49,667 research outputs found
Models with second order properties, V: A General principle
We present a general framework for carrying out some constructions. The
unifying factor is a combinatorial principle which we present in terms of a
game in which the first player challenges the second player to carry out
constructions which would be much easier in a generic extension of the
universe, and the second player cheats with the aid of Diamond. Section 1
contains an axiomatic framework suitable for the description of a number of
related constructions, and the statement of the main theorem in terms of this
framework. In section 2 we illustrate the use of our combinatorial principle.
The proof of the main result is then carried out in sections 3-5
Covering by discrete and closed discrete sets
Say that a cardinal number is \emph{small} relative to the space
if , where is the least cardinality of a
non-empty open set in . We prove that no Baire metric space can be covered
by a small number of discrete sets, and give some generalizations. We show a
ZFC example of a regular Baire -space and a consistent example of a
normal Baire Moore space which can be covered by a small number of discrete
sets. We finish with some remarks on linearly ordered spaces.Comment: 12 pages, to appear on Topology and its Application
Computability of simple games: A complete investigation of the sixty-four possibilities
Classify simple games into sixteen "types" in terms of the four conventional
axioms: monotonicity, properness, strongness, and nonweakness. Further classify
them into sixty-four classes in terms of finiteness (existence of a finite
carrier) and algorithmic computability. For each such class, we either show
that it is empty or give an example of a game belonging to it. We observe that
if a type contains an infinite game, then it contains both computable ones and
noncomputable ones. This strongly suggests that computability is logically, as
well as conceptually, unrelated to the conventional axioms.Comment: 25 page
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