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Saharon Shelah

Publications and source records attributed to Saharon Shelah.

At least 379 records · Page 21Linked to original sources
arXiv2008

Decisive creatures and large continuum

For $f,g\inω\ho$ let $\mycfa_{f,g}$ be the minimal number of uniform $g$-splitting trees needed to cover the uniform $f$-splitting tree, i.e. for every branch $ν$ of the $f$-tree, one of the $g$-trees contains $ν$. $\myc_{f,g}$ is the dual notion: For every branch $ν$, one of the $g$-trees guesses $ν(m)$ infinitely often. It is consistent that $\myc_{f_ε,g_ε}=\mycfa_{f_ε,g_ε}=κ_ε$ for $\al1$ many pairwise different cardinals $κ_ε$ and suitable pairs $(f_ε,g_ε)$. For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision.

math.LO↗
arXiv2008

Model theory without choice: Categoricity

We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in aleph_alpha for some (equivalently for every) ordinal alpha>0. Another central result here is, in this context: the number of models of a countable first order T of cardinality aleph_alpha is either >=|alpha| for every aleph_αor it has a small upper bound (close to beth_2).

math.LO↗
arXiv2008

Advances in Cardinal Arithmetic

If cf(kappa) = kappa, kappa^+< cf(lambda) = λ, then there is a stationary subset S of {delta , C_delta a club of lambda, otp(C_delta)=kappa, guessing clubs and for each alpha<lambda we have: {C_delta \cap alpha: alpha \in nacc(C_delta)} has cardinality <lambda. Also, we prove that e.g. there is a stationary subset of S_{<aleph_1}(lambda) of cardinality cf(S_{<aleph_1}(lambda),subseteq) Then we prove the existence of nice filters when instead being normal filters on omega_1 they are normal filters with larger domains, which can increase during a play. They can help us transfer situation on aleph_1-complete filters to normal ones

math.LO↗
arXiv2008

What majority decisions are possible

Suppose we are given a family of choice functions on pairs from a given finite set (with at least three elements) closed under permutations of the given set. The set is considered the set of alternatives (say candidates for an office). The question is, what are the choice functions c on pairs of this set of the following form: for some (finite) family of ``voters'', each having a preference, i.e., a choice from each pair from the given family, c{x,y} is chosen by the preference of the majority of voters. We give full characterization.

math.CO↗
arXiv2008

Depth of Boolean algebras

We prove a Theorem about the relationship between the Depth of the ultraproduct of Boolean algebras, divided by an ultrafilter, and the products of the depths of each component. This answers (partly) an open problem of Monk.

math.LO↗
arXiv2007

Absolutely Indecomposable Modules

A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about R-modules over a large class of commutative rings R with endomorphism ring R which remains the same when passing to a generic extension of the universe. It turns out that `large' in this context has the precise meaning, namely being smaller then the first omega-Erdos cardinal defined below. We will first apply result on large rigid trees with a similar property established by Shelah in 1982, and will prove the existence of related ` R_omega-modules' (R-modules with countably many distinguished submodules) and finally pass to R-modules. The passage through R_omega-modules has the great advantage that the proofs become very transparent essentially using a few `linear algebra' arguments accessible also for graduate students. The result gives a new construction of indecomposable modules in general using a counting argument.

math.LO↗
arXiv2007

Karp height of models of stable theories

A trichotomy theorem for countable, stable, unsuperstable theories is offered. We develop the notion of a `regular ideal' of formulas and study types that are minimal with respect to such an ideal.

math.LO↗
arXiv2007

Generalized E-Algebras via lambda-Calculus I

An R-algebra A is called E(R)-algebra if the canonical homomorphism from A to the endomorphism algebra End_RA of the R-module {}_R A, taking any a in A to the right multiplication a_r in End_R A by a is an isomorphism of algebras. In this case {}_R A is called an E(R)-module. E(R)-algebras come up naturally in various topics of algebra, so it's not surprising that they were investigated thoroughly in the last decade. Despite some efforts it remained an open question whether proper generalized E(R)-algebras exist. These are R-algebras A isomorphic to End_R A but not under the above canonical isomorphism, so not E(R)-algebras. This question was raised about 30 years ago (for R=Z) by Phil Schultz and we will answer it. For PIDs R of characteristic 0 that are neither quotient fields nor complete discrete valuation rings - we will establish the existence of generalized E(R)-algebras. It can be shown that E(R)-algebras over rings R that are complete discrete valuation rings or fields must trivial (copies of R). The main tool is an interesting connection between lambda-calculus (used in theoretical computer sciences) and algebra. It seems reasonable to divide the work into two parts, in this paper we will work in V=L (Godel universe) hence stronger combinatorial methods make the final arguments more transparent. The proof based entirely on ordinary set theory (the axioms of ZFC) will appear in a subsequent paper.

math.LO↗
arXiv2007

Basic Subgroups and Freeness, A Counterexample

We construct a non-free but aleph_1-separable, torsion-free abelian group G with a pure free subgroup B such that all subgroups of G disjoint from B are free and such that G/B is divisible. This answers a question of Irwin and shows that a theorem of Blass and Irwin cannot be strengthened so as to give an exact analog for torsion-free groups of a result proved for p-groups by Benabdallah and Irwin.

math.LO↗
arXiv2007

Tie-points and fixed-points in N^*

A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of betaN setminus N and in the recent study of (precisely) 2-to-1 maps on betaN setminus N . In these cases the tie-points have been the unique fixed point of an involution on betaN setminus N. This paper is motivated by the search for 2-to-1 maps and obtaining tie-points of strikingly differing characteristics.

math.LO↗
arXiv2007

More on Tie-points and homeomorphism in N^*

A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of betaN setminus N= N^* and in the recent study of (precisely) 2-to-1 maps on N^*. In these cases the tie-points have been the unique fixed point of an involution on N^*. One application of the results in this paper is the consistency of there being a 2-to-1 continuous image of N^* which is not a homeomorph of N^* .

math.LO↗
arXiv2007

Reflection implies the SCH

We prove that, e.g., if mu >cf(mu)= aleph_0 and mu>2^{aleph_0} and every stationary family of countable subsets of mu^+ reflect in some subset of mu^+ of cardinality aleph_1, then the SCH for mu^+ (moreover, for mu^+, any scale for mu^+ has a bad stationary set of cofinality aleph_1). This answers a question of Foreman and Todorcevic who got such conclusion from the simultaneous reflection of four stationary sets.

math.LO↗
arXiv2007

EF equivalent not isomorphic models

We construct non-isomorphic models M, N, e.g. of cardinality aleph_1 such that in the Ehrenfeucht-Fraisse game of length zeta < omega_1 the isomorphism player wins

math.LO↗
arXiv2007

Generating ultrafilters in a reasonable way

We continue investigations of reasonable ultrafilters on uncountable cardinals defined in Shelah math.LO/0407498 and studied also in math.LO/0605067. We introduce a general scheme of generating a filter on lambda from filters on smaller sets and we investigate the combinatorics of objects obtained this way.

math.LO↗
arXiv2007

On some problems in general topology

We prove that Arhangelskii's problem has a consistent positive answer: if V\models CH, then for some aleph_1-complete aleph_2-c.c. forcing notion P of cardinality aleph_2 we have that P forces ``CH and there is a Lindelof regular topological space of size aleph_2 with clopen basis with every point of pseudo-character aleph_0 (i.e. each singleton is the intersection of countably many open sets)''. Also, we prove the consistency of: CH+ 2^{aleph_1} > \aleph_2 + "there is no space as above with aleph_2 points" (starting with a weakly compact cardinal).

math.LO↗