Fast Fourier transforms by Michael Clausen

By Michael Clausen

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N 1 ) Enthiilt eine Teilmenge N von N das Element 0 und mit n auch v(n), so gilt N = N. 1 Bemerkungen (a) Fur n E N heiBt v(n) Nachfolger von n, und v ist die Nachfolgerfunktion. , die Abbildung v: N ----+ N X ist surjektiv, also, wegen (No), bijektiv. Beweis Es sei N:= {n EN; :In' EN: v(n') = n} U {O} = im(lI) U {O} . Fur n E N gilt lI(n) E im(lI) eN. Wegen 0 E N erhalten wir so mit aus (N 1 ), daB N = N gilt. • (b) Statt 0, v(O), v(v(O)), v(v(v(O))), ... schreibt man ublicherweise 0, 1,2,3, ...

Dann ist

Schreibt man ublicherweise 0, 1,2,3, ... (c) Manche Autoren ziehen es vor, den durch v beschriebenen "Ziihlvorgang" bei 1 und nicht bei 0 beginnen zu lassen. Dies ist natUrlich mathematisch ohne Bedeutung. 5 Die natiirlichen Zahlen 33 (d) Axiom (N d beschreibt das Prinzip der vollstandigen Induktion. 8 ausfiihrlich diskutieren. 2 Bemerkungen (a) 1m folgenden werden wir sehen, daB aus den Peano-Axiomen alle Aussagen iiber das Rechnen mit Zahlen, an das wir von der Schule her gewohnt sind, deduziert werden konnen.

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