相关论文: Long low iterations

Let $\mathcal{N}$ be the $\sigma$-ideal of the null sets of reals. We introduce a new property of forcing notions that enable control of the additivity of $\mathcal{N}$ after finite support iterations. This is applied to answer some open…

We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The…

We augment LP with a strong conditional operator, to yield a logic we call "strong LP," or LP=>. The resulting logic can speak of consistency in more discriminating ways, but introduces new possibilities for trivializing paradoxes.

We prove the consistency result from the title. By forcing we construct a model of g=aleph_1, b=cf(Sym(omega))=aleph_2.

We show that, consistently, every MAD family has cardinality strictly bigger than the dominating number, that is a > d, thus solving one of the oldest problems on cardinal invariants of the continuum. The method is a contribution to the…

Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic…

We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof is by interpolation and uses the Mapping Reflection Principle.

The results of the previous version are impoved. This basically completes the study of consistency strength of various gaps between a strong limit singular cardinal of cofinality omega and its power under GCH type assumptions below.

This paper explores the potential of Lagrangian duality for learning applications that feature complex constraints. Such constraints arise in many science and engineering domains, where the task amounts to learning optimization problems…

We study relationships between various set theoretic compactness principles, focusing on the interplay between the three families of combinatorial objects or principles mentioned in the title. Specifically, we show the following. (1) Strong…

In this paper we show that forcings which are strongly proper for stationarily many countable elementary submodels preserve each of the following properties of topological spaces: countably tight; Lindel\"of; Rothberger; Menger; and a…

We continue here [She88] but we do not rely on it. The motivation was a conjecture of Galvin stating that 2^{omega} >= omega_2 + omega_2-> [omega_1]^{n}_{h(n)} is consistent for a suitable h: omega-> omega. In section 5 we disprove this and…

A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing…

We consider natural cardinal invariants hm_n and prove several duality theorems, saying roughly: if I is a suitably definable ideal and provably cov(I)>=hm_n, then non(I) is provably small. The proofs integrate the determinacy theory,…

Let k be a definable L-cardinal. Then there is a set of reals X, class-generic over L, such that L(X) and L have the same cardinals, X has size k in L(X) and some pi-1-2 formula defines X in all set-generic extensions of L(X). Two…

We answer a question of Moore by building a forcing extension satisfying measuring together with CH. The construction works over any model of ZFC and can be described as a forcing iteration with countable structures as side conditions and…

We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing…

We show that well-chosen Lagrangians for a class of two-dimensional integrable lattice equations obey a closure relation when embedded in a higher dimensional lattice. On the basis of this property we formulate a Lagrangian description for…

I explore two separate topics: the concept of jointness for set-theoretic guessing principles, and the notion of grounded forcing axioms. A family of guessing sequences is said to be joint if all of its members can guess any given family of…

We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using our method we prove that the forcing axiom for the class of all the small finitely proper posets is compatible with a…