16,750 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
Cardinal characteristics and countable Borel equivalence relations
Boykin and Jackson recently introduced a property of countable Borel
equivalence relations called Borel boundedness, which they showed is closely
related to the union problem for hyperfinite equivalence relations. In this
paper, we introduce a family of properties of countable Borel equivalence
relations which correspond to combinatorial cardinal characteristics of the
continuum in the same way that Borel boundedness corresponds to the bounding
number . We analyze some of the basic behavior of these
properties, showing for instance that the property corresponding to the
splitting number coincides with smoothness. We then settle many
of the implication relationships between the properties; these relationships
turn out to be closely related to (but not the same as) the Borel Tukey
ordering on cardinal characteristics
Isomorphism versus commensurability for a class of finitely presented groups
We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are first examples of such a contrastive complexity behaviour with respect to the isomorphism problem
Analogs of Schur functions for rank two Weyl groups obtained from grid-like posets
In prior work, the authors, along with M. McClard, R. A. Proctor, and N. J.
Wildberger, studied certain distributive lattice models for the "Weyl
bialternants" (aka "Weyl characters") associated with the rank two root
systems/Weyl groups. These distributive lattices were uniformly described as
lattices of order ideals taken from certain grid-like posets, although the
arguments connecting the lattices to Weyl bialternants were case-by-case
depending on the type of the rank two root system. Using this connection with
Weyl bialternants, these lattices were shown to be rank symmetric and rank
unimodal, and their rank generating functions were shown to have beautiful
quotient-of-products expressions. Here, these results are re-derived from
scratch using completely uniform and elementary combinatorial reasoning in
conjunction with some new combinatorial methodology developed elsewhere by the
second listed author.Comment: 15 page
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