arXiv subjects
Publications and source records attributed to Saharon Shelah.
Let l^0 and m^0 be the ideals associated with Laver and Miller forcing, respectively. We show that add (l^0) < cov(l^0) and add (m^0) < cov(m^0) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal <= h .
Assuming 0# does not exist, we present a combinatorial approach to Jensen's method of coding by a real. The forcing uses combinatorial consequences of fine structure (including the Covering Lemma, in various guises), but makes no direct appeal to fine structure itself.
We give a model where there is a ccc Souslin forcing which does not satisfy the Knaster condition. Next, we present a model where there is a sigma-linked not sigma-centered Souslin forcing such that all its small subsets are sigma-centered but Martin Axiom fails for this order. Furthermore, we construct a totally nonhomogeneous Souslin forcing and we build a Souslin forcing which is proper but not ccc that does not contain a perfect set of mutually incompatible conditions. Finally we show that ccc Sigma^1_2-notions of forcing may not be indestructible ccc.
Upon presenting the proof of Theorem 3.3 in "Maximal chains in $$ and ultrapowers of the integers" I discovered that it is not entirely correct and certainly some details should be added. I have therefore written an addendum to the paper and made it available by ftp. Unfortunately the published version will be somewhat wanting.
A canary tree is a tree of cardinality the continuum which has no uncountable branch, but gains a branch whenever a stationary set is destroyed (without adding reals). Canary trees are important in infinitary model theory. The existence of a canary tree is independent of ZFC + GCH.
We consider the question of which valuation domains (of cardinality aleph_1) have non-standard uniserial modules. We show that a criterion conjectured by Osofsky is independent of ZFC + GCH.
We prove the consistency of ``CH + 2^{aleph_1} is arbitrarily large + 2^{aleph_1} not-> (omega_1 x omega)^2_2''. If fact, we can get 2^{aleph_1} not-> [omega_1 x omega]^2_{aleph_0}. In addition to this theorem, we give generalizations to other cardinals.
It is proved consistent with either CH or the negation of CH that there is an aleph_1-separable group of cardinality aleph_1 which does not have a coherent system of projections. It had previously been shown that it is consistent with not CH that every aleph_1-separable group of cardinality aleph_1 does have a coherent system of projections.
It is shown to be consistent that there is a non-trivial autohomeomorphism of beta N while all such autohomeomorphisms are trivial on some open set. The model used is one due to Velickovic in which, coincidentally, Martin's Axiom also holds.
A graph is called dominating if its vertices can be labelled with integers in such a way that for every function f: omega-> omega the graph contains a ray whose sequence of labels eventually exceeds f. We obtain a characterization of these graphs by producing a small family of dominating graphs with the property that every dominating graph must contain some member of the family.
A model M of cardinality lambda is said to have the small index property if for every G subseteq Aut(M) such that [Aut(M):G] <= lambda there is an A subseteq M with |A|< lambda such that Aut_A(M) subseteq G. We show that if M^* is a saturated model of an unsuperstable theory of cardinality > Th(M), then M^* has the small index property.
Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a countable base and is not countable, then T has cardinality at least continuum.
By an omega_1 --tree we mean a tree of power omega_1 and height omega_1. We call an omega_1 --tree a Jech--Kunen tree if it has kappa --many branches for some kappa strictly between omega_1 and 2^{omega_1}. In this paper we construct the models of CH plus 2^{omega_1}> omega_2, in which there are Jech--Kunen trees and there are no Kurepa trees.
Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We show here that it is consistent that there is no interpretation even in the monadic second-order theory of all chains.
In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D, either C <= D or D <= C. However, in ZF this is no longer so. For a given infinite set A consider Seq(A), the set of all sequences of A without repetition. We compare |Seq(A)|, the cardinality of this set, to |P(A)|, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF |- for all A: |Seq(A)| not= |P(A)| and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then |fin(B)|<|P(B)|, even though the existence for some infinite set B^* of a function f from fin(B^*) onto P(B^*) is consistent with ZF.
We give some existence/nonexistence statements on universal graphs, which under GCH give a necessary and sufficient condition for the existence of a universal graph of size lambda with no K(kappa), namely, if either kappa is finite or cf(kappa)>cf(lambda). (Here K(kappa) denotes the complete graph on kappa vertices.) The special case when lambda^{< kappa}= lambda was first proved by F. Galvin. Next, we investigate the question that if there is no universal K(kappa)-free graph of size lambda then how many of these graphs embed all the other. It was known, that if lambda^{< lambda}= lambda (e.g., if lambda is regular and the GCH holds below lambda), and kappa = omega, then this number is lambda^+. We show that this holds for every kappa <= lambda of countable cofinality. On the other hand, even for kappa = omega_1, and any regular lambda >= omega_1 it is consistent that the GCH holds below lambda, 2^{lambda} is as large as we wish, and the above number is either lambda^+ or 2^{lambda}, so both extremes can actually occur.
We show that there is always a uniformly antisymmetric f:A-> {0,1} if A subset R is countable. We prove that the continuum hypothesis is equivalent to the statement that there is an f:R-> omega with |S_x| <= 1 for every x in R. If the continuum is at least aleph_n then there exists a point x such that S_x has at least 2^n-1 elements. We also show that there is a function f:Q-> {0,1,2,3} such that S_x is always finite, but no such function with finite range on R exists
If ZFC is consistent, then each of the following are consistent with ZFC + 2^{aleph_0}= aleph_2 : 1.) X subseteq R is of strong measure zero iff |X| <= aleph_1 + there is a generalized Sierpinski set. 2.) The union of aleph_1 many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size aleph_2.