Intensionale Funktionalinterpretation der Analysis

Diller, Justus; Vogel, H.

doi:10.1007/bfb0079547

Cited by 10 publications

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“…Here (BR) is the schema of (simultaneous) bar recursion of Spector (see [23,3,18]) extended to the types T X (see [15]) and A ω [X, d, X , Z ] results from A ω [X, d] by adding new constants X and Z of type…”

Section: Remark 37mentioning

confidence: 99%

A Logical Uniform Boundedness Principle for Abstract Metric and Hyperbolic Spaces

Kohlenbach

2006

Electronic Notes in Theoretical Computer Science

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We extend the principle Σ 0 1 -UB of uniform Σ 0 1 -boundedness introduced earlier by the author to a uniform boundedness principle ∃-UB X for abstract bounded metric and hyperbolic spaces which are not assumed to be compact. Despite the fact that this principle implies numerous results which in general are true only for compact spaces (and continuous functions) we can prove that for a large class K of such consequences A the conclusion A is true in arbitrary bounded spaces even when ∃-UB X is used to facilitate the proof of A. For a somewhat more restricted class of sentences A even effective uniform bounds can be extracted from such proofs.

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Section: Remark 37mentioning

confidence: 99%

A Logical Uniform Boundedness Principle for Abstract Metric and Hyperbolic Spaces

Kohlenbach

2006

Electronic Notes in Theoretical Computer Science

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“…So there is something to be explained (for other comments, cf. Diller and Vogel 1975, Burr 2000a, Burr 2002). We start with point 2.…”

Section: Constructive Set Theory In All Finite Types Czfω ‐ and Imentioning

confidence: 99%

Functional Interpretations of Constructive Set Theory in All Finite Types

Diller

2008

Dialectica

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Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit ‘interpreting’ instances that make the implication valid. For proofs in constructive set theory CZF ‐, it may not always be possible to find just one such instance, but it must suffice to explicitly name a set consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ∧‐interpretation theorem for constructive set theory in all finite types CZFω ‐ shows. By changing to a hybrid interpretation ∧q, we show closure of CZFω ‐ under rules that – in stronger forms – have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZFω ‐ and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation.

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Über ein mit der Bar‐Induktion Verwandtes Schema

Vogel¹

1979

Mathematical Logic Qtrly

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Einleitung B R~U W E R S Bar-Theorem ist in Gestalt des Prinzips der Bar-Induktion grundlegend fur die formale Behandlung der intuitionistischen Analysis in KLEENE-VESLEY [6]. Dagegen wird in der Monographie von KREISEL-TROELSTRA [S] eine induktiv definierte Menge K von BRouwER-Operationen an den Anfang gestellt und die Bar-Induktion (vom Typ 0) daraus (und einem Stetigkeitsprinzip) abgeleitet. In Analogie daeu definieren wir induktiv zu jedem Typ c eine Menge von C-Objekten vom TypInduktion uber C, wobei wir im Funktionalkontext die Menge C durch ein Funktional C, das C aufzahlt, ersetzen. Um triviale Bedeutungen von C auszuschliefien, fiigen wir das Axiom CC bei mit der Aussage: Y(c( Yc) 5 Yc fur Y E C, c vom Typ 0 -+ cr. Wir erhalten, daB in der HEYTma-Arithmetik endlicher Typen mit Extensionalitat (E-HA,), vollem Auswahlaxiom ( A C ) und CI, CC die monotone Bar-Induktion herleitbar ist, so daB E-HA, + AC + CI + CC nach einer Bemerkung von SCARPELLINI [9] rnindestens so stark ist wie die klassische Analysis. Der transfiniten Rekursion iiber C entspricht die Bar-Rekursion. Nach dem Vorbild von HOWARD [4]und DILLER-VOGEL [3] funktionalinterpretieren wir in 9 3 die C-Induktion mit der C-Rekursion. Der offenbaren Ahnlichkeit mit der Theorie T, von ZUCKER in [l2] gehen wir in $ 4 nach. Es zeigt sich, daB T, in ein Funktionalsystem iiber dem Grundtyp der natiirliclien Zahlen eingebettet werden kann, wenn nur gemal3 der Definition der endlichen Baumklassen auch C erblich definiert wird (als C').Mein Dank gilt den Herren J. DTLLER und H. LUCKHARDT fur wertvolle Gespriiche uber den Gegenstand dieser Arbeit.

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Intensionale Funktionalinterpretation der Analysis

Cited by 10 publications

References 7 publications

A Logical Uniform Boundedness Principle for Abstract Metric and Hyperbolic Spaces

A Logical Uniform Boundedness Principle for Abstract Metric and Hyperbolic Spaces

Functional Interpretations of Constructive Set Theory in All Finite Types

Über ein mit der Bar‐Induktion Verwandtes Schema

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