Paweł Zbierski

On Herbrand consistency in weak arithmetic

Abstract. We prove that the Gödel incompleteness theorem holds for a weak arithmetic T = IΔ0 + Ω2 in the form where ConsH(T) is an arithmetic formula expressing the consistency of T with respect to the Herbrand notion of provability.

Logic of mathematics : a modern course of classical logic

Partial table of contents: MATHEMATICAL STRUCTURES AND THEIR THEORIES. Relational Systems. Boolean Algebras. Terms and Formulas. Substitution of Terms. Theorems and Proofs. Generalization Rule and Elimination of Constants. Peano Arithmetic. Ultraproducts. Supplementary Questions. SELECTED TOPICS. Total Functions. Incompleteness of Arithmetic. Tarskis Theorem. Matiyasevichs Theorem. Guide to Further Reading. References. Index.

On closed P-sets with ccc in the space ω*

It is proved that -- consistently -- there can be no ccc closed P-sets in the remainder space omega^* .

On complexity reduction of Σ1 formulas

Abstract. For a fixed q  ℕ and a given Σ1 definition φ(d,x), where d is a parameter, we construct a model M of 1 Δ0 + ¬ exp and a non standard d  M such that in M either φ has no witness smaller than d or phgr; is equivalent to a formula ϕ(d,x) having no more than q alternations of blocks of quantifiers.