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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 613 records · Page 34Linked to original sources
arXiv1999

On the existence of large subsets of [lambda]^{<kappa} which contain no unbounded non-stationary subsets

Here we deal with some problems posed by Matet. The first section deals with the existence of stationary subsets of [lambda]^{<kappa} with no unbounded subsets which are not stationary, where, of course, kappa is regular uncountable less or equal than lambda. In the second section we deal with the existence of such clubs. The proofs are easy but the result seems to be very surprising. Theorem 1.2 was proved some time ago by Baumgartner and is presented here for the sake of completeness.

math.LO↗
arXiv1999

Strongly meager and strong measure zero sets

We show that the following are consistent with ZFC: 1. Strongly meager sets form an ideal with the same additivity as the ideal of meager sets. 2. There exists a strong measure zero set of size > d (dominating number).

math.LO↗
arXiv1999

On what I do not understand (and have something to say): Part I

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (``see ... '' means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The other half, concentrating on model theory, will subsequently appear.

math.LO↗
arXiv1999

Few non-minimal types and non-structure

We pay two debts from [Sh:576] (math.LO/9805146). The main demands little knowledge from [Sh:576], just quoting a model theoretic consequence of the weak diamond. We assume that K has amalgamation in lambda, and that the minimal types are not dense to get many non-isomorphic models in lambda^+. For this also pcf considerations are relevant. The minor debt was the use in one point of [Sh:576] of lambda not= aleph_0, it is minor as for this case by [Sh:88] we ``usually'' know more.

math.LO↗
arXiv1999

On Arhangelskii's Problem

We prove the consistency (modulo supercompact) of a negative answer to Arhangelskii's problem (some Hausdorff compact space cannot be partitioned to two sets not containing a closed copy of Cantor discontinuum). In this model we have CH. Without CH we get consistency results using a pcf assumption, close relatives of which are necessary for such results.

math.LO↗
arXiv1999

Category analog of sup-measurability problem

A function F:R^2->R is sup-measurable if F_f:R->R given by F_f(x)=F(x,f(x)), x in R, is measurable for each measurable function f:R->R. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analog. In this paper we will show that the existence of category analog of sup-measurable non-measurable functions is independent of ZFC. A similar result for the original measurable case is a subject of a work in prepartion by Roslanowski and Shelah.

math.LO↗
arXiv1999

The Cichon diagram

We conclude the discussion of additivity, Baire number, uniformity and covering for measure and category by constructing the remaining 5 models. Thus we complete the analysis of Cichon's diagram.

math.LO↗
arXiv1999

Closed measure zero sets

We study the relationship between the sigma-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the ideal of meager sets.

math.LO↗
arXiv1999

On absolutely divergent series

We show that in the aleph_2-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(omega)/fin. This complements Vojtas' result, that under cf(c)=p the two algebras are isomorphic

math.LO↗
arXiv1999

On incomparability and related cardinal functions on ultraproducts of Boolean algebras

Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, pi-character, hereditary Lindelof number, hereditary density. It is shown to be consistent that there exists a sequence of Boolean algebras and an ultrafilter D on kappa such that C(prod_{i<kappa}B_i/D)<|prod_{i<kappa}C(B_i)/D|. This answers a number of problems posed by Monk.

math.LO↗
arXiv1999

Antichains in products of linear orders

1. For many regular cardinals lambda (in particular, for all successors of singular strong limit cardinals, and for all successors of singular omega-limits), for all n in {2,3,4, ...} : There is a linear order L such that L^n has no (incomparability-)antichain of cardinality lambda, while L^{n+1} has an antichain of cardinality lambda . 2. For any nondecreasing sequence (lambda2,lambda3, ...) of infinite cardinals it is consistent that there is a linear order L such that L^n has an antichain of cardinality lambda_n, but not one of cardinality lambda_n^+ .

math.LO↗
arXiv1999

Changing cardinal characteristics without changing omega-sequences or cofinalities

We show: There are pairs of universes V_1 subseteq V_2 and there is a notion of forcing P in V_1 such that the change mentioned in the title occurs when going from V_1[G] to V_2[G] for a P-generic filter G over V_2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.

math.LO↗
arXiv1998

Gently Killing S--spaces

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^{aleph_0}<2^{aleph_1}. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size aleph_1, answering a question of the second author.

math.LO↗
arXiv1998

Strongly almost disjoint families, II

The relations M(kappa,lambda,mu)->B [resp. B(sigma)] meaning that if A subset [kappa]^lambda with |A|=kappa is mu-almost disjoint then A has property B [resp. has a sigma-transversal] had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following: Assume GCH and rho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above rho. Then there is a rho-closed forcing P such that, in V^P, we have both GCH and M(rho^{(+rho+1)},rho^+,rho) not-> B [resp. M(rho^{(+rho+1)},lambda,rho) not-> B(rho^+) for all lambda =< rho^{(+rho+1)}].

math.LO↗
arXiv1998

A Delta^2_2 well-order of the reals and incompactness of L(Q^{MM})

A forcing poset of size 2^{2^{aleph_1}} which adds no new reals is described and shown to provide a Delta^2_2 definable well-order of the reals (in fact, any given relation of the reals may be so encoded in some generic extension). The encoding of this well-order is obtained by playing with products of Aronszajn trees: Some products are special while other are Suslin trees. The paper also deals with the Magidor-Malitz logic: it is consistent that this logic is highly non compact.

math.LO↗
arXiv1998

The Yellow Cake

We consider the following property: (*) For every function f from R^2 to R there are functions g_n,h_n from R to R (for n<omega) such that for all real numbers x and y, f(x,y) = sum_{n<omega} g_n(x)h_n(y). We show that, despite some expectation suggested by [Sh:675], (*) does not imply MA(sigma-centered). Next, we introduce cardinal characteristics of the continuum responsible for the failure of (*).

math.LO↗