On the spectrum of characters of ultrafilters
We show that the character spectrum $Sp_χ(λ)$, for a singular cardinal $λ$ of countable cofinality, may include any prescribed set of regular cardinals between $λ$ and $2^λ$.
math.LO↗
arXiv subjects
Menachem Magidor
Publications and source records attributed to Menachem Magidor.
On supercompactness and the continuum function
Given a cardinal $κ$ that is $λ$-supercompact for some regular cardinal $λ\geqκ$ and assuming $\GCH$, we show that one can force the continuum function to agree with any function $F:[κ,λ]\cap\REG\to\CARD$ satisfying $\forallα,β\in\dom(F)$ $α<\cf(F(α))$ and $α<β$ $\implies$ $F(α)\leq F(β)$, while preserving the $λ$-supercompactness of $κ$ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding $j:V\to M$ with critical point $κ$ such that $M^λ\subseteq M$ and $j(κ)>F(λ)$. Our argument extends Woodin's technique of surgically modifying a generic filter to a new case: Woodin's key lemma applies when modifications are done on the range of $j$, whereas our argument uses a new key lemma to handle modifications done off of the range of $j$ on the ghost coordinates. This work answers a question of Friedman and Honzik [FH2012]. We also discuss several related open questions.
math.LO↗
Independence of the existence of Pitowsky spin models
In 1982 I. Pitowsky used Continuum Hypothesis to construct hidden variable models for spin-1/2 and spin-1 particles in quantum mechanics. We show that the existence of Pitowsky models is independent from ZFC.
quant-ph↗
Nonmonotonic Reasoning, Preferential Models and Cumulative Logics
Many systems that exhibit nonmonotonic behavior have been described and studied already in the literature. The general notion of nonmonotonic reasoning, though, has almost always been described only negatively, by the property it does not enjoy, i.e. monotonicity. We study here general patterns of nonmonotonic reasoning and try to isolate properties that could help us map the field of nonmonotonic reasoning by reference to positive properties. We concentrate on a number of families of nonmonotonic consequence relations, defined in the style of Gentzen. Both proof-theoretic and semantic points of view are developed in parallel. The former point of view was pioneered by D. Gabbay, while the latter has been advocated by Y. Shoham in. Five such families are defined and characterized by representation theorems, relating the two points of view. One of the families of interest, that of preferential relations, turns out to have been studied by E. Adams. The "preferential" models proposed here are a much stronger tool than Adams' probabilistic semantics. The basic language used in this paper is that of propositional logic. The extension of our results to first order predicate calculi and the study of the computational complexity of the decision problems described in this paper will be treated in another paper.
cs.AI↗
What does a conditional knowledge base entail?
This paper presents a logical approach to nonmonotonic reasoning based on the notion of a nonmonotonic consequence relation. A conditional knowledge base, consisting of a set of conditional assertions of the type "if ... then ...", represents the explicit defeasible knowledge an agent has about the way the world generally behaves. We look for a plausible definition of the set of all conditional assertions entailed by a conditional knowledge base. In a previous paper, S. Kraus and the authors defined and studied "preferential" consequence relations. They noticed that not all preferential relations could be considered as reasonable inference procedures. This paper studies a more restricted class of consequence relations, "rational" relations. It is argued that any reasonable nonmonotonic inference procedure should define a rational relation. It is shown that the rational relations are exactly those that may be represented by a "ranked" preferential model, or by a (non-standard) probabilistic model. The rational closure of a conditional knowledge base is defined and shown to provide an attractive answer to the question of the title. Global properties of this closure operation are proved: it is a cumulative operation. It is also computationally tractable. This paper assumes the underlying language is propositional.
cs.AI↗
Distance Semantics for Belief Revision
A vast and interesting family of natural semantics for belief revision is defined. Suppose one is given a distance d between any two models. One may then define the revision of a theory K by a formula a as the theory defined by the set of all those models of a that are closest, by d, to the set of models of K. This family is characterized by a set of rationality postulates that extends the AGM postulates. The new postulates describe properties of iterated revisions.
cs.AI↗
Length of Boolean algebras and ultraproducts
We prove the consistency with ZFC of ``the length of an ultraproduct of Boolean algebras is smaller than the ultraproduct of the lengths''. Similarly for some other cardinal invariants of Boolean algebras.
math.LO↗
The tree property at successors of singular cardinals
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
math.LO↗
Bext^2(G,T) can be nontrivial, even assuming GCH
Using the consistency of some large cardinals we produce a model of Set Theory in which the generalized continuum hypothesis holds and for some torsion-free abelian group G of cardinality aleph_{omega +1} and for some torsion group T, Bext^2(G,T) not =0 .
math.LO↗