Ordinal analysis of the formal theory for noniterated inductive definitions

2006 ◽  
pp. 109-186
Keyword(s):  
Formal Theory ◽  
Ordinal Analysis ◽  
Inductive Definitions

Ordinals connected with formal theories for transfinitely iterated inductive definitions

1978 ◽  
Vol 43 (2) ◽  
pp. 161-182 ◽  
Author(s):  
W. Pohlers
Keyword(s):  
Fixed Point ◽  
Number Theory ◽  
Formal Theory ◽  
Order Type ◽  
General Notion ◽  
Inductive Definitions ◽  
Primitive Recursive ◽  
Formal Theories

Let Th be a formal theory extending number theory. Call an ordinal ξ provable in Th if there is a primitive recursive ordering which can be proved in Th to be a wellordering and whose order type is > ξ. One may define the ordinal ∣ Th ∣ of Th to be the least ordinal which is not provable in Th. By [3] and [12] we get , where IDN is the formal theory for N-times iterated inductive definitions. We will generalize this result not only to the case of transfinite iterations but also to a more general notion of ‘the ordinal of a theory’.For an X-positive arithmetic formula [X,x] there is a natural norm ∣x∣: = μξ (x ∈ Iξ), where Iξ is defined as {x: [∪μ<ξIμ, x]} by recursion on ξ (cf. [7], [17]). By P we denote the least fixed point of [X,x]. Then P = ∪ξξ holds. If Th allows the formation of P, we get the canonical definitions ∥Th()∥ = sup{∣x∣ + 1: Th ⊢ x ∈ P} and ∥Th∥ = sup{∥Th()∥: P is definable in Th} (cf. [17]). If ≺ is any primitive recursive ordering define Q≺[X,x] as the formula ∀y(y ≺ x → y ∈ X) and ∣x∣≺:= ∣x∣O≺. Then ∣x∣≺ turns out to be the order type of the ≺ -predecessors of x and P is the accessible part of ≺. So Th ⊢ x ∈ P implies the provability of ∣x∣≺ in Th.

A proof-irrelevant model of Martin-Löf's logical framework

2002 ◽  
Vol 12 (6) ◽  
pp. 771-795 ◽  
Author(s):  
DANIEL FRIDLENDER
Keyword(s):  
Elementary Proof ◽  
Lambda Calculus ◽  
Formal Theory ◽  
Level Of Detail ◽  
Logical Framework ◽  
Inductive Definitions ◽  
Definition Of

We extend the proof-irrelevant model defined in Smith (1988) to the whole of Martin-Löf's logical framework. The main difference here is the existence of a type whose objects themselves represent types rather than proof-objects. This means that the model must now be able to distinguish between objects with different degree of relevance: those that denote proofs are irrelevant whereas those that denote types are not. In fact a whole hierarchy of relevance exists.Another difference is the higher level of detail in the formulation of the formal theory, such as the explicit manipulation of contexts and substitutions. This demands an equally detailed definition of the model, including interpreting contexts and substitutions.We are thus led to a whole reformulation of the proof-irrelevant model. We present a model that is built up from an arbitrary model of the untyped lambda calculus. We also show how to extend it when the logical framework itself is enlarged with inductive definitions. In doing so, a variant of Church numerals is introduced.As in Smith (1988), the model can only be defined in the absence of universes, and it is useful to obtain an elementary proof of consistency and to prove the independence of Peano's fourth axiom.

Review of Status Passage: A Formal Theory.

1972 ◽  
Vol 17 (6) ◽  
pp. 358-359
Author(s):  
KURT W. BACK
Keyword(s):  
Formal Theory ◽  
Status Passage

Introduction

2018 ◽  
Author(s):  
Jacob Stegenga
Keyword(s):  
Medical Practice ◽  
Evidence Based Medicine ◽  
Inductive Inference ◽  
Formal Theory ◽  
Practice Research ◽  
Evidence Based ◽  
Master Argument ◽  
Medical Interventions ◽  
Concepts Of Disease ◽  
Based Medicine

This chapter introduces the book, describes the key arguments of each chapter, and summarizes the master argument for medical nihilism. It offers a brief survey of prominent articulations of medical nihilism throughout history, and describes the contemporary evidence-based medicine movement, to set the stage for the skeptical arguments. The main arguments are based on an analysis of the concepts of disease and effectiveness, the malleability of methods in medical research, and widespread empirical findings which suggest that many medical interventions are barely effective. The chapter-level arguments are unified by our best formal theory of inductive inference in what is called the master argument for medical nihilism. The book closes by considering what medical nihilism entails for medical practice, research, and regulation.

scholarly journals REFLECTION RANKS AND ORDINAL ANALYSIS

2020 ◽  
pp. 1-34
Author(s):  
FEDOR PAKHOMOV ◽  
JAMES WALSH
Keyword(s):  
Ordinal Analysis
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