Philosophy Lexicon of Arguments

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Recursion, theory of science, philosophy: recursion is a certain form in which rules are formulated, and which makes it possible to produce infinitely many possible cases from the application of a finite system of rules. See also inserting, embedding, infinity, systems, models, theories.

Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.

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IX 58
Recursive definition/recursion/sum/product/potency/arithmetic/Quine: recursion scheme: x + 0 = x - x + S°y = S°(x + y); - x times 0 = 0; - x times (S°y) = x + x times y (s) difference to the successor for x u y equal)>; - x0 = S°0 (=1) ; - x S°y = x times x y. - "Plus"/plus sign/Quine: so we can eliminate "+" completely from "x + 3": c - but not from "x + y" (Because we do not know how often we need the successor of x) - multiplication: we can eliminate the "times" from "x 3 times": "x + (x + (x + 0))" but not from "x times y" - recursions are real definitions if we regard the characters as scheme letters for numbers, not as bound variables.
IX 126
Transfinite recursion/sum/product/potency/Quine: c transformed into a real or direct definition: a " 0 = k, a " (S "z) = b "(a "z) - a " y = b Iy " k - from the last element: a = U{w: w e Seq u e w u w I S ^w < b}. - Advanced, liberal recursion: not only from the last previous element - instead totality of the previous elements - a = U{w: w e Seq u "y(y e ^w " " J >

Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.

W.V.O. Quine
Wort und Gegenstand Stuttgart 1980

W.V.O. Quine
Theorien und Dinge Frankfurt 1985

W.V.O. Quine
Grundzüge der Logik Frankfurt 1978

W.V.O. Quine
Mengenlehre und ihre Logik Wiesbaden 1967

W.V.O. Quine
Die Wurzeln der Referenz Frankfurt 1989

W.V.O. Quine
Unterwegs zur Wahrheit Paderborn 1995

W.V.O. Quine
From a logical point of view Cambridge, Mass. 1953

W.V.O. Quine
Bezeichnung und Referenz
Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982

W.V.O. Quine
Philosophie der Logik Bamberg 2005

W.V.O. Quine
Ontologische Relativität Frankfurt 2003

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Ed. Martin Schulz, access date 2017-11-21