Reverse mathematics (original) (raw)

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Die reverse Mathematik, ein Teilgebiet der mathematischen Logik, versucht zu bestimmen, welche Axiome notwendig sind, um bestimmte Theoreme zu beweisen. Reverse Mathematik ist damit gewissermaßen die Umkehrung der gewöhnlichen Mathematik, die versucht, Theoreme aus Axiomen herzuleiten. Die reverse Mathematik wurde 1974 von Harvey Friedman als mathematisches Projekt aufgebracht. Die Idee dazu entstand aus Ergebnissen der Mengenlehre, unter anderem dem klassischen Theorem, dass das Auswahlaxiom und das Lemma von Zorn über der Zermelo-Fraenkel-Mengenlehre ZF äquivalent sind.

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dbo:abstract Die reverse Mathematik, ein Teilgebiet der mathematischen Logik, versucht zu bestimmen, welche Axiome notwendig sind, um bestimmte Theoreme zu beweisen. Reverse Mathematik ist damit gewissermaßen die Umkehrung der gewöhnlichen Mathematik, die versucht, Theoreme aus Axiomen herzuleiten. Die reverse Mathematik wurde 1974 von Harvey Friedman als mathematisches Projekt aufgebracht. Die Idee dazu entstand aus Ergebnissen der Mengenlehre, unter anderem dem klassischen Theorem, dass das Auswahlaxiom und das Lemma von Zorn über der Zermelo-Fraenkel-Mengenlehre ZF äquivalent sind. (de) Les mathématiques à rebours sont une branche des mathématiques qui pourrait être définie simplement par l'idée de « remonter aux axiomes à partir des théorèmes », contrairement au sens habituel (des axiomes vers les théorèmes). Un peu plus précisément, il s'agit d'évaluer la robustesse logique d'un ensemble de résultats mathématiques usuels en déterminant exactement quels axiomes sont nécessaires et suffisants pour les prouver. (fr) Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory. Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. In higher-order reverse mathematics, the focus is on subsystems of higher-order arithmetic, and the associated richer language. The program was founded by Harvey Friedman and brought forward by Steve Simpson. A standard reference for the subject is , while an introduction for non-specialists is . An introduction to higher-order reverse mathematics, and also the founding paper, is . (en) 逆数学とは、数学の定理の証明に必要な公理を決定しようとする数理論理学のプログラムである。簡単に言えば、通常の数学が公理から定理を導くのとは逆に、「定理から公理を証明する」手法を用いることが特徴である。「選択公理とツォルンの補題はZF上で同値である」、というような集合論の古典的定理は、逆数学プログラムの予兆となるものだった。しかし、実際の逆数学では主に、集合論の公理ではなく、通常の数学の定理を研究するのを目的とする。 逆数学は大抵の場合、について実行され、定理がと証明論に動機付けられたの部分体系のうち、どれに対応するのかを研究する。2階算術を使うことで、再帰理論からの多くの技術も利用できる。実際、逆数学の結果の多くは、計算可能性解析学の結果を反映している。 逆数学は、 によってはじめて言及された。基本文献はを参照。 (ja) La matematica inversa è un ramo della matematica che si occupa di determinare quali sono gli assiomi minimi necessari per dimostrare un particolare teorema e più in generale cerca di determinare la teoria base che costituisce la matematica nel suo complesso. Partendo da una base di assiomi debole, si può scoprire che molte proposizioni matematiche sono equivalenti all'assioma aggiunto ad essa per dimostrarlo, come ad esempio il lemma di Zorn rispetto all'assioma della scelta. La maggior parte della matematica può essere formalizzata usando l'aritmetica del second'ordine e nei famosi teoremi dimostrati in ACA0, che è definita nell'aritmetica di Peano anche se questa è sovrabbondante come assiomi necessari per le dimostrazioni. Insiemi più ampi dei numeri reali, compresi tutti gli insiemi di Borel, possono essere codificati per mezzo di numeri reali con le relazioni di appartenenza esprimibili con l'. La differenza primaria fra la matematica classica nella teoria degli insiemi (ZFC) e nell'aritmetica del second'ordine è che in quest'ultima si usano codici degli insiemi invece che gli insiemi stessi (tranne che per gli insiemi di numeri interi). Con una formalizzazione corretta, la maggior parte dei teoremi generali sono effettivamente equivalenti all'assioma canonico minimo richiesto per la loro dimostrazione. La maggior parte dei risultati di base nell'analisi e nell'algebra sono provabili in WKL0, la cui consistenza logica equivale a quella dell'aritmetica ricorsiva primitiva e in cui il repertorio di funzioni dimostrabilmente ricorsive consiste delle funzioni ricorsive primitive. I teoremi aritmetici di base possono essere dimostrati nell'aritmetica di funzione esponenziale (EFA), che oltre agli assiomi di base per somma, moltiplicazione e l'elevamento a potenza, include l'assioma di induzione per le formule limitate da quantificatori. EFA basta, tra l'altro, per dimostrare che la teoria dei campi reali chiusi, e quindi anche la geometria classica, è completa. (it) 逆数学（Reverse mathematics）是数学的一个分支，大致可以看成是“从定理导向公理”而不是通常的方向(从公理到定理)。更精确一点，它试图通过找出证明所需的充分和必要的公理来评价一批常用数学结果的逻辑有效性。 该领域由Harvey Friedman在其文章“二阶算术系统及其应用（Some systems of second order arithmetic and their use）”中创立。它被Stephen G. Simpson和他的学生以及其他一些人所追随。Simpson写了关于该主题的参考教科书二阶算数的子系统(Subsystems of Second Order Arithmetic);本条目大部分内容取自该书的简介性质的第一章。其他参考读物的细节参看。 (zh)
dbo:wikiPageExternalLink http://www.math.psu.edu/simpson/sosoa/ https://rmzoo.math.uconn.edu/ https://www2.mathematik.tu-darmstadt.de/~kohlenbach/ http://www.math.psu.edu/simpson/ http://www.math.ohio-state.edu/~friedman/ http://www.brics.dk/RS/00/49/BRICS-RS-00-49.pdf https://www.math.wisc.edu/~lempp/theses/hunter.pdf%7Ctype=PhD
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rdfs:comment Die reverse Mathematik, ein Teilgebiet der mathematischen Logik, versucht zu bestimmen, welche Axiome notwendig sind, um bestimmte Theoreme zu beweisen. Reverse Mathematik ist damit gewissermaßen die Umkehrung der gewöhnlichen Mathematik, die versucht, Theoreme aus Axiomen herzuleiten. Die reverse Mathematik wurde 1974 von Harvey Friedman als mathematisches Projekt aufgebracht. Die Idee dazu entstand aus Ergebnissen der Mengenlehre, unter anderem dem klassischen Theorem, dass das Auswahlaxiom und das Lemma von Zorn über der Zermelo-Fraenkel-Mengenlehre ZF äquivalent sind. (de) Les mathématiques à rebours sont une branche des mathématiques qui pourrait être définie simplement par l'idée de « remonter aux axiomes à partir des théorèmes », contrairement au sens habituel (des axiomes vers les théorèmes). Un peu plus précisément, il s'agit d'évaluer la robustesse logique d'un ensemble de résultats mathématiques usuels en déterminant exactement quels axiomes sont nécessaires et suffisants pour les prouver. (fr) 逆数学とは、数学の定理の証明に必要な公理を決定しようとする数理論理学のプログラムである。簡単に言えば、通常の数学が公理から定理を導くのとは逆に、「定理から公理を証明する」手法を用いることが特徴である。「選択公理とツォルンの補題はZF上で同値である」、というような集合論の古典的定理は、逆数学プログラムの予兆となるものだった。しかし、実際の逆数学では主に、集合論の公理ではなく、通常の数学の定理を研究するのを目的とする。 逆数学は大抵の場合、について実行され、定理がと証明論に動機付けられたの部分体系のうち、どれに対応するのかを研究する。2階算術を使うことで、再帰理論からの多くの技術も利用できる。実際、逆数学の結果の多くは、計算可能性解析学の結果を反映している。 逆数学は、 によってはじめて言及された。基本文献はを参照。 (ja) 逆数学（Reverse mathematics）是数学的一个分支，大致可以看成是“从定理导向公理”而不是通常的方向(从公理到定理)。更精确一点，它试图通过找出证明所需的充分和必要的公理来评价一批常用数学结果的逻辑有效性。 该领域由Harvey Friedman在其文章“二阶算术系统及其应用（Some systems of second order arithmetic and their use）”中创立。它被Stephen G. Simpson和他的学生以及其他一些人所追随。Simpson写了关于该主题的参考教科书二阶算数的子系统(Subsystems of Second Order Arithmetic);本条目大部分内容取自该书的简介性质的第一章。其他参考读物的细节参看。 (zh) Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. (en) La matematica inversa è un ramo della matematica che si occupa di determinare quali sono gli assiomi minimi necessari per dimostrare un particolare teorema e più in generale cerca di determinare la teoria base che costituisce la matematica nel suo complesso. Partendo da una base di assiomi debole, si può scoprire che molte proposizioni matematiche sono equivalenti all'assioma aggiunto ad essa per dimostrarlo, come ad esempio il lemma di Zorn rispetto all'assioma della scelta. (it)
rdfs:label Reverse Mathematik (de) Mathématiques à rebours (fr) Matematica inversa (it) 逆数学 (ja) Reverse mathematics (en) 逆数学 (zh)
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