Zofia Adamowicz

On Herbrand consistency in weak arithmetic

Abstract. We prove that the Gödel incompleteness theorem holds for a weak arithmetic T = IΔ0 + Ω2 in the form where ConsH(T) is an arithmetic formula expressing the consistency of T with respect to the Herbrand notion of provability.

Existentially closed structures and gödel's second incompleteness theorem

We prove that any 1-closed (see def 1.1) model of the Π 2 consequences of PA satisfies ¬ Cons PA which gives a proof of the second Godel incompleteness theorem without the use of the Godel diagonal lemma. We prove a few other theorems by the same method.

Logic of mathematics : a modern course of classical logic

Partial table of contents: MATHEMATICAL STRUCTURES AND THEIR THEORIES. Relational Systems. Boolean Algebras. Terms and Formulas. Substitution of Terms. Theorems and Proofs. Generalization Rule and Elimination of Constants. Peano Arithmetic. Ultraproducts. Supplementary Questions. SELECTED TOPICS. Total Functions. Incompleteness of Arithmetic. Tarskis Theorem. Matiyasevichs Theorem. Guide to Further Reading. References. Index.

A contribution to the end-extension problem and the Π1 conservativeness problem

Abstract We formulate a Π1 sentence τ which is a version of the Tableau consistency of GlΔ0. The sentence τ is true and is provable in GlΔ0 + exp. We construct a model M of GlΔ0+Ω1+τ+BGs1 which has no proper end-extension to a model of GlΔ0+Ω1+τ. Also we prove that GlΔ0+Ω1+τ is not Π1 conservative over GlΔ0+τ.

On Finite Lattices of Degrees of Constructibility of Reals

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice . The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lermans paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lermans representation. Let K be a given finite lattice. Assume that the universe of K is an integer l . Let ≤ K be the ordering in K . A sequential representation of K is a sequence U i ⊆ U i+1 of finite subsets of ω i such that the following holds: (1) For any s, s ′ Є U i , i Є ω, k, m Є l , k ≤ K m & s(m) = s ′ (m) → s(k) = s ′ (k) . (2) For any s Є U i , i Є ω, s (0) = 0 where 0 is the least element of K . (3) For any s , s ′ Є i Є ω, k,j Є l , if k y K j = m and s(k) = s ′ (k) & s(j) = s ′ (j) → s(m) = s ′ (m) , where v K denotes the join in K .

On maximal theories

Let S be a recursive theory. Let a theory T * consisting of Σ 1 sentences be called maximal (with respect to S ) if T * is maximal consistent with S , i.e. there is no Σ 1 sentence consistent with T * + S which is not in T *. A maximal theory with respect to I Δ 0 was considered by Wilkie and Paris in [WP] in connection with the end-extension problem. Let us recall that I Δ 0 is the fragment of Peano arithmetic consisting of the finite collection of algebraic axioms PA − together with the induction scheme restricted to bounded formulas. The main open problem concerning the end-extendability of models of I Δ 0 is the following: (*) Does every model of I Δ 0 + B Σ 1 have a proper end-extension to a model of I Δ 0 ? Here B Σ 1 is the following collection scheme: where φ runs over bounded formulas and may contain parameters. It is well known(see [KP]) that if I is a proper initial segment of a model M of I Δ 0 , then I satisfies I Δ 0 + B Σ 1 . For a wide discussion of the problem (*) see [WP]. Wilkie and Paris construct in [WP] a model M of I Δ 0 + B Σ 1 which has no proper end-extension to a model of I Δ 0 under the assumption I Δ 0 ⊢¬Δ 0 H (see [WP] for an explanation of this assumption). Their model M is a model of a maximal theory T * where S = I Δ 0 . Moreover, T *, which is the set Σ 1 ( M ) of all Σ 1 sentences true in M , is not codable in M .

Well-behaved principles alternative to bounded induction

Abstract We introduce some Π1-expressible combinatorial principles which may be treated as axioms for some bounded arithmetic theories. The principles, denoted Sk(Σ n b ,length log k ) and Sk(Σ n b ,depth log k ) (where ‘Sk’ stands for ‘Skolem’), are related to the consistency of Σnb induction: for instance, they provide models for Σnb induction. However, the consistency is expressed indirectly, via the existence of evaluations for sequences of terms. The evaluations do not have to satisfy Σnb induction, but must determine the truth value of Σnb statements. Our principles have the property that Sk(Σ n b ,depth log k ) proves Sk(Σ n+1 b ,length log k ) . Additionally, Sk(Σ n b ,length log k−2 ) proves Sk(Σ n+1 b ,length log k ) . Thus, some provability is involved where conservativity is known in the case of Σnb induction on an initial segment and induction for higher Σmb classes on smaller segments.

Existentially closed models in the framework of arithmetic

We prove that the standard cut is definable in each existentially closed model of IΔ0 +exp by a (parameter free) Π1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic. §

Partial collapses of the Σ1 complexity hierarchy in models for fragments of bounded arithmetic

Abstract For any n , we construct a model of T 2 n + ¬ exp in which each ∃ s Π n + 1 b formula is equivalent to an ∃ Π n b formula.

On complexity reduction of Σ1 formulas

Abstract. For a fixed q  ℕ and a given Σ1 definition φ(d,x), where d is a parameter, we construct a model M of 1 Δ0 + ¬ exp and a non standard d  M such that in M either φ has no witness smaller than d or phgr; is equivalent to a formula ϕ(d,x) having no more than q alternations of blocks of quantifiers.