The absorption law: Or: how to Kreisel a Hilbert–Bernays–Löb

In this paper, we show how to construct for a given consistent theory U a Σ01-predicate that both satisfies the Löb Conditions and the Kreisel Condition—even if U is unsound. We do this in such a way that U itself can verify satisfaction of an internal version of the Kreisel Condition

On Notions of Provability

In this thesis, we study notions of provability, i.e. formulas B(x,y) such that a formula ϕ is provable in T if, and only if, there is m ∈ N such that T ⊢ B(⌜ϕ⌝,m) (m plays the role of a parameter); the usual notion of provability, k-step provability (also known as k-provability), s-symbols provability are examples of notions of provability. We develop general results concerning notions of provability, but we also study in detail concrete notions. We present partial results concerning the decidability of kprovability for Peano Arithmetic (PA), and we study important problems concerning k-provability, such as Kreisel’s Conjecture and Montagna’s Problem: (∀n ∈ N.T ⊢k steps ϕ(n)) =⇒ T ⊢ ∀x.ϕ(x), [Kreisel’s Conjecture] and Does PA ⊢k steps PrPA(⌜ϕ⌝)→ϕ imply PA ⊢k steps ϕ? [Montagna’s Problem] Incompleteness, Undefinability of Truth, and Recursion are different entities that share important features; we study this in detail and we trace these entities to common results. We present numeral forms of completeness and consistency, numeral completeness and numeral consistency, respectively; numeral completeness guarantees that, whenever a Σb 1(S12 )-formula ϕ(⃗x ) is such that ⃗Q ⃗x .ϕ(⃗x ) is true (where ⃗Q is any array of quantifiers), then this very fact can be proved inside S12 , more precisely S12 ⊢ ⃗Q ⃗x .Prτ (⌜ϕ( •⃗ x )⌝). We examine these two results from a mathematical point of view by presenting the minimal conditions to state them and by finding consequences of them, and from a philosophical point of view by relating them to Hilbert’s Program. The derivability condition “provability implies provable provability” is one of the main derivability conditions used to derive the Second Incompleteness Theorem and is known to be very sensitive to the underlying theory one has at hand. We create a weak theory G2 to study this condition; this is a theory for the complexity class FLINSPACE. We also relate properties of G2 to equality between computational classes.O tema desta tese são noções de demonstração; estas últimas são fórmulas B(x,y) tais que uma fórmula ϕ é demonstrável em T se, e só se, existe m ∈ N tal que T ⊢ B(⌜ϕ⌝,m) (m desempenha o papel de um parâmetro). A noção usual de demonstração, demonstração em k-linhas (demonstração-k), demonstração em s-símbolos são exemplos de noções de demonstração. Desenvolvemos resultados gerais sobre noções de demonstração, mas também estudamos exemplos concretos. Damos a conhecer resultados parciais sobre a decidibilidade da demonstração-k para a Aritmética de Peano (PA), e estudamos dois problemas conhecidos desta área, a Conjectura de Kreisel e o Problema de Montagna: (∀n ∈ N.T ⊢k steps ϕ(n)) =⇒ T ⊢ ∀x.ϕ(x), [Conjectura de Kreisel] e PA ⊢k steps PrPA(⌜ϕ⌝)→ϕ implica PA ⊢k steps ϕ? [Problema de Montagna] A Incompletude, a Incapacidade de Definir Verdade, e Recursão são entidades que têm em comum características relevantes; nós estudamos estas entidades em detalhe e apresentamos resultados que são simultaneamente responsáveis pelas mesmas. Além disso, apresentamos formas numerais de completude e consistência, a completude numeral e a consistência numeral, respectivamente; a completude numeral assegura que, quando uma fórmula-Σb 1(S12) ϕ(⃗x ) é tal que ⃗Q ⃗x .ϕ(⃗x ) é verdadeira, então este facto pode ser verificado dentro de S12, mais precisamente S12 ⊢ ⃗Q ⃗x .Prτ (⌜ϕ( •⃗ x )⌝). Este dois resultados são analisados de um ponto de vista matemático onde apresentamos as condições mínimas para os demonstrar e apresentamos consequências dos mesmos, e de um ponto de vista filosófico, onde relacionamos os mesmos com o Programa de Hilbert. A condição de derivabilidade “demonstração implica demonstrabilidade da demonstração” é uma das condições usadas para derivar o Segundo Teorema da Incompletude e sabemos ser muito sensível à teoria de base escolhida. Nós criámos uma teoria fraca G2 para estudar esta condição; esta é uma teoria para a classe de complexidade FLINSPACE. Também relacionámos propriedades de G2 com igualdades entre classes de complexidade computacional

On Notions of Provability

In this thesis, we study notions of provability, i.e. formulas B(x,y) such that a formula ' is provable in T if, and only if, there is m 2 N such that T ` B(p'q,m) (m plays the role of a parameter); the usual notion of provability, k-step provability (also known as k-provability), s-symbols provability are examples of notions of provability. We develop general results concerning notions of provability, but we also study in detail concrete notions. We present partial results concerning the decidability of k- provability for Peano Arithmetic (PA), and we study important problems concerning k-provability, such as Kreisel’s Conjecture and Montagna’s Problem: (8n 2 N.T `k steps '(n)) =) T ` 8x.'(x), [Kreisel’s Conjecture] Does PA `k steps PrPA(p'q) ! ' imply PA `k steps '? [Montagna’s Problem] Incompleteness, Undefinability of Truth, and Recursion are di↵erent entities that share important features; we study this in detail and we trace these entities to common results. We present numeral forms of completeness and consistency, numeral completeness and numeral consistency, respectively; numeral completeness guarantees that, whenever a⌃b1(S12)-formula'(x~)issuchthatQ~x~.'(x~)istrue(whereQ~ isanyarrayofquantifiers), then this very fact can be proved inside S12, more precisely S12 ` Q~ x~.Pr⌧(p'(x~• )q). We examine these two results from a mathematical point of view by presenting the minimal conditions to state them and by finding consequences of them, and from a philosophical point of view by relating them to Hilbert’s Program. The derivability condition “provability implies provable provability” is one of the main derivability conditions used to derive the Second Incompleteness Theorem and is known to be very sensitive to the underlying theory one has at hand. We create a weak theory G2 to study this condition; this is a theory for the complexity class FLINSPACE. We also relate properties of G2 to equality between computational classes

Interpretability suprema in Peano Arithmetic

This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic ILM of Peano Arithmetic (PA). It is well-known that any theories extending PA have a supremum in the interpretability ordering. While provable in PA, this fact is not reflected in the theorems of the modal system ILM, due to limited expressive power. Our goal is to enrich the language of ILM by adding to it a new modality for the interpretability supremum. We explore different options for specifying the exact meaning of the new modality. Our final proposal involves a unary operator, the dual of which can be seen as a (nonstandard) provability predicate satisfying the axioms of the provability logic GL

Interpretability suprema in Peano Arithmetic

This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic ILM of Peano Arithmetic (PA). It is well-known that any theories extending PA have a supremum in the interpretability ordering. While provable in PA, this fact is not reflected in the theorems of the modal system ILM, due to limited expressive power. Our goal is to enrich the language of ILM by adding to it a new modality for the interpretability supremum. We explore different options for specifying the exact meaning of the new modality. Our final proposal involves a unary operator, the dual of which can be seen as a (nonstandard) provability predicate satisfying the axioms of the provability logic GL