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.     

Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes

A new case of Shelah's eventual categoricity conjecture is established:

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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.     

A Groszek‐Laver pair of undistinguishable ‐classes

A generic extension of the constructible universe by reals is defined, in which the union of ‐classes of x and y is a lightface set, but neither of these two ‐classes is separately ordinal‐definable.     

Many symmetrically indivisible structures

A structure in a first‐order language is indivisible if for every colouring of its universe M in two colours, there is a monochromatic such that . Additionally, we say that is symmetrically indivisible if can be chosen to be symmetrically embedded in (that is, every automorphism of can be extended to an automorphism of ). In the following paper we give a general method for constructing new symmetrically indivisible structures out of existing ones. Using this method, we construct many non‐isomorphic symmetrically indivisible countable structures in given (elementary) classes and answer negatively the following question from 6 : Let be a symmetrically indivisible structure in a language . Let . Is symmetrically indivisible?     

The length of an intersection

A poset is well‐partially ordered (WPO) if all its linear extensions are well orders; the supremum of ordered types of these linear extensions is the length , of p . We prove that if the vertex set X is infinite, of cardinality κ, and the ordering ⩽ is the intersection of finitely many well partial orderings of X , , then, letting , with , denote the euclidian division by κ (seen as an initial ordinal) of the length of each corresponding poset: where denotes the least initial ordinal greater than the ordinal . This inequality is optimal. This result answers questions of Forster.     

On Schauder equivalence relations

We study Schauder equivalence relations, which are Borel equivalence relations induced by actions of Banach spaces with Schauder bases. Firstly, we show that and are minimal Schauder equivalence relations. Then, we prove that neither of them is Borel reducible to the quotient where T is the Tsirelson space. This implies that they cannot form a basis for the Schauder equivalence relations. In addition, we apply an argument of Farah to show that every basis for the Schauder equivalence relations, if such exist, has to be of cardinality .     

On the strength of a weak variant of the axiom of counting

In this paper is used to denote Jensen's modification of Quine's ‘new foundations’ set theory () fortified with a type‐level pairing function but without the axiom of choice. The axiom is the variant of the axiom of counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that proves the consistency of the simple theory of types with infinity (). This result implies that proves that consistency of , and that proves the consistency of .     

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 .     

Models of : when two elements are necessarily order automorphic

We are interested in the question of how much the order of a non‐standard model of can determine the model. In particular, for a model M, we want to characterize the complete types of non‐standard elements such that the linear orders and are necessarily isomorphic. It is proved that this set includes the complete types such that if the pair realizes it (in M) then there is an element c such that for all standard n, , , , and . We prove that this is optimal, because if holds, then there is M of cardinality ?₁ for which we get equality. We also deal with how much the order in a model of may determine the addition.     

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 .     

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.     

A wild model of linear arithmetic and discretely ordered modules

Linear arithmetics are extensions of Presburger arithmetic () by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages. In this paper, we construct a model of the 2‐linear arithmetic (linear arithmetic with two scalars) in which an infinitely long initial segment of “Peano multiplication” on is ‐definable. This shows, in particular, that is not model complete in contrast to theories and that are known to satisfy quantifier elimination up to disjunctions of primitive positive formulas. As an application, we show that , as a discretely ordered module over the discretely ordered ring generated by the two scalars, does not have the NIP, answering negatively a question of Chernikov and Hils.     

Fermat's last theorem and Catalan's conjecture in weak exponential arithmetics

We study Fermat's last theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions of models of arithmetical theories (in the language ) by a binary (partial or total) function e intended as an exponential. We provide a general construction of such expansions and prove that it is universal for the class of all exponentials e which satisfy a certain natural set of axioms . We construct a model and a substructure with e total and (Presburger arithmetic) such that in both and Fermat's last theorem for e is violated by cofinally many exponents n and (in all coordinates) cofinally many pairwise linearly independent triples . On the other hand, under the assumption of ABC conjecture (in the standard model), we show that Catalan's conjecture for e is provable in (even in a weaker theory) and thus holds in and . Finally, we also show that Fermat's last theorem for e is provable (again, under the assumption of ABC in ) in “coprimality for e ”.     

Two new equivalents of Lindelöf metric spaces

In the realm of Lindelöf metric spaces the following results are obtained in : (i) If is a Lindelöf metric space then it is both densely Lindelöf and almost Lindelöf. In addition, under the countable axiom of choice , the three notions coincide. (ii) The statement “every separable metric space is almost Lindelöf” implies that every infinite subset of has a countably infinite subset). (iii) The statement “every almost Lindelöf metric space is quasi totally bounded implies . (iv) The proposition “every quasi totally bounded metric space is separable” lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem and . Likewise, the statement “every pre‐Lindelöf (or Lindelöf) metric space is separable” lies strictly between and .     

The Hypothesis and a supercompact cardinal

In this paper, we prove that: if κ is supercompact and the Hypothesis holds, then there is a proper class of regular cardinals in which are measurable in . Woodin also proved this result independently 11 . As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the Hypothesis and supercompact cardinals, large cardinals in are reflected to be large cardinals in in a local way, and reveals the huge difference between ‐supercompact cardinals and supercompact cardinals under the Hypothesis.     

The grounded Martin's axiom

We introduce a variant of Martin's axiom, called the grounded Martin's axiom, or , which asserts that the universe is a c.c.c. forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of the combinatorial consequences of . The new axiom is shown to be consistent with the failure of and a singular continuum. We prove that is preserved in a strong way when adding a Cohen real and that adding a random real to a model of preserves (even though it destroys itself). We also consider the analogous variant of the proper forcing axiom.     

The Structure of an SL2‐module of finite Morley rank

We consider a universe of finite Morley rank and the following definable objects: a field , a non‐trivial action of a group on a connected abelian group V , and a torus T of G such that . We prove that every T‐minimal subgroup of V has Morley rank . Moreover V is a direct sum of ‐minimal subgroups of the form , where W is T‐minimal and ζ is an element of G of order 4 inverting T .     

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 .     

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”.