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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 505 records · Page 28Linked to original sources
arXiv2002

Pcf theory and Woodin cardinals

We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|} aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists. Theorem B: Let kappa be a singular cardinal of uncountable cofinality. If {alpha<kappa| 2^alpha=alpha^+} is stationary as well as co-stationary then for all n< omega and for all bounded X subset kappa, M_n^#(X) exists. Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound for an assertion discussed in Gitik, M., Introduction to Prikry type forcing notions, in: Handbook of set theory, Foreman, Kanamori, Magidor (see Problem 4 there). The proofs of these theorems combine pcf theory with core model theory. Along the way we establish some ZFC results in cardinal arithmetic, motivated by Silver's theorem and we obtain results of core model theory, motivated by the task of building a ``stable core model.'' Both sets of results are of independent interest.

math.LO↗
arXiv2002

The Future of Set Theory

Haim Judah has asked me to speak on the future of set theory, so, as the next millennium is coming, to speak on set theory in the next millennium. But we soon cut this down to set theory in the next century. Later on I thought I had better cut it down to dealing with the next decade, but I suspect I will speak on what I hope to try to prove next year, or worse - what I have done in the last year (or twenty). It seems I am not particularly suitable for such a lecture, as I have repeatedly preferred to try to prove another theorem rather than to prepare the lecture (or article); so why did I agree at all to such a doubtful endeavor? Well, under the hypothesis that I had some moral obligation to help Haim in the conference (and the proceedings) and you should not let a friend down, had I been given the choice to help with organizing the dormitories, writing a lengthy well written expository paper or risking making a fool of myself in such a lecture, I definitely prefer the last.

math.LO↗
arXiv2002

Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic.

math.LO↗
arXiv2002

Some notes concerning the homogeneity of Boolean algebras and Boolean spaces

We consider homogeneity properties of Boolean algebras that have nonprincipal ultrafilters which are countably generated.It is shown that a Boolean algebra B is homogeneous if it is the union of countably generated nonprincipal ultrafilters and has a dense subset D such that for every a in D the relative algebra B restriction a:= {b in B:b <= a} is isomorphic to B. In particular, the free product of countably many copies of an atomic Boolean algebra is homogeneous. Moreover, a Boolean algebra B is homogeneous if it satisfies the following conditions: (i) B has a countably generated ultrafilter, (ii) B is not c.c.c., and (iii) for every a in B setminus {0} there are finitely many automorphisms h_1, ...,h_n of B such that 1=h_1(a) cup ... cup h_n(a).

math.LO↗
arXiv2002

Cofinality of the nonstationary ideal

We show that the reduced cofinality of the nonstationary ideal NS_kappa on a regular uncountable cardinal kappa may be less than its cofinality, where the reduced cofinality of NS_kappa is the least cardinality of any family F of nonstationary subsets of kappa such that every nonstationary subset of kappa can be covered by less than $κ$ many members of F.

math.LO↗
arXiv2002

Successors of singular cardinals and coloring theorems. I

We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.

math.LO↗
arXiv2001

Forcing axiom failure for any lambda>aleph_1

David Aspero asks on the possibility of having Forcing axiom FA_{aleph_2}(K), where K is the class of forcing notions preserving stationarity of subsets of aleph_1 and of aleph_2. We answer negatively, in fact we show the negative result for any regular lambda>aleph_1 even demanding adding no new sequence of ordinals of length<lambda.

math.LO↗
arXiv2001

Potential isomorphism and semi-proper trees

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notions of semi-proper and weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We prove the existence of semi-proper trees under certain cardinal arithmetic assumptions. We also show the consistency of the non-existence of weakly semi-proper trees assuming the consistency of some large cardinals.

math.LO↗
arXiv2001

Characterizing automorphism groups of ordered abelian groups

We characterize the groups isomorphic to full automorphism groups of ordered abelian groups. The result will follow from classical theorems on ordered groups adding an argument from proofs used to realize rings as endomorphism rings of abelian groups.

math.LO↗
arXiv2001

Van der Waerden spaces and Hindman spaces are not the same

A Hausdorff topological space X is van der Waerden if for every sequence (x_n)_n in X there is a converging subsequence (x_n)_{n in A} where subset A of omega contains arithmetic progressions of all finite lengths. A Hausdorff topological space X is Hindman if for every sequence (x_n)_n in X there is an IP-converging subsequence (x_n)_{n in FS(B)} for some infinite subset B of omega. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.

math.GN↗
arXiv2001

On universal and epi-universal locally nilpotent groups

In this paper we mainly consider the class LN of all locally nilpotent groups. We first show that there is no universal group in LN_lambda if lambda is a cardinal such that lambda = lambda^{aleph_0}; here we call a group G universal (in LN_lambda) if any group H in LN_lambda can be embedded into G where LN_lambda denotes the class of all locally nilpotent groups of cardinality at most lambda. However, our main interest is the construction of torsion-free epi-universal groups in LN_lambda, where G in LN_lambda is said to be epi-universal if any group H in LN_lambda is an epimorphic image of G. Thus we give an affirmative answer to a question by Plotkin. To prove the torsion-freeness of the constructed locally nilpotent group we adjust the well-known commutator collecting process due to P. Hall to our situation. Finally, we briefly discuss how to use the same methods as for the class LN for other canonical classes of groups to construct epi-universal objects.

math.LO↗
arXiv2001

Kulikov's problem on universal torsion-free abelian groups

Let T be an abelian group and lambda an uncountable regular cardinal. We consider the question of whether there is a lambda-universal group G^* among all torsion-free abelian groups G of cardinality less than or equal to lambda satisfying Ext(G,T)=0. Here G^* is said to be lambda-universal for T if, whenever a torsion-free abelian group G of cardinality less than or equal to lambda satisfies Ext(G,T)=0, then there is an embedding of G into G^*. For large classes of abelian groups T and cardinals lambda it is shown that the answer is consistently no. In particular, for T torsion, this solves a problem of Kulikov.

math.LO↗
arXiv2001

Vive la difference III

We show that, consistently, there is an ultrafilter F on omega such that if N^l_n=(P^l_n cup Q^l_n,P^l_n,Q^l_n,R^l_n) (for l =1,2, n< omega), P^l_n cup Q^l_n subseteq omega, and prod_{n<omega} N^1_n/F and prod_{n< omega}N^2_n/F are isomorphic models of the canonical theory t^ind of the strong independence property, then every isomorphism from prod_{n<omega} N^1_n/F onto prod_{n<omega} N^2_n/F is a product isomorphism.

math.LO↗
arXiv2001

Long iterations for the continuum

We deal with an iteration theorem of forcing notion with a kind of countable support of nice enough forcing notion which is proper aleph_2-c.c. forcing notions. We then look at some special cases (Q_D 's preceded by random forcing).

math.LO↗
arXiv2001

On the Arrow property

Let X be a finite set of alternatives. A choice function c is a mapping which assigns to nonempty subsets S of X an element c(S) of S. A rational choice function is one for which there is a linear ordering on the alternatives such that c(S) is the maximal element of S according to that ordering. Arrow's impossibility theorem asserts that under certain natural conditions, if there are at least three alternatives then every non-dictatorial social choice gives rise to a non-rational choice function. Gil Kalai asked if Arrow's theorem can be extended to the case when the individual choices are not rational but rather belong to an arbitrary non-trivial symmetric class of choice functions. The main theorem of this paper gives an affirmative answer in a very general setting.

math.LO↗
arXiv2001

Almost-free E-rings of cardinality aleph_1

An E-ring is a unital ring R such that every endomorphism of the underlying abelian group R^+ is multiplication by some ring-element. The existence of almost-free E-rings of cardinality greater than 2^{aleph_0} is undecidable in ZFC. While they exist in Goedel's universe, they do not exist in other models of set theory. For a regular cardinal aleph_1 <= lambda <= 2^{aleph_0} we construct E-rings of cardinality lambda in ZFC which have aleph_1-free additive structure. For lambda = aleph_1 we therefore obtain the existence of almost-free E-rings of cardinality aleph_1 in ZFC.

math.LO↗