On the number of countable models of complete theories with finite Rudin-Keisler preorders.
On the number of countable models of stable theories
We prove: Theorem. If T is a countable, complete, stable, first-order theory having an infinite set of constants with different interpretations, then I(T,ℵ₀) ≥ ℵ₀.
On the number of generic models
On the number of models of the Kelley-Morse theory of classes
On the number of monotonic functions from two-valued logic to -valued logic
On the number of non-isomorphic subspaces of a Banach space
We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, , or for a residual set of infinite subsets A of ℕ, is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to for any D ⊂ ℕ, and isomorphic to a denumerable Schauder decomposition...
On the number of Russell’s socks or
The following question is analyzed under the assumption that the Axiom of Choice fails badly: Given a countable number of pairs of socks, then how many socks are there? Surprisingly this number is not uniquely determined by the above information, thus giving rise to the concept of Russell-cardinals. It will be shown that: • some Russell-cardinals are even, but others fail to be so; • no Russell-cardinal is odd; • no Russell-cardinal is comparable with any cardinal of the form or ; • finite sums...
On the open-open game
We modify a game due to Berner and Juhász to get what we call “the open-open game (of length ω)”: a round consists of player I choosing a nonempty open subset of a space X and II choosing a nonempty open subset of I’s choice; I wins if the union of II’s open sets is dense in X, otherwise II wins. This game is of interest for ccc spaces. It can be translated into a game on partial orders (trees and Boolean algebras, for example). We present basic results and various conditions under which I or II...
On the Order Type L-valued Relations on L-powersets.
On the orthogonalization of arbitrary Boolean formulae.
On the Paradox of Confirmation.
On the planarity of the equilateral, isogonal pentagon.
On the ◊ principle
On the principle square : coding and extending embeddings
On the problem of axiomatization of tame representation type
Associative algebras of fixed dimension over algebraically closed fields of fixed characteristic are considered. It is proved that the class of algebras of tame representation type is axiomatizable. Moreover, finite axiomatizability of this class is equivalent to the conjecture that the algebras of tame representation type form a Zariski-open subset in the variety of algebras.
On the problem of finite signature.
On the products of quantum logics
On the propositional system A of Sobocinski
On the Quantifier of Limiting Realizability
On the range of a Boolean transformation.