218 research outputs found
The Strawberry Statement: Notes of a College Revolutionary
Personal narrative excerpted from The Strawberry Statement: Notes of a College Revolutionary (New York: Random House, 1969): 26-45
Topological partition relations to the form omega^*-> (Y)^1_2
Theorem: The topological partition relation omega^{*}-> (Y)^{1}_{2}
(a) fails for every space Y with |Y| >= 2^c ;
(b) holds for Y discrete if and only if |Y| <= c;
(c) holds for certain non-discrete P-spaces Y ;
(d) fails for Y= omega cup {p} with p in omega^{*} ;
(e) fails for Y infinite and countably compact
A simultaneous generalization of independence and disjointness in boolean algebras
We give a definition of some classes of boolean algebras generalizing free
boolean algebras; they satisfy a universal property that certain functions
extend to homomorphisms. We give a combinatorial property of generating sets of
these algebras, which we call n-independent. The properties of these classes
(n-free and omega-free boolean algebras) are investigated. These include
connections to hypergraph theory and cardinal invariants on these algebras.
Related cardinal functions, Ind, which is the supremum of the cardinalities
of n-independent subsets; i_n, the minimum size of a maximal n-independent
subset; and i_omega, the minimum size of an omega-independent subset, are
introduced and investigated. The values of i_n and i_omega on P(omega)/fin are
shown to be independent of ZFC.Comment: Sumbitted to Orde
A convergence on Boolean algebras generalizing the convergence on the Aleksandrov cube
We compare the forcing related properties of a complete Boolean algebra B
with the properties of the convergences (the algebraic convergence)
and on B generalizing the convergence on the Cantor and
Aleksandrov cube respectively. In particular we show that is a
topological convergence iff forcing by B does not produce new reals and that
is weakly topological if B satisfies condition
(implied by the -cc). On the other hand, if is a
weakly topological convergence, then B is a -cc algebra or in
some generic extension the distributivity number of the ground model is greater
than or equal to the tower number of the extension. So, the statement "The
convergence on the collapsing algebra B=\ro
((\omega_2)^{<\omega}) is weakly topological" is independent of ZFC
A Generalization of Martin's Axiom
We define the chain condition. The corresponding forcing axiom
is a generalization of Martin's Axiom and implies certain uniform failures of
club--guessing on that don't seem to have been considered in the
literature before.Comment: 36 page
Quantum energy inequalities and local covariance II: Categorical formulation
We formulate Quantum Energy Inequalities (QEIs) in the framework of locally
covariant quantum field theory developed by Brunetti, Fredenhagen and Verch,
which is based on notions taken from category theory. This leads to a new
viewpoint on the QEIs, and also to the identification of a new structural
property of locally covariant quantum field theory, which we call Local
Physical Equivalence. Covariant formulations of the numerical range and
spectrum of locally covariant fields are given and investigated, and a new
algebra of fields is identified, in which fields are treated independently of
their realisation on particular spacetimes and manifestly covariant versions of
the functional calculus may be formulated.Comment: 27 pages, LaTeX. Further discussion added. Version to appear in
General Relativity and Gravitatio
Set Theory and its Place in the Foundations of Mathematics:a new look at an old question
This paper reviews the claims of several main-stream candidates to be the foundations of mathematics, including set theory. The review concludes that at this level of mathematical knowledge it would be very unreasonable to settle with any one of these foundations and that the only reasonable choice is a pluralist one
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
A method for finding new sets of axioms for classes of semigroups
We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices
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