Cardinal characteristics for Menger-bounded subgroups

Machura, Shelah and Tsaban showed in [M. Machura, S. Shelah, B. Tsaban, Squares of Menger-bounded groups, Trans. Amer. Math. Soc., in press, http://arxiv.org/pdf/math.GN/0611353, 2007] that under the condition, that a relative d^′(P) of the dominating number is at least d, there are subgroups of the Baer-Specker group whose kth power is Menger-bounded and whose (k+1)st power is not. We show that the sufficient condition implies r?d and indeed can be replaced by r?d. This result includes an affirmative answer to a question by Tsaban on a possibly weaker still sufficient condition. We show that it is consistent relative to ZFC that g?r<d and there are subgroups of the Baer-Specker group whose kth power is Menger-bounded and whose (k+1)st power is not.     

Projective mad families

Using almost disjoint coding we prove the consistency of the existence of a definable ω-mad family of infinite subsets of ω (resp. functions from ω to ω) together with b=2^ω=ω₂.     

The γ-borel conjecture

In this paper we prove that it is consistent that every -set is countable while not every strong measure zero set is countable. We also show that it is consistent that every strong -set is countable while not every -set is countable. On the other hand we show that every strong measure zero set is countable iff every set with the Rothberger property is countable.Thanks to Boise State University for support during the time this paper was written and to Alan Dow for some helpful discussions and to Boaz Tsaban for some suggestions to improve an earlier version.     

Covering with universally Baire operators

We introduce a covering conjecture and show that it holds below A_DR+“Θ is regular”$A{D}_{\mathbb{R}}+\text{“}\Theta \text{is regular”}$. We then use it to show that in the presence of mild large cardinal axioms, PFA   implies that there is a transitive model containing the reals and ordinals and satisfying A_DR+“Θ is regular”$A{D}_{\mathbb{R}}+\text{“}\Theta \text{is regular”}$. The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of AD⁺+_θ0<Θ$A{D}^{+}+{\theta }_0<\Theta$.     

Constructibility in higher order arithmetics

Summary We define and investigate constructibility in higher order arithmetics. In particular we get an interpretation ofn-order arithmetic inn-order arithmetic without the scheme of choice such that and the property to be a well-ordering are absolute in it and such that this interpretation is minimal among such interpretations.     

Cardinal sequences of LCS spaces under GCH

Let C(α) denote the class of all cardinal sequences of length α associated with compact scattered spaces. Also put

C_λ(α)={f∈C(α):f(0)=λ=min[f(β):β<α]}.     

Fusion and large cardinal preservation

In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ?κ   not only does not collapse κ⁺${\kappa }^{+}$ but also preserves the strength of κ (after a suitable preparatory forcing). This provides a general theory covering the known cases of tree iterations which preserve large cardinals (cf. Dobrinen and Friedman (2010) [3], Friedman and Halilovi? (2011) [5], Friedman and Honzik (2008) [6], Friedman and Magidor (2009) [8], Friedman and Zdomskyy (2010) [10], Honzik (2010) [12]).     

Covering a bounded set of functions by an increasing chain of slaloms

A slalom is a sequence of finite sets of length ω. Slaloms are ordered by coordinatewise inclusion with finitely many exceptions. Improving earlier results of Mildenberger, Shelah and Tsaban, we prove consistency results concerning existence and non-existence of an increasing sequence of a certain type of slaloms which covers a bounded set of functions in ωω.     

Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ?ℵ₂.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ?ℵ₁ are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ?ℵ₁ are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).     

Distinguishing groupwise density numbers

We show that is consistent, where is the groupwise density number and is the groupwise density number for ideals. This answers a question of Heike Mildenberger. Partially supported by Grant-in-Aid for Scientific Research (C) 17540116, Japan Society for the Promotion of Science.     

Weak covering and the tree property

Suppose that there is no transitive model of ZFC + there is a strong cardinal, and let K denote the core model. It is shown that if has the tree property then and is weakly compact in K. Received: 11 June 1997     

More infinity for a better finitism

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed in around 1995 by Patrick Suppes and Richard Sommer. It is based on an earlier system developed by Rolando Chuaqui and Patrick Suppes. Here, we discuss the inherent problems and limitations of the classical nonstandard framework and propose a much-needed refinement of ERNA, called ERNA^A, in the spirit of Karel Hrbacek’s stratified set theory. We study the metamathematics of ERNA^A and its extensions. In particular, we consider several transfer principles, both classical and ‘stratified’, which turn out to be related. Finally, we show that the resulting theory allows for a truly general, elegant and elementary treatment of basic analysis.     

Some Remarks on the Simultaneous Chromatic Number

We present several partial results, variants, and consistency results concerning the following (as yet unsolved) conjecture. If X is a graph on the ground set V with then X has an edge coloring F with colors such that if V is decomposed into parts then there is one in which F assumes all values.Due to some unfortunate misunderstandings, this paper appeared much later than we expected.* Research partially supported by NSF grants DMS-9704477 and DMS-0072560. Research partially supported by Hungarian National Research Grant T 032455.     

Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or D

We discuss relationships in Lindelöf spaces among the properties “Menger”, “Hurewicz”, “Alster”, “productive”, and “D”.     

Enumeration of one-nodal rational curves in projective spaces

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. The formula involves intersections of tautological classes on moduli spaces of stable rational maps. We combine the methods and results from three different papers.     

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.