The uncountable spectra of countable theories.
The Undecidability of Pseudo Real Closed Fields.
The undecidability of the existence of a non-separable normal Moore space satisfying the countable chain condition
The unique existential quantifier.
The uniqueness of Haar measure and set theory
Let G be a group of homeomorphisms of a nondiscrete, locally compact, σ-compact topological space X and suppose that a Haar measure on X exists: a regular Borel measure μ, positive on nonempty open sets, finite on compact sets and invariant under the homeomorphisms from G. Under some mild assumptions on G and X we prove that the measure completion of μ is the unique, up to a constant factor, nonzero, σ-finite, G-invariant measure defined on its domain iff μ is ergodic and the G-orbits of all points...
The unsolvability of the emptiness problem of the two deterministic finite transducer mappings
The use of injective-like structures in model theory
The variational principle of fixed point theorems in certain fuzzy topological spaces
The main purpose of this paper is to introduce the concept of -type fuzzy topological spaces. Further variational principle and Caristi’s fixed point theorem have been extended in the -type fuzzy topological spaces.
The weak extension property and finite axiomatizability for quasivarieties
We define and compare a selection of congruence properties of quasivarieties, including the relative congruence meet semi-distributivity, RSD(∧), and the weak extension property, WEP. We prove that if 𝒦 ⊆ ℒ ⊆ ℒ' are quasivarieties of finite signature, and ℒ' is finitely generated while 𝒦 ⊨ WEP, then 𝒦 is finitely axiomatizable relative to ℒ. We prove for any quasivariety 𝒦 that 𝒦 ⊨ RSD(∧) iff 𝒦 has pseudo-complemented congruence lattices and 𝒦 ⊨ WEP. Applying these results and other results...
The Wholeness Axioms and the Class of Supercompact Cardinals
We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.
The word problem in polycyclic groups is elementary
The "World's Simplest Axiom of Choice" Fails.
The Zero-one law for Products of Internal *-finitely Additive Probabiltty Spaces
The Ziegler spectrum of a tame hereditary algebra
The -calculus alternation-depth hierarchy is strict on binary trees
The μ-calculus alternation-depth hierarchy is strict on binary trees
In this paper we give a simple proof that the alternation-depth hierarchy of the μ-calculus for binary trees is strict. The witnesses for this strictness are the automata that determine whether there is a winning strategy for the parity game played on a tree.
The Σ* approach to the fine structure of L
We present a reformulation of the fine structure theory from Jensen [72] based on his Σ* theory for K and introduce the Fine Structure Principle, which captures its essential content. We use this theory to prove the Square and Fine Scale Principles, and to construct Morasses.
The -property in
The -property of a Riesz space (real vector lattice) is: For each sequence of positive elements of , there is a sequence of positive reals, and , with for each . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “” obtains for a Riesz space of continuous real-valued functions . A basic result is: For discrete , has iff the cardinal , Rothberger’s bounding number. Consequences and...
The σ-complete MV-algebras which have enough states
We characterize Łukasiewicz tribes, i.e., collections of fuzzy sets that are closed under the standard fuzzy complementation and the Łukasiewicz t-norm with countably many arguments. As a tool, we introduce σ-McNaughton functions as the closure of McNaughton functions under countable MV-algebraic operations. We give a measure-theoretical characterization of σ-complete MV-algebras which are isomorphic to Łukasiewicz tribes.
The σ-ideal of closed smooth sets does not have the covering property
We prove that the σ-ideal I(E) (of closed smooth sets with respect to a non-smooth Borel equivalence relation E) does not have the covering property. In fact, the same holds for any σ-ideal containing the closed transversals with respect to an equivalence relation generated by a countable group of homeomorphisms. As a consequence we show that I(E) does not have a Borel basis.