Saeed Salehi

HERBRAND CONSISTENCY OF SOME ARITHMETICAL THEORIES

G¨ odels second incompleteness theorem is proved for Herbrand consistency of some arith- metical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz (Herbrand consistency and bounded arithmetic,Fundamenta Mathematicae, vol. 171 (2002), pp. 279-292). In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories I±0 + ∫m withm > 2, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theoryT ⊇ I±0 + ∫2 inT itself. In this paper, the above results are generalized for I±0 + ∫1. Also after tailoring the definition of Herbrand consistency for I±0 we prove the corresponding theorems for I±0. Thus the Herbrand version of G¨ odels second incompleteness theorem follows for the theories I±0 + ∫1 and I±0.

Separating bounded arithmetical theories by Herbrand consistency

The problem of Π1–separating the hierarchy of bounded arithmetic has been studied in the article. It is shown that the notion of Herbrand consistency, in its full generality, cannot Π1–separate the theory IΔ0+⋀jΩj from IΔ0; though it can Π1–separate IΔ0+Exp from IΔ0. Namely, we show the unprovability of the Herbrand consistency of IΔ0 in the theory IΔ0+⋀jΩj. This partially extends a result of L. A. Kolodziejczyk who showed that for a finite fragment S⊆IΔ0, the Herbrand consistency of S+Ω1 is not provable in IΔ0+⋀jΩj.

On Axiomatizability of the Multiplicative Theory of Numbers

The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the complex, real and (positive) rational numbers. These theories (and also the multiplicative theories of natural and integer numbers) are known to be decidable (i.e., there exists an algorithm that decides whether a given sentence is derivable form the theory); here we present explicit axiomatizations for them and show that they are not finitely axiomatizable. For each of these sets (of complex, real and [positive] rational numbers) a language, including the multiplication operation, is introduced in a way that it allows quantifier elimination (for the theory of that set).

Gödel–Rosser's Incompleteness Theorem, generalized and optimized for definable theories

Godels First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true

Intuitionistic axiomatizations for bounded extension Kripke models

Pi_n

On decidability and axiomatizability of some ordered structures

-sentences or equivalently the

Kripke semantics for fuzzy logics

Sigma_n

On the Arithmetical Truth of Self-Referential Sentences: ARITHMETICAL TRUTH OF SELF-REFERENTIAL SENTENCES

-soundness of the theory, and the other is

On constructivity and the Rosser property: a closer look at some Gödelean proofs

n

Herbrand consistency of some finite fragments of bounded arithmetical theories

-consistency the restriction of