Factorials and the finite sequences of sets

We write for the cardinality of the set of finite sequences of a set which is of cardinality . With the Axiom of Choice (), for every infinite cardinal where is the cardinality of the permutations on a set which is of cardinality . In this paper, we show that “ for every cardinal ”  is provable in and this is the best possible result in the absence of . Similar results are also obtained for : the cardinality of the set of finite sequences without repetition of a set which is of cardinality .     

Jump inversions of algebraic structures and Σ‐definability

It is proved that for every countable structure and a computable successor ordinal α there is a countable structure which is ‐least among all countable structures such that is Σ‐definable in the αth jump . We also show that this result does not hold for the limit ordinal . Moreover, we prove that there is no countable structure with the degree spectrum for .     

The cofinality of the least Berkeley cardinal and the extent of dependent choice

This paper is concerned with the possible values of the cofinality of the least Berkeley cardinal. Berkeley cardinals are very large cardinal axioms incompatible with the Axiom of Choice, and the interest in the cofinality of the least Berkeley arises from a result in [1], showing it is connected with the failure of . In fact, by a theorem of Bagaria, Koellner and Woodin, if γ is the cofinality of the least Berkeley cardinal then γ‐ fails. We shall prove that this result is optimal for or . In particular, it will follow that the cofinality of the least Berkeley is independent of .     

Reductions on equivalence relations generated by universal sets

Let X, Y be Polish spaces, , . We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by , where . The following results are shown:

• (1) If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and .
• (2) If A is a set universal for all countable subsets of Y, then is a equivalence relation, and
  • (i) and ;
  • (ii) if , then ;
  • (iii) if every set is Lebesgue measurable or has the Baire property, then .
  • (iv) for , if every set has the Baire property, and E is any equivalence relation, then .

     

On the relative strengths of fragments of collection

Let be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ₀‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to . We focus on two common parameterisations of the collection: ‐collection, which is the usual collection scheme restricted to ‐formulae, and strong ‐collection, which is equivalent to ‐collection plus ‐separation. The main result of this paper shows that for all ,

1. proves that there exists a transitive model of Zermelo Set Theory plus ‐collection,
2. the theory is ‐conservative over the theory .

It is also shown that (2) holds for when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus ) that does not include the powerset axiom.     

A variant of Shelah's characterization of Strong Chang's Conjecture

Shelah considered a certain version of Strong Chang's Conjecture which we denote , and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted , and prove an analogous characterization of it. In particular, is equivalent to the assertion that the the Friedman‐Krueger poset is semiproper. This strengthens and sharpens results by Cox and sheds some light on problems posed by Usuba, Torres‐Perez and Wu.     

Neutrally expandable models of arithmetic

A subset of a model of is called neutral if it does not change the relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non‐existence of neutral sets in various models of . We show that cofinal extensions of prime models are neutrally expandable, and ω₁‐like neutrally expandable models exist, while no recursively saturated model is neutrally expandable. We also show that neutrality is not a first‐order property. In the last section, we study a local version of neutral expandability.     

Indivisible sets and well‐founded orientations of the Rado graph

Every set can been thought of as a directed graph whose edge relation is ∈ . We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ?₁ that are distinct up to siblinghood.     

Some properties of infinite factorials

With the Axiom of Choice , for any infinite cardinal but, without , we cannot conclude any relationship between and for an arbitrary infinite cardinal . In this paper, we give some properties of in the absence of and compare them to those of for an infinite cardinal . Among our results, we show that “ for any infinite cardinal and any natural number n” is provable in although “ for any infinite cardinal ” is not.     

The tree property and the continuum function below

We say that a regular cardinal κ, , has the tree property if there are no κ‐Aronszajn trees; we say that κ has the weak tree property if there are no special κ‐Aronszajn trees. Starting with infinitely many weakly compact cardinals, we show that the tree property at every even cardinal , , is consistent with an arbitrary continuum function below which satisfies , . Next, starting with infinitely many Mahlo cardinals, we show that the weak tree property at every cardinal , , is consistent with an arbitrary continuum function below which satisfies , . Thus the tree property has no provable effect on the continuum function below except for the trivial requirement that the tree property at implies for every infinite κ.     

Building prime models in fully good abstract elementary classes

We show how to build prime models in classes of saturated models of abstract elementary classes (AECs) having a well‐behaved independence relation: Let be an almost fully good AEC that is categorical in and has the ‐existence property for domination triples. For any , the class of Galois saturated models of of size λ has prime models over every set of the form . This generalizes an argument of Shelah, who proved the result when λ is a successor cardinal.     

Generic at

In this paper we introduce a generic large cardinal akin to , together with the consequences of being such a generic large cardinal. In this case is Jónsson, and in a choiceless inner model many properties hold that are in contrast with pcf theory in .     

Algebraic numbers with elements of small height

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers.     

The maximum number of columns in supersaturated designs with

Cheng and Tang [Biometrika, 88 (2001), pp. 1169–1174] derived an upper bound on the maximum number of columns that can be accommodated in a two‐symbol supersaturated design (SSD) for a given number of rows () and a maximum in absolute value correlation between any two columns (). In particular, they proved that for (mod ) and . However, the only known SSD satisfying this upper bound is when . By utilizing a computer search, we prove that for , and . These results are obtained by proving the nonexistence of certain resolvable incomplete blocks designs. The combinatorial properties of the RIBDs are used to reduce the search space. Our results improve the lower bound for SSDs with rows and columns, for , and . Finally, we show that a skew‐type Hadamard matrix of order can be used to construct an SSD with rows and columns that proves . Hence, we establish for and for all (mod ) such that . Our result also implies that when is a prime power and (mod ). We conjecture that for all and (mod ), where is the maximum number of equiangular lines in with pairwise angle .     

On the finite axiomatizability of

The question of whether the bounded arithmetic theories and are equal is closely connected to the complexity question of whether is equal to . In this paper, we examine the still open question of whether the prenex version of , , is equal to . We give new dependent choice‐based axiomatizations of the ‐consequences of and . Our dependent choice axiomatizations give new normal forms for the ‐consequences of and . We use these axiomatizations to give an alternative proof of the finite axiomatizability of and to show new results such as is finitely axiomatized and that there is a finitely axiomatized theory, , containing and contained in . On the other hand, we show that our theory for splits into a natural infinite hierarchy of theories. We give a diagonalization result that stems from our attempts to separate the hierarchy for .     

A complete solution to existence of H designs

An H design is a triple , where is a set of points, a partition of into disjoint sets of size , and a set of ‐element transverses of , such that each ‐element transverse of is contained in exactly one of them. In 1990, Mills determined the existence of an H design with . In this paper, an efficient construction shows that an H exists for any integer with . Consequently, the necessary and sufficient conditions for the existence of an H design are , , and , with a definite exception .     

On Turán problems for Cartesian products of graphs

Let be disjoint sets of sizes and . Let be a family of quadruples, having elements from and from , such that any subset with and contains one of the quadruples. We prove that the smallest size of is as . We also solve asymptotically a more general two‐partite Turán problem for quadruples.     

Free sets for set‐mappings relative to a family of sets

Given a family of subsets of , we try to compute the least natural number n such that for every function there exists a bijection such that for all .     

Fractal graphs

The lexicographic sum of graphs is defined as follows. Let be a graph. With each associate a graph . The lexicographic sum of the graphs over is obtained from by substituting each by . Given distinct , we have all the possible edges in the lexicographic sum between and if , and none otherwise. When all the graphs are isomorphic to some graph , the lexicographic sum of the graphs over is called the lexicographic product of by and is denoted by . We say that a graph is fractal if there exists a graph , with at least two vertices, such that . There is a simple way to construct fractal graphs. Let be a graph with at least two vertices. The graph is defined on the set of functions from to as follows. Given distinct is an edge of if is an edge of , where is the smallest integer such that . The graph is fractal because . We prove that a fractal graph is isomorphic to a lexicographic sum over an induced subgraph of , which is itself fractal.     

A note on the deductive strength of the Nielsen‐Schreier theorem

We show that the Boolean Prime Ideal Theorem () does not imply the Nielsen‐Schreier Theorem () in , thus strengthening the result of Kleppmann from “Nielsen‐Schreier and the Axiom of Choice” that the (strictly weaker than ) Ordering Principle () does not imply in . We also show that is false in Mostowski's Linearly Ordered Model of . The above two results also settle the corresponding open problems from Howard and Rubin's “Consequences of the Axiom of Choice”.