Proof-theoretic Ordinals Research Articles

PROOF-THEORETIC STRENGTHS OF WEAK THEORIES FOR POSITIVE INDUCTIVE DEFINITIONS

AbstractIn this article the lightface ${\rm{\Pi }}_1^1$-Comprehension axiom is shown to be proof-theoretically strong even over ${\rm{RCA}}_0^{\rm{*}}$, and we calibrate the proof-theoretic ordinals of weak fragments of the theory ${\rm{I}}{{\rm{D}}_1}$ of positive inductive definitions over natural numbers. Conjunctions of negative and positive formulas in the transfinite induction axiom of ${\rm{I}}{{\rm{D}}_1}$ are shown to be weak, and disjunctions are strong. Thus we draw a boundary line between predicatively reducible and impredicative fragments of ${\rm{I}}{{\rm{D}}_1}$.

Notes on some second-order systems of iterated inductive definitions and [formula omitted]-comprehensions and relevant subsystems of set theory

Pohlers's ordinal analysis in his monograph [12] contains some flaws and thereby ends up with incorrect proof-theoretic ordinals of several systems. The present paper determines their correct proof-theoretic ordinals and also supplements [12] with the ordinal analysis of some other relevant impredicative systems.

現代集合論における巨大基数(あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

現代集合論における巨大基数(<特集>あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

特集への序文:あたらしい数理論理学の揺籃 : 証明論的な順序数と集合論的な順序数

特集への序文:あたらしい数理論理学の揺籃 : 証明論的な順序数と集合論的な順序数

The strength of extensionality I — weak weak set theories with infinity

We measure, in the presence of the axiom of infinity, the proof-theoretic strength of the axioms of set theory which make the theory look really like a “theory of sets”, namely, the axiom of extensionality Ext, separation axioms and the axiom of regularity Reg (and the axiom of choice AC). We first introduce a weak weak set theory Basic (which has the axioms of infinity and of collapsing) as a base over which to clarify the strength of these axioms. We then prove the following results about proof-theoretic ordinals: 1. | Basic | = ω ω and | Basic + Ext | = ε 0 , 2. | Basic + Δ 0 - Sep | = ε 0 and | Basic + Δ 0 - Sep + Ext | = Γ 0 . We also show that neither Reg nor AC affects the proof-theoretic strength, i.e., | T | = | T + Reg | = | T + AC | = | T + Reg + AC | where T is Basic plus any combination of Ext and Δ 0 - Sep .

Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results

This paper is intended to give for a general mathematical audience (including non-logicians) a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below ε 0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible (logarithmic) compression technique we give applications to phase transitions for independence results, Hilbert’s basis theorem, local number theory, Ramsey theory, Hydra games, and Goodstein sequences. We discuss briefly universality and renormalization issues in this context. Finally, we indicate how regularity properties of ordinal count functions can be used to prove logical limit laws.

Provability algebras and proof-theoretic ordinals, I

We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra , that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic (syntax-sensitive) information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to ε 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysis and the work which has been done in the last two decades on provability logic and reflection principles. Because of its abstract algebraic nature, we hope that it will also be of interest for non-prooftheorists.

Provability algebras and proof-theoretic ordinals, I

We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic (syntax-sensitive) information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to ε 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysis and the work which has been done in the last two decades on provability logic and reflection principles. Because of its abstract algebraic nature, we hope that it will also be of interest for non-prooftheorists.

Proof-theoretic analysis by iterated reflection

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. Moreover, they provide a uniform definition of a proof-theoretic ordinal for any arithmetical complexity \(\Pi _{n}^{0}\). We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity \(\Pi _{1}^{0}\). We provide a more general version of the fine structure relationships for iterated reflection principles (due to Ulf Schmerl). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including \(I\Sigma _{n}\), \(I\Sigma _{n}^{-}\), \(I\Pi _{n}^{-}\) and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform \(\Sigma _{1}\)-reflection principle for T is \(\Sigma _{2}\)-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl’s theorem.

The model-theoretic ordinal analysis of theories of predicative strength

AbstractWe use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first- and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ0.

The proof-theoretic analysis of transfinitely iterated fixed point theories

AbstractThis article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories and ; the exact proof-theoretic ordinals of these systems are presented.

About the proof-theoretic ordinals of weak fixed point theories

AbstractThis paper presents several proof-theoretic results concerning weak fixed point theories over second order number theory with arithmetic comprehension and full or restricted induction on the natural numbers. It is also shown that there are natural second order theories which are proof-theoretically equivalent but have different proof-theoretic ordinals.

The slow-growing and the Graegorczyk hierarchies

We give here an elementary proof of a recent result of Girard [4] comparing the rates of growth of the two principal (and extreme) examples of a spectrum of “majorization hierarchies”—i.e. hierarchies of increasing number-theoretic functions, indexed by (systems of notations for) initial segments I of the countable ordinals so that if α &lt; β ∈ I then the βth function dominates the αth one at all but finitely-many positive integers x.Hardy [5] was perhaps the first to make use of a majorization hierarchy—the Hα's below—in “exhibiting” a set of reals with cardinality ℵ1. More recently such hierarchies have played important roles in mathematical logic because they provide natural classifications of recursive functions according to their computational complexity. (All the functions considered here are “honest” in the sense that the size of their values gives a measure of the number of steps needed to compute them.)The hierarchies we are concerned with fall into three main classes depending on their mode of generation at successor stages, the other crucial parameter being the initial choice of a particular (standard) fundamental sequence λ0 &lt; λ1 &lt; λ2 &lt; … to each limit ordinal λ under consideration which, by a suitable diagonalization, will then determine the generation at stage λ.Our later comparisons will require the use of a “large” initial segment I of proof-theoretic ordinals, extending as far as the “Howard ordinal”. However we will postpone a precise description of these ordinals and their associated fundamental sequences until later.

A simplification of the Bachmann method for generating large countable ordinals

In [2] Bachmann showed how a hierarchy of normal functions on the countable ordinals Ω can be constructed using certain uncountable ordinals together with appropriate fundamental sequences for limit ordinals, as indexing ordinals for the hierarchy. This method, explained in more detail in §1 below, has been generalized by Pfeiffer [6] and Isles [4] to produce larger hierarchies. Using the hierarchies constructed in this way it is possible, as an immediate consequence of the definitions, to assign ω-sequences 〈αn〉n∈ω to limit ordinals α in an initial segment I of Ω such that . Indeed this was the motivation for Bachmann's original construction. However the initial segments I constructed in this way are of interest even without the fundamental sequences because they occur naturally as the proof-theoretic ordinals associated with particular formal theories. The fundamental sequences necessary for the Bachmann method are not intrinsically of interest proof-theoretically and only serve to obscure the definitions of the associated initial segments. Feferman and Weyhrauch [9] suggested an alternative definition for a sequence of functions on the countable ordinals. This definition was generalized by Aczel [1] who showed how the new functions corresponded to those in Bachmann's hierarchy. (Independently, Weyhrauch established the same results for an initial segment of the sequence of Bachmann functions, i.e. those functions indexed by α &lt; εΩ+1.)