative way by showing that “ and every Aronszajn tree is special” is consistent relative to ZFC.

Let be a finite relational language and α = (αR: R ∈ ) a tuple with 0 < αR ≤ 1 for each R ∈ . Consider a dimension function

where each eR(A) is the number of realizations of R in A. Let Kα be the class of finite structures A such that δα (X) ≥ 0 for any substructure X of A. We show that the theory of the generic model of Kα is AE-axiomatizable for any α.

Many classical statements of set theory are settled by the existence of generic elementary embeddings that are analogous the elementary embeddings posited by large cardinals. [2] The embeddings analogous to measurable cardinals are determined by uniform, κ-complete precipitous ideals on cardinals κ. Stronger embeddings, analogous to those originating from supercompact or huge cardinals are encoded by normal fine ideals on sets such as [κ]<λ or [κ]λ.

The embeddings generated from these ideals are limited in ways analogous to conventional large cardinals. Explicitly, if j: V → M is a generic elementary embedding with critical point κ and λ supnЄωjn(κ) and the forcing yielding j is λ-saturated then j“λ+ ∉ M. (See [2].)

Ideals that yield embeddings that are analogous to strongly compact cardinals have more puzzling behavior and the analogy is not as straightforward. Some natural ideal properties of this kind have been shown to be inconsistent:

Theorem 1 (Kunen). There is no ω2-saturated, countably complete uniform ideal on any cardinal in the interval [ℵω, ℵω).

Generic embeddings that arise from countably complete, ω2-saturated ideals have the property that sup . So the Kunen result is striking in that it apparently allows strong ideals to exist above the conventional large cardinal limitations. The main result of this paper is that it is consistent (relative to a huge cardinal) that such ideals exist.

Assuming every real has a sharp, we prove that for any equivalence relation either Borel reduces E0 or in a manner allows the assignment of bounded subsets of ω1 as complete invariants.

We find the model completion of the theory modules over , where is a finitely generated commutative algebra over a field K. This is done in a context where the field K and the module are represented by sorts in the theory, so that constructible sets associated with a module can be interpreted in this language. The language is expanded by additional sorts for the Grassmanians of all powers of Kn, which are necessary to achieve quantifier elimination.

The result turns out to be that the model completion is the theory of a certain class of “big” injective modules. In particular, it is shown that the class of injective modules is itself elementary. We also obtain an explicit description of the types in this theory.

It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G.

The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).

Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.

We show that if ℱ is any “well-behaved” subset of the Borel functions and we assume the Axiom of Determinacy then the hierarchy of degrees on (ωω) induced by ℱ turns out to look like the Wadge hierarchy (which is the special case where ℱ is the set of continuous functions).

Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that Jκ[E] is iterable above t with respect to iteration trees of length less than κ.

As an application we show L[E] to be a model of the following two cardinals versions of the diamond principle. If λ > κ > ω1 are cardinals, then holds true, and if in addition λ is regular, then holds true.

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 investigate various strong notions of rigidity for Souslin trees, separating them under ⟡ into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under ⟡ that there is a group whose automorphism tower is highly malleable by forcing.

We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.

We show that the ℵ0-categorical structures produced by Hrushovski's predimension construction with a control function fit neatly into Shelah's SOPn hierarchy: if they are not simple, then they have SOP3 and NSOP4. We also show that structures produced without using a control function can be undecidable and have SOP.

Equivalences and translations between consequence relations abound in logic. The notion of equivalence can be denned syntactically, in terms of translations of formulas, and order-theoretically, in terms of the associated lattices of theories. W. Blok and D. Pigozzi proved in [4] that the two definitions coincide in the case of an algebraizable sentential deductive system. A refined treatment of this equivalence was provided by W. Blok and B. Jónsson in [3]. Other authors have extended this result to the cases of κ-deductive systems and of consequence relations on associative, commutative, multiple conclusion sequents. Our main result subsumes all existing results in the literature and reveals their common character. The proofs are of order-theoretic and categorical nature.

Over 25 years ago, the first author conjectured in [15] that the existence of arbitrarily large primes is provable from the axioms IΔ0(π) + def(π), where π(x) is the number of primes not exceeding x, IΔ0(π) denotes the theory of Δ0 induction for the language of arithmetic including the new function symbol π, and def(π) is an axiom expressing the usual recursive definition of π. We prove a modified version in which π is replaced by a more general function ξ that counts some of the primes below x (which primes depends on the values of parameters in ξ), and has the property that π is provably Δ0(ξ) definable.

A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n ≥ 3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary.

Our proof is based on the following construction. From an arbitrary undirected, loop-free graph Γ, we construct an n-dimensional atom structure , and prove, for infinite Γ, that is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdős shows that there are graphs Γk(k < ω) with infinite chromatic number, but having a non-principal ultraproduct ΠDΓk whose chromatic number is just two. It follows that is strongly representable (each k < ω) but is not.

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and and are generic extensions of V satisfying CH then and agree on all Σ12-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ12. Moreover, CH is the unique Σ12-statement with this feature in the sense that any other Σ12-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment Vλ of the universe of sets (for example, one might take Vλ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of Vλ. If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies ¬-CH.

For f, g ∈ ωω let be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch v of the f-tree, one of the g-trees contains v. is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often.

It is consistent that for ℵ1 many pairwise different cardinals κ∊ and suitable pairs (f∊, g∊).

For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision.

In the model theory of modules the Ziegler spectrum, the space of indecomposable pure-injective modules, has played a key role. We investigate the possibility of defining a similar space in the context of G-sets where G is a group.