Ordinals Connected with Formal Theories for Transfinitely Iterated Inductive Definitions.

1983 ◽  
Vol 48 (3) ◽  
pp. 878
Author(s):  
Kurt Schutte ◽  
W. Pohlers
Keyword(s):  
Inductive Definitions ◽  
Formal Theories

scholarly journals Provable Wellorderings of Formal Theories for Transfinitely Iterated Inductive Definitions.

1983 ◽  
Vol 48 (3) ◽  
pp. 878
Author(s):  
Kurt Schutte ◽  
W. Buchholz ◽  
W. Pohlers
Keyword(s):  
Inductive Definitions ◽  
Formal Theories

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.

scholarly journals Provable wellorderings of formal theories for transfinitely iterated inductive definitions

1978 ◽  
Vol 43 (1) ◽  
pp. 118-125 ◽  
Author(s):  
W. Buchholz ◽  
W. Pohlers
Keyword(s):  
Transfinite Induction ◽  
Crucial Point ◽  
Inductive Definitions ◽  
Image Position ◽  
Intuitionistic Version ◽  
Formal Theories

By [12] we know that transfinite induction up to ΘεΩN+10 is not provable in IDN, the theory of N-times iterated inductive definitions. In this paper we will show that conversely transfinite induction up to any ordinal less than ΘεΩN+10 is provable in IDNi, the intuitionistic version of IDN, and extend this result to theories for transfinitely iterated inductive definitions.In [14] Schütte proves the wellordering of his notational systems using predicates is wellordered) with Mκ ≔ {x ∈ and 0 ≤ κ ≤ N. Obviously the predicates are definable in IDNi with the defining axioms:where Prog [Mκ, X] means that X is progressive with respect to Mκ, i.e.The crucial point in Schütte's wellordering proof is Lemma 19 [14, p. 130] which can be modified towhere TI[Mκ + 1, a] is the scheme of transfinite induction over Mκ + 1 up to a.

Is Rationality Normative?

2017 ◽  
Author(s):  
Ralph Wedgwood
Keyword(s):  
Rational Choice ◽  
Rational Belief ◽  
False Beliefs ◽  
Ought Implies Can ◽  
Normative Concept ◽  
Original Meaning ◽  
Formal Theories

In its original meaning, the word ‘rational’ referred to the faculty of reason—the capacity for reasoning. It is undeniable that the word later came also to express a normative concept—the concept of the proper use of this faculty. Does it express a normative concept when it is used in formal theories of rational belief or rational choice? Reasons are given for concluding that it does express a normative concept in these contexts. But this conclusion seems to imply that we ought always to think rationally. Four objections can be raised. (1) What about cases where thinking rationally has disastrous consequences? (2) What about cases where we have rational false beliefs about what we ought to do? (3) ‘Ought’ implies ‘can’—but is it true that we can always think rationally? (4) Rationality requires nothing more than coherence—but why does coherence matter?

scholarly journals Equivalents of the finitary non-deterministic inductive definitions

2019 ◽  
Vol 170 (10) ◽  
pp. 1256-1272 ◽  
Author(s):  
Ayana Hirata ◽  
Hajime Ishihara ◽  
Tatsuji Kawai ◽  
Takako Nemoto
Keyword(s):  
Inductive Definitions

scholarly journals Scoping review assessing the evidence used to support the adoption of mobile health (mHealth) technologies for the education and training of community health workers (CHWs) in low-income and middle-income countries

BMJ Open ◽  
2018 ◽  
Vol 8 (7) ◽  
pp. e019827 ◽  
Author(s):  
Niall Winters ◽  
Laurenz Langer ◽  
Anne Geniets
Keyword(s):  
Systematic Reviews ◽  
Scoping Review ◽  
Low Income ◽  
Health Workers ◽  
Education And Training ◽  
Evidence Base ◽  
Mobile Technologies ◽  
Middle Income ◽  
And Training ◽  
Formal Theories

ObjectivesUndertake a systematic scoping review to determine how a research evidence base, in the form of existing systematic reviews in the field of mobile health (mHealth), constitutes education and training for community health workers (CHWs) who use mobile technologies in everyday work. The review was informed by the following research questions: does educational theory inform the design of the education and training component of mHealth interventions? How is education and training with mobile technology by CHWs in low-income and middle-income countries categorised by existing systematic reviews? What is the basis for this categorisation?SettingThe review explored the literature from 2000 to 2017 to investigate how mHealth interventions have been positioned within the available evidence base in relation to their use of formal theories of learning.ResultsThe scoping review found 24 primary studies that were categorised by 16 systematic reviews as supporting CHWs’ education and training using mobile technologies. However, when formal theories of learning from educational research were used to recategorise these 24 primary studies, only four could be coded as such. This identifies a problem with how CHWs’ education and training using mobile technologies is understood and categorised within the existing evidence base. This is because there is no agreed on, theoretically informed understanding of what counts as learning.ConclusionThe claims made by mHealth researchers and practitioners regarding the learning benefits of mobile technology are not based on research results that are underpinned by formal theories of learning. mHealth suffers from a reductionist view of learning that underestimates the complexities of the relationship between pedagogy and technology. This has resulted in miscategorisations of what constitutes CHWs’ education and training within the existing evidence base. This can be overcome by informed collaboration between the health and education communities.

μ-definable sets of integers

1993 ◽  
Vol 58 (1) ◽  
pp. 291-313 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Fixed Point ◽  
Free Variable ◽  
Order Logic ◽  
First Order Logic ◽  
Inductive Definition ◽  
Natural Class ◽  
First Order ◽  
Inductive Definitions ◽  
Definable Sets ◽  
The Universe

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

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