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1
Content available remote

Herbrand consistency and bounded arithmetic

100%
Adamowicz Z.
Fundamenta Mathematicae
|
2002
|
tom 171
|
nr 3
279-292
EN
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 $Tₘ ⊬ HCons^{Iₘ}(Tₘ)$, where $HCons^{Iₘ}$ 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.
2
Content available remote

Diagonal reasonings in mathematical logic

100%
Adamowicz Z.
Banach Center Publications
|
1995
|
tom 34
|
nr 1
9-18
EN
First we show a few well known mathematical diagonal reasonings. Then we concentrate on diagonal reasonings typical for mathematical logic.
3
Content available remote

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

64%
Adamowicz Z., Zdanowski K.
Fundamenta Mathematicae
|
2011
|
tom 212
|
nr 3
191-216
EN
We prove that for i ≥ 1, the arithmetic $IΔ₀ + Ω_i$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1+ε)log^{i+2}$, 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 $log^{i+3}$.
4
Content available remote

An application of a reflection principle

51%
Adamowicz Z., Kołodziejczyk L., Zbierski P.
Fundamenta Mathematicae
|
2003
|
tom 180
|
nr 2
139-159
EN
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.
5
Content available remote

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

50%
Adamowicz Z.
Fundamenta Mathematicae
|
1988
|
tom 129
|
nr 3
231-236
6
Content available remote

Axiomatization of the forcing relation with an application to Peano Arithmetic

50%
Adamowicz Z.
Fundamenta Mathematicae
|
1984
|
tom 120
|
nr 2
167-186
7
Content available remote

Perfect set theorems for $Π^1_2$ in the universe without choice

44%
Adamowicz Z.
Fundamenta Mathematicae
|
1983
|
tom 118
|
nr 1
11-31
8
Content available remote

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

38%
Adamowicz Z., Bigorajska T.
Fundamenta Mathematicae
|
1989
|
tom 132
|
nr 3
189-194
9
Content available remote

End-extending models of $IΔ_0 + exp + ΒΣ_1$

38%
Adamowicz Z.
Fundamenta Mathematicae
|
1990
|
tom 136
|
nr 3
133-145
10
Content available remote

A generalization of Shoenfield theorem on $(Σ^1)_2$ sets

38%
Adamowicz Z.
Fundamenta Mathematicae
|
1984
|
tom 123
|
nr 2
81-90
11
Content available remote

Continuous relations and generalized $G_δ$ sets

38%
Adamowicz Z.
Fundamenta Mathematicae
|
1984
|
tom 123
|
nr 2
91-107
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