Modelltheorie: Eine Einführung in Die Mathematische Logik Und Grundlagentheorie
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Wir betrachten hier jenen Teil der Grundlagenforschung, der die "intui- tive" oder "inhaltliche" Mathematik, d. h. das, was ein gew6hnlicher Mathematiker unter Mathematik versteht, systematisch beschreibt und analysiert. Im deskriptiven Teil wird die informale Mathematik in einer formal en Sprache (z. B. der der Mengenlehre) neu formuliert. Eine solche Sprache hat, verglichen mit der Sprache der informalen Mathematik, ein sehr eingeschr nktes Vokabular und eine vollkommen exakte Grammatik; dadurch wird naturlich die Pr zision erh6ht und der Blick von Unwesentlichem befreit. Im Gegensatz zu einer weitverbreiteten Ansicht, die weiter unten diskutiert wird, ist die Neuformulierung (die wie jede Beschreibung eines intuitiv erfa ten Gegenstandes wesentlich von unserer Auffassung seiner Natur abh ngt) nur ein Hilfsmittel der Grundlagenforschung: es handelt sich n mlich darum, die Bedeutung von S tzen der informalen Mathematik richtig wiederzugeben und nicht deren syntaktische Struktur; denn der u eren Form nach haben die formale und die informale Sprache (glucklicherweisel) wenig gemeinsam. Auch sollte man bemerken, da die durch die Umformulierung erzielte Pr zision zwar die technische Entwick- lung f6rdert, aber kaum geeignet ist, Schwierigkeiten zu beseitigen, die aus Unzul nglichkeiten der ursprunglichen Begriffe entstehen (gerade das Gegenteil ist der Fall: durch Nachdenken uber informale Begriffe werden wir zu einer guten Formalisierung gefuhrt). In den wohlbekannten "Krisen" (siehe z. B. Teil A, Abschnitt 1 weiter unten) ruhrten die Widerspruche von durchaus expliziten Prinzipien (Axiomen, Regeln) her, so da diese Schwierigkeiten nichts mit ungenugender formaler Pr zision zu tun hatten; das Problem lag vielmehr darin, unter verschiedenen, formal pr zisen Prinzipien die gultigen herauszufinden.
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