EUDML

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 $T ₘ ⊬ H C o n s^{I ₘ} (T ₘ)$ , where $H C o n s^{I ₘ}$ 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.

Functions provably total in $I^{-} Σ_{n}$

Zofia Adamowicz; Teresa Bigorajska — 1989

Fundamenta Mathematicae

Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics

Zofia Adamowicz; Konrad Zdanowski — 2011

Fundamenta Mathematicae

We prove that for i ≥ 1, the arithmetic $I Δ ₀ + Ω_{i}$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1 + ε) l o g^{i + 2}$ , where ε is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $l o g^{i + 3}$ .

An application of a reflection principle

Zofia Adamowicz; Leszek Aleksander Kołodziejczyk; Paweł Zbierski — 2003

Fundamenta Mathematicae

We define a recursive theory which axiomatizes a class of models of IΔ₀ + Ω ₃ + ¬ exp all of which share two features: firstly, the set of Δ₀ definable elements of the model is majorized by the set of elements definable by Δ₀ formulae of fixed complexity; secondly, Σ₁ truth about the model is recursively reducible to the set of true Σ₁ formulae of fixed complexity.

Page 1

Download Results (CSV)

Advanced Search

Formula preview

Currently displaying 1 – 11 of 11

Diagonal reasonings in mathematical logic

End-extending models of $I Δ_{0} + e x p + Β Σ_{1}$

Axiomatization of the forcing relation with an application to Peano Arithmetic

A generalization of Shoenfield theorem on ${(Σ^{1})}_{2}$ sets

Continuous relations and generalized $G_{δ}$ sets

A recursion-theoretic characterization of instances of $Β Σ_{n}$ provable in $П_{n + 1} (N)$

Perfect set theorems for $Π_{2}^{1}$ in the universe without choice

Herbrand consistency and bounded arithmetic

Functions provably total in $I^{-} Σ_{n}$

Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics

An application of a reflection principle