Progress in computer science and applied logic, 2016
Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive ... more Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jäger in the late 1970s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and characterize the strength of the theory in terms of provable heights of the cumulative hierarchy. Here this technology is applied to intuitionistic Kripke-Platek set theories IKP(P) and IKP(E), where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. In particular, IKP(P) proves the powerset axiom whereas IKP(E) proves the exponentiation axiom. The latter expresses that given any sets A and B, the collection of all functions from A to B is a set, too. While IKP(P) can be dealt with in a similar vein as its classical cousin, the treatment of IKP(E) posed considerable obstacles. One of them was that in the infinitary system the levels of terms become a moving target as they cannot be assigned a fixed level in the formal cumulative hierarchy solely based on their syntactic structure. As adumbrated in an earlier paper, the results of this paper are an important tool in showing that several intuitionistic set theories with the collection axiom possess the existence property, i.e., if they prove an existential theorem then a witness can be provably described in the theory, one example being intuitionistic Zermelo-Fraenkel set theory with bounded separation. * Dedicated to Gerhard Jäger on the occasion of his 60th birthday. We call a formula of L ∈ ∆ P 0 if all its quantifiers are of the form Q x ⊆ y or Q x∈y where Q is ∀ or ∃ and x and y are distinct variables. Let Fun(f, x, y) be a acronym for the bounded formula expressing that f is a function with domain x and co-domain y. We use exponentiation bounded quantifiers ∃f ∈ x y. .. and ∀f ∈ x y. .. as abbreviations for ∃f (Fun(f, x, y) ∧. . .) and ∀x(Fun(f, x, y) →. . .), respectively. Definition 1.2. The ∆ P 0-formulae are the smallest class of formulae containing the atomic formulae closed under ∧, ∨, →, ¬ and the quantifiers ∀x ∈ a, ∃x ∈ a, ∀x ⊆ a, ∃x ⊆ a. The ∆ E 0-formulae are the smallest class of formulae containing the atomic formulae closed under ∧, ∨, →, ¬ and the quantifiers ∀x ∈ a, ∃x ∈ a, ∀f ∈ a b, ∃f ∈ a b. Definition 1.3. IKP(E) has the same language and logic as IKP.
This article is concerned with classifying the provably total set-functions of Kripke-Platek set ... more This article is concerned with classifying the provably total set-functions of Kripke-Platek set theory, KP, and Power Kripke-Platek set theory, KP(P), as well as proving several (partial) conservativity results. The main technical tool used in this paper is a relativisation technique where ordinal analysis is carried out relative to an arbitrary but fixed set x. A classic result from ordinal analysis is the characterisation of the provably recursive functions of Peano Arithmetic, PA, by means of the fast growing hierarchy [10]. Whilst it is possible to formulate the natural numbers within KP, the theory speaks primarily about sets. For this reason it is desirable to obtain a characterisation of its provably total set functions. We will show that KP proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised constructible hierarchy stretching up in length to any ordinal below the Bachmann-Howard ordinal. As a consequence of this result we obtain that IKP + ∀x∀y (x ∈ y ∨ x / ∈ y) is conservative over KP for Π 2-formulae, where IKP stands for intuitionistic Kripke-Platek set theory. In a similar vein, utilising [56], it is shown that KP(P) proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised von Neumann hierarchy of the same length. The relativisation technique applied to KP(P) with the global axiom of choice, AC global , also yields a parameterised extension of a result in [58], showing that KP(P) + AC global is conservative over KP(P) + AC and CZF + AC for Π P 2 statements. Here AC stands for the ordinary axiom of choice and CZF refers to constructive Zermelo-Fraenkel set theory. Proof. This follows immediately from 2.8. ⊓ ⊔ Our goal is to turn R(θ) into a formal representation system, the main obstacle to this is that it is not immediately clear how to deal with the constraint γ ∈ B(γ) in a computable way. This problem leads to the following definition. Definition 2.11. To each α ∈ R(θ) we assign a set Kα of ordinal terms by induction on the complexity Gα: (K1) K0 := KΩ := KΓ β := ∅ for all β ≤ θ. (K2) If α = N F α 1 +. .. + α n then Kα := Kα 1 ∪. .. ∪ Kα n. (K3) If α = N F ϕγδ then Kα := Kγ ∪ Kδ. (K4) If α = N F ψγ then Kα := {γ} ∪ Kγ. Kα consists of the ordinals that occur as arguments of the ψ function in the normal form representation of α. Note that each ordinal in Kα belongs to R(θ) itself and has complexity lower than Gα. Lemma 2.12. For any α, η ∈ R(θ) α ∈ B(η) if and only if (∀ξ ∈ Kα)(ξ < η) Proof. The proof is by induction on Gα. If Gα = 0 then α ∈ B(η) for any η, and Kα = ∅ by (K1) so the result holds. Case 1. If α = N F α 1 +. .. + α n then α ∈ B(η) iff α 1 ,. .. , α n ∈ B(η) by 2.8i). Now inductively α 1 ,. .. , α n ∈ B(η) iff (∀ξ ∈ Kα 1 ∪. .. ∪ Kα n)(ξ < η), but by (K2) Kα = Kα 1 ∪. .. ∪ Kα n .
This special issue contains selected papers from the summer school "Proof, Truth, Computation (PT... more This special issue contains selected papers from the summer school "Proof, Truth, Computation (PTC)" which was held from 21st to 25th July 2014 in Frauenwörth, Chiemsee, Germany, within the Volkswagen Foundation's programme Symposia and Summer Schools. With its focus on the interactions between modern foundations of mathematics and contemporary philosophy, the topics of the summer school included truth theories, predicativity, constructivity, proof theory, formal epistemology and set-theoretic truth. Mathematical methods are about to shape some branches of contemporary philosophy just as they have formed most of the natural and many of the social sciences. The thread of the school was to mirror this development, known as mathematical philosophy or formal epistemology; to highlight the challenges that arise from it; and in particular to display its repercussions in mathematics. As for theoretical computer science, a quite comparable spin-off of mathematics, the principal counterpart within mathematics is mathematical logic. Since many of the objects of study lie beyond the typical commitment of contemporary mathematics, it is decisive to include non-classical issues such as predicativity and constructivity. Proof theory does indeed play a pivotal role: as the area of mathematical logic that is closest to the understanding of logic as the science of formal languages and reasoning, it is predestined for interaction both with philosophical and computer science logic. A hot topic that crossed over wide ranges of the school is whether axiomatic theories of truth and of related notions, such as provability and knowledge, are possible at all in the stress field between syntax and semantics. Rational belief and rational choice, epistemic issues of principal philosophical relevance, are currently put under mathematical scrutiny by applying probabilism: that is, the thesis that a rational agent's degrees of belief should conform to the axioms of probability theory. The summer school was thought to help to bridge the gap between two of the most fundamental faculties of human intellect, mathematics and philosophy, right at their natural point of contact: that is, logic. The gap, which would have been inconceivable for Leibniz, say, has opened when mathematics went abstract in the
Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive ... more Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jager in the late 1970 s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and characterize the strength of the theory in terms of provable heights of the cumulative hierarchy. Here this technology is applied to intuitionistic Kripke-Platek set theories \(\mathbf{IKP }({\mathcal {P}})\) and \(\mathbf{IKP }({\mathcal {E}})\), where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. In particular, \(\mat...
In the early 1980s, the forum of ordinal analysis switched from analysing subsystems of second or... more In the early 1980s, the forum of ordinal analysis switched from analysing subsystems of second order arithmetic and theories of inductive definitions to set theories. The new results were much more uniform and elegant than their predecessors. This thesis uses techniques for the ordinal analysis of set theories developed over the past 30 years to extract some useful information about Kripke Platek set theory, KP and some related theories. First I give a classification of the provably total set functions of KP, this result is reminiscent of a classic theorem of ordinal analysis, characterising the provably total recursive functions of Peano Arithmetic, PA. For the remainder of the thesis the focus switches to intuitionistic theories. Firstly, a detailed rendering of the ordinal analysis of intuitionistic Kripke-Platek set theory, IKP, is given. This is done in such a way as to demonstrate that IKP has the existence property for its verifiable ⌃ sentences. Combined with the results of [40...
Progress in computer science and applied logic, 2016
Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive ... more Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jäger in the late 1970s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and characterize the strength of the theory in terms of provable heights of the cumulative hierarchy. Here this technology is applied to intuitionistic Kripke-Platek set theories IKP(P) and IKP(E), where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. In particular, IKP(P) proves the powerset axiom whereas IKP(E) proves the exponentiation axiom. The latter expresses that given any sets A and B, the collection of all functions from A to B is a set, too. While IKP(P) can be dealt with in a similar vein as its classical cousin, the treatment of IKP(E) posed considerable obstacles. One of them was that in the infinitary system the levels of terms become a moving target as they cannot be assigned a fixed level in the formal cumulative hierarchy solely based on their syntactic structure. As adumbrated in an earlier paper, the results of this paper are an important tool in showing that several intuitionistic set theories with the collection axiom possess the existence property, i.e., if they prove an existential theorem then a witness can be provably described in the theory, one example being intuitionistic Zermelo-Fraenkel set theory with bounded separation. * Dedicated to Gerhard Jäger on the occasion of his 60th birthday. We call a formula of L ∈ ∆ P 0 if all its quantifiers are of the form Q x ⊆ y or Q x∈y where Q is ∀ or ∃ and x and y are distinct variables. Let Fun(f, x, y) be a acronym for the bounded formula expressing that f is a function with domain x and co-domain y. We use exponentiation bounded quantifiers ∃f ∈ x y. .. and ∀f ∈ x y. .. as abbreviations for ∃f (Fun(f, x, y) ∧. . .) and ∀x(Fun(f, x, y) →. . .), respectively. Definition 1.2. The ∆ P 0-formulae are the smallest class of formulae containing the atomic formulae closed under ∧, ∨, →, ¬ and the quantifiers ∀x ∈ a, ∃x ∈ a, ∀x ⊆ a, ∃x ⊆ a. The ∆ E 0-formulae are the smallest class of formulae containing the atomic formulae closed under ∧, ∨, →, ¬ and the quantifiers ∀x ∈ a, ∃x ∈ a, ∀f ∈ a b, ∃f ∈ a b. Definition 1.3. IKP(E) has the same language and logic as IKP.
This article is concerned with classifying the provably total set-functions of Kripke-Platek set ... more This article is concerned with classifying the provably total set-functions of Kripke-Platek set theory, KP, and Power Kripke-Platek set theory, KP(P), as well as proving several (partial) conservativity results. The main technical tool used in this paper is a relativisation technique where ordinal analysis is carried out relative to an arbitrary but fixed set x. A classic result from ordinal analysis is the characterisation of the provably recursive functions of Peano Arithmetic, PA, by means of the fast growing hierarchy [10]. Whilst it is possible to formulate the natural numbers within KP, the theory speaks primarily about sets. For this reason it is desirable to obtain a characterisation of its provably total set functions. We will show that KP proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised constructible hierarchy stretching up in length to any ordinal below the Bachmann-Howard ordinal. As a consequence of this result we obtain that IKP + ∀x∀y (x ∈ y ∨ x / ∈ y) is conservative over KP for Π 2-formulae, where IKP stands for intuitionistic Kripke-Platek set theory. In a similar vein, utilising [56], it is shown that KP(P) proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised von Neumann hierarchy of the same length. The relativisation technique applied to KP(P) with the global axiom of choice, AC global , also yields a parameterised extension of a result in [58], showing that KP(P) + AC global is conservative over KP(P) + AC and CZF + AC for Π P 2 statements. Here AC stands for the ordinary axiom of choice and CZF refers to constructive Zermelo-Fraenkel set theory. Proof. This follows immediately from 2.8. ⊓ ⊔ Our goal is to turn R(θ) into a formal representation system, the main obstacle to this is that it is not immediately clear how to deal with the constraint γ ∈ B(γ) in a computable way. This problem leads to the following definition. Definition 2.11. To each α ∈ R(θ) we assign a set Kα of ordinal terms by induction on the complexity Gα: (K1) K0 := KΩ := KΓ β := ∅ for all β ≤ θ. (K2) If α = N F α 1 +. .. + α n then Kα := Kα 1 ∪. .. ∪ Kα n. (K3) If α = N F ϕγδ then Kα := Kγ ∪ Kδ. (K4) If α = N F ψγ then Kα := {γ} ∪ Kγ. Kα consists of the ordinals that occur as arguments of the ψ function in the normal form representation of α. Note that each ordinal in Kα belongs to R(θ) itself and has complexity lower than Gα. Lemma 2.12. For any α, η ∈ R(θ) α ∈ B(η) if and only if (∀ξ ∈ Kα)(ξ < η) Proof. The proof is by induction on Gα. If Gα = 0 then α ∈ B(η) for any η, and Kα = ∅ by (K1) so the result holds. Case 1. If α = N F α 1 +. .. + α n then α ∈ B(η) iff α 1 ,. .. , α n ∈ B(η) by 2.8i). Now inductively α 1 ,. .. , α n ∈ B(η) iff (∀ξ ∈ Kα 1 ∪. .. ∪ Kα n)(ξ < η), but by (K2) Kα = Kα 1 ∪. .. ∪ Kα n .
This special issue contains selected papers from the summer school "Proof, Truth, Computation (PT... more This special issue contains selected papers from the summer school "Proof, Truth, Computation (PTC)" which was held from 21st to 25th July 2014 in Frauenwörth, Chiemsee, Germany, within the Volkswagen Foundation's programme Symposia and Summer Schools. With its focus on the interactions between modern foundations of mathematics and contemporary philosophy, the topics of the summer school included truth theories, predicativity, constructivity, proof theory, formal epistemology and set-theoretic truth. Mathematical methods are about to shape some branches of contemporary philosophy just as they have formed most of the natural and many of the social sciences. The thread of the school was to mirror this development, known as mathematical philosophy or formal epistemology; to highlight the challenges that arise from it; and in particular to display its repercussions in mathematics. As for theoretical computer science, a quite comparable spin-off of mathematics, the principal counterpart within mathematics is mathematical logic. Since many of the objects of study lie beyond the typical commitment of contemporary mathematics, it is decisive to include non-classical issues such as predicativity and constructivity. Proof theory does indeed play a pivotal role: as the area of mathematical logic that is closest to the understanding of logic as the science of formal languages and reasoning, it is predestined for interaction both with philosophical and computer science logic. A hot topic that crossed over wide ranges of the school is whether axiomatic theories of truth and of related notions, such as provability and knowledge, are possible at all in the stress field between syntax and semantics. Rational belief and rational choice, epistemic issues of principal philosophical relevance, are currently put under mathematical scrutiny by applying probabilism: that is, the thesis that a rational agent's degrees of belief should conform to the axioms of probability theory. The summer school was thought to help to bridge the gap between two of the most fundamental faculties of human intellect, mathematics and philosophy, right at their natural point of contact: that is, logic. The gap, which would have been inconceivable for Leibniz, say, has opened when mathematics went abstract in the
Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive ... more Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jager in the late 1970 s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and characterize the strength of the theory in terms of provable heights of the cumulative hierarchy. Here this technology is applied to intuitionistic Kripke-Platek set theories \(\mathbf{IKP }({\mathcal {P}})\) and \(\mathbf{IKP }({\mathcal {E}})\), where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. In particular, \(\mat...
In the early 1980s, the forum of ordinal analysis switched from analysing subsystems of second or... more In the early 1980s, the forum of ordinal analysis switched from analysing subsystems of second order arithmetic and theories of inductive definitions to set theories. The new results were much more uniform and elegant than their predecessors. This thesis uses techniques for the ordinal analysis of set theories developed over the past 30 years to extract some useful information about Kripke Platek set theory, KP and some related theories. First I give a classification of the provably total set functions of KP, this result is reminiscent of a classic theorem of ordinal analysis, characterising the provably total recursive functions of Peano Arithmetic, PA. For the remainder of the thesis the focus switches to intuitionistic theories. Firstly, a detailed rendering of the ordinal analysis of intuitionistic Kripke-Platek set theory, IKP, is given. This is done in such a way as to demonstrate that IKP has the existence property for its verifiable ⌃ sentences. Combined with the results of [40...
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