The alternation hierarchy for the theory of -lattices.
The Amalgamation Property, the Universal-Homogeneous Models, and the Generic Models.
The arithmetic and turing degrees are not elementary equivalent.
The Arkhangel’skiĭ–Tall problem: a consistent counterexample
We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in , and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.
The Arkhangel'skiĭ–Tall problem under Martin’s Axiom
We show that MA implies that normal locally compact metacompact spaces are paracompact, and that MA() implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.
The automorphism group of the random lattice is not amenable
We prove that the automorphism group of the random lattice is not amenable, and we identify the universal minimal flow for the automorphism group of the random distributive lattice.
The axiom of choice and linearly ordered sets
The axiom of choice for linearly ordered families
The Axiom of Choice, the Löwenheim-Skolem Theorem and Borel models
The Axiom of Determinateness and canonical measures
The axiom of determinateness implies has precisely two countably complete, uniform, weakly normal ultrafilters
The axiom of reflection
The axiomatic melting point. Teaching probability theory in Prague during the 1930's.
The axiomatic method and Ernst Schröder's algebraic approach to logic
The Axiomatization of Propositional Linear Time Temporal Logic
The article introduces propositional linear time temporal logic as a formal system. Axioms and rules of derivation are defined. Soundness Theorem and Deduction Theorem are proved [9].
The Axiomatization of Propositional Logic
This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes α ⇒ (β ⇒ α), (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)), (¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β). Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved....
The axioms for implication in orthologic
We set up axioms characterizing logical connective implication in a logic derived by an ortholattice. It is a natural generalization of an orthoimplication algebra given by J. C. Abbott for a logic derived by an orthomodular lattice.
The Backward and Forward Summation of Infinite Series of Isols.
The Banach-Tarski Paradox
The Banach-Tarski paradox for the hyperbolic plane (II)
The second author found a gap in the proof of the main theorem in [J. Mycielski, Fund. Math. 132 (1989), 143-149]. Here we fill that gap and add some remarks about the geometry of the hyperbolic plane ℍ².