Elimination theory for the ring of algebraic integers.
End-extending models of
Existentially closed domains with radical relations.
Expansions of models of ZFC.
Fragments of complete extensions of PA and McDowell-Specker's theorem.
Herbrand consistency and bounded arithmetic
We prove that the Gödel incompleteness theorem holds for a weak arithmetic Tₘ = IΔ₀ + Ωₘ, for m ≥ 2, in the form Tₘ ⊬ HCons(Tₘ), where HCons(Tₘ) is an arithmetic formula expressing the consistency of Tₘ with respect to the Herbrand notion of provability. Moreover, we prove , where is HCons relativised to the definable cut Iₘ of (m-2)-times iterated logarithms. The proof is model-theoretic. We also prove a certain non-conservation result for Tₘ.
If it looks and smells like the reals...
Given a topological space ⟨X,T⟩ ∈ M, an elementary submodel of set theory, we define to be X ∩ M with topology generated by U ∩ M:U ∈ T ∩ M. We prove that if is homeomorphic to ℝ, then . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted.
More on extending automorphisms of models of Peano Arithmetic
Continuing the earlier research [Fund. Math. 129 (1988) and 149 (1996)] we give some information about extending automorphisms of models of PA to cofinal extensions.
More reflections on compactness
We consider the question of when , where is the elementary submodel topology on X ∩ M, especially in the case when is compact.
Mutually generic classes and incompatible expansions
n - Infinite Forcing via Infinite Forcing
O struktuře modelů omezené -indukce
On elementary cuts in recursively saturated models of Peano Arithmetic
On embedding models of arithmetic of cardinality ℵ₁ into reduced powers
In the early 1970’s S. Tennenbaum proved that all countable models of PA₁¯ + ∀₁ -Th(ℕ) are embeddable into the reduced product , where ℱ is the cofinite filter. In this paper we show that if M is a model of PA¯ + ∀₁ - Th(ℕ), and |M| = ℵ₁, then M is embeddable into , where D is any regular filter on ω.
On expandability of models of arithmetic and set theory to models of weak second-order theories
On extending automorphisms of models of Peano Arithmetic
Continuing the earlier research in [10] we give some information on extending automorphisms of models of PA to end extensions and cofinal extensions.
On regular interstices and selective types in countable arithmetically saturated models of Peano Arithmetic
We continue the earlier research of [1]. In particular, we work out a class of regular interstices and show that selective types are realized in regular interstices. We also show that, contrary to the situation above definable elements, the stabilizer of an element inside M(0) whose type is selective need not be maximal.
On the Leibniz-Mycielski axiom in set theory
Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that satisfies φ(x) ∧¬ φ(y). We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows: 1. In the presence of ZF, the following are equivalent: (a) LM. (b)...
Order-types of models of arithmetic and a connection with arithmetic saturation.
Ramified analysis and the minimal β-models of higher order arithmetics