Approximations of -classes and -classes
Äquivalenz- und Ordnungsrelationen
Archimedean and geodetical biequivalences
Archimedean atomic lattice effect algebras in which all sharp elements are central
We prove that every Archimedean atomic lattice effect algebra the center of which coincides with the set of all sharp elements is isomorphic to a subdirect product of horizontal sums of finite chains, and conversely. We show that every such effect algebra can be densely embedded into a complete effect algebra (its MacNeille completion) and that there exists an order continuous state on it.
Archimedean atomic lattice effect algebras with complete lattice of sharp elements.
Archimedean frames, revisited
This paper extends the notion of an archimedean frame to frames which are not necessarily algebraic. The new notion is called joinfitness and is Choice-free. Assuming the Axiom of Choice and for compact normal algebraic frames, the new and the old coincide. There is a subfunctor from the category of compact normal frames with skeletal maps with joinfit values, which is almost a coreflection. Conditions making it so are briefly discussed. The concept of an infinitesimal element arises naturally,...
Archimedean unital groups with finite unit intervals.
Arhangel'skiĭ sheaf amalgamations in topological groups
We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property is equivalent to Arhangel’skiĭ’s formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space of continuous real-valued functions on X with the topology...
Arithmetic of cuts and cuts of classes
Arithmetic over the rings of all algebraic integers.
Arithmetical classification of the set of all provably recursive functions
The set of all indices of all functions provably recursive in any reasonable theory is shown to be recursively isomorphic to , where is -complete set.
Arithmetical m-degrees.
Arithmetical transfinite induction and hierarchies of functions
We generalize to the case of arithmetical transfinite induction the following three theorems for PA: the Wainer Theorem, the Paris-Harrington Theorem, and a version of the Solovay-Ketonen Theorem. We give uniform proofs using combinatorial constructions.
Arithmetically independent integers and values of rational functions.
Arithmetische Prädikate über einem Bereich endlicher Automaten.
Arithmetization of the field of reals with exponentiation extended abstract
(1) Shepherdson proved that a discrete unitary commutative semi-ring A+ satisfies IE0 (induction scheme restricted to quantifier free formulas) iff A is integral part of a real closed field; and Berarducci asked about extensions of this criterion when exponentiation is added to the language of rings. Let T range over axiom systems for ordered fields with exponentiation; for three values of T we provide a theory in the language of rings plus exponentiation such that the ...
Aronszajn orderings.
Around cofin
We show the consistency of "there is a nice σ-ideal ℐ on the reals with add(ℐ) = ℵ₁ which cannot be represented as the union of a strictly increasing sequence of length ω₁ of σ-subideals". This answers [Borodulin-Nadzieja and Głąb, Math. Logic Quart. 57 (2011), 582-590, Problem 6.2(ii)].
Around Podewski's conjecture
A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski's conjecture to the (partially) ordered case, and we conjecture that such fields do not exist. We prove the conjecture in case the incomparability relation is transitive (the almost linear case). We also study minimal groups with a (partial) order, and give a complete classification of...
Around splitting and reaping
We prove several results on some cardinal invariants of the continuum which are closely related to either the splitting number or its dual, the reaping number .