A note on the theory of positive induction, $${{\rm ID}^*_1}$$

Afshari, Bahareh; Rathjen, Michael

doi:10.1007/s00153-009-0168-9

Cited by 8 publications

(20 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Given a functional f : LO → LO, statements of the form ∀X ∈ LO (WO(X ) =⇒ WO(f (X ))) have been studied for different functionals f coming from ordinal notations in proof theory such as X → ω X , X → X , etc. and shown to have interesting strengths [Gir87,Hir94,FMW,AR10,RW,MM,Rat]. For a survey of recent results, see the Introduction of [MM].…”

Section: Lavmentioning

confidence: 99%

Open Questions in Reverse Mathematics

Montalbán¹

2011

Bull. symb. log

View full text Add to dashboard Cite

show abstract

Section: Lavmentioning

confidence: 99%

Open Questions in Reverse Mathematics

Montalbán¹

2011

Bull. symb. log

View full text Add to dashboard Cite

show abstract

“…The following theorem is shown by D. Probst [7], and independently by B. Afshari and M. Rathjen [1]. [7], and as ID * [1], resp. Theorem 1.6 (Probst [7], Afshari and Rathjen [1])…”

Section: Weak Fragmentsmentioning

confidence: 85%

Proof-Theoretic Strengths of Weak Theories for Positive Inductive Definitions

Arai¹

2018

J. symb. log.

View full text Add to dashboard Cite

In this paper the lightface Π 1 1 -Comprehension axiom is shown to be proof-theoretically strong even over RCA * 0 , and we calibrate the prooftheoretic ordinals of weak fragments of the theory ID1 of positive inductive definitions over natural numbers. Conjunctions of negative and positive formulas in the transfinite induction axiom of ID1 are shown to be weak, and disjunctions are strong. Thus we draw a boundary line between predicatively reducible and impredicative fragments of ID1.

show abstract

“…In particular, we shall exploit the Π 1 1 definability of a least fixed-point. A similar approach has been taken in [AR10] and [Pro06] for the treatment of the theories Π 1 2 -RFN 0 and ID * 1 (a subsystem of ID 1 that allows only positive induction for the predicates P A that are assigned to each positive operator form A). Below, we shall provide an upper bound for FIT by embedding it directly into Π 1 3 -RFN 0 .…”

Section: Upper Bound ϑω ω For Fit and Tidmentioning

confidence: 99%

A flexible type system for the small Veblen ordinal

Ranzi

Strahm

2019

Arch. Math. Logic

View full text Add to dashboard Cite

We introduce and analyze two theories for typed (accessible part) inductive definitions and establish their proof-theoretic ordinal to be the small Veblen ordinal ϑΩ ω. We investigate on the one hand the applicative theory FIT of functions, (accessible part) inductive definitions, and types. It includes a simple type structure and is a natural generalization of S. Feferman's system QL(F 0-IR N). On the other hand, we investigate the arithmetical theory TID of typed (accessible part) inductive definitions, a natural subsystem of ID 1 , and carry out a wellordering proof within TID that makes use of fundamental sequences for ordinal notations in an ordinal notation system based on the finitary Veblen functions. The essential properties for describing the ordinal notation system are worked out.

show abstract

A note on the theory of positive induction, $${{\rm ID}^*_1}$$

Cited by 8 publications

References 7 publications

Open Questions in Reverse Mathematics

Open Questions in Reverse Mathematics

Proof-Theoretic Strengths of Weak Theories for Positive Inductive Definitions

A flexible type system for the small Veblen ordinal

Contact Info

Product

Resources

About