Diagonal reasonings in mathematical logic
First we show a few well known mathematical diagonal reasonings. Then we concentrate on diagonal reasonings typical for mathematical logic.
End-extending models of
Axiomatization of the forcing relation with an application to Peano Arithmetic
A generalization of Shoenfield theorem on sets
Continuous relations and generalized sets
A recursion-theoretic characterization of instances of provable in
Perfect set theorems for in the universe without choice
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.
Functions provably total in
Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics
We prove that for i ≥ 1, the arithmetic does not prove a variant of its own Herbrand consistency restricted to the terms of depth in , 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 .
An application of a reflection principle
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