2. DEDEKIND CUTS
3. Existence and Uniqueness of the Real Numbers. We now show that the axiom system (R1)—(R3) for the real numbers characterizes them
there would be an x E M with x < c, and since every < z we should
have 6,, — a,, c —a,, c —x which would contradict lim(6,, —a,,)
=
0.This c is the greatest of the lower bounds, because if 6> c were a lower bound, we should have to have 6,, > 6 and b,, —a,, > 6 —a,, > b — c in contradiction to lim(b,, —a,,) = 0.
The list (a)—(e) of equivalent statements by no means exhausts all the possible formulations. One could for example also mention the HEINE—
BOREL covering property or the fact that every bounded infinite subset contains a limit point. The student learns about these and other results, as consequences of the property of completeness, in every introductory course on analysis.
There are totally ordered fields in which every fundamental sequence converges, but in which the ordering is not Archimedean. An example of this will be given in Chapter 12 where the real numbers will be extended to the field *R of non-standard numbers. In this extended field there are infinitely small and infinitely large numbers, and for this reason is not Archimedean, while every fundamental sequence is constant and therefore convergent. Just how much the Archimedean axiom restricts the possibili- ties is shown clearly by the following result due to HoLDER [13a], see also CARTAN [6]: An ordered group is Archimedean if and only if it is isomor- phic to a subgroup of the additive group of real numbers. One does not even have to assume that the group is commutative; it follows from the other hypotheses.
3. Existence and Uniqueness of the Real Numbers. We now show
The field of real numbers has no automorphisms apart from the identity mapping.
By the "field of real numbers" is here meant any field K which satisfies the (R1)—(R3). To prove this we start from the fact that K must contain the field Q of
the rationals. Every automorphism a of K maps Q identically on to itself, since a(O) =
0and o(1) =
1 and it followstherefore by complete induction that a N =
IdN. As every element of Q can be expressed in the form (a —b)/c with a,b, c E N, it then follows that0 Q = ide.
The ordering relation in K can be defined on the basis of the field struc- ture alone. We have z
y if and only if there exists a z E K such that
=
— y. It follows that every automorphi8m a is order preserving. If now a sequence (xv) converges to z in K, the image sequence must con- verge to a(z), or in other words a is continuous. As Qis dense in K, there is, for every z K, a sequence in Q which converges to x. This sequence is mapped identically on to itself by a. Regarded as an image sequence it converges to o(z). Since a limit is uniquely defined, a(x) = z.0
In Chapters 1 and 2, we have created starting from an infinite set, and using the methods of set theory to construct in succession the sets N, Z and Q on the way. The existence of Ilk is therefore assured, provided that we accept the validity of this set theory. Expressed in other words we may say that the axioms (R1)—(R3) are consistent (that is, free from contradiction), provided that the set theory we have used is consistent. The problem of the consistency of set theory is dealt with in the last chapter.
REFERENCES
[1] BACHMANN, P.: Vorlesungen über die Theorie der Irrationalzahien, Leipzig 1892
[21 BECKER, 0.: Grundlagen der Mathematik in geschichtlicher Entwick- lung, Freiburg/Munchen 1954
[3] BISHOP, E.: Foundations of Constructive Analysis, New York 1967 [4] BOLZANO, B.: Rein analytischer Beweis des Lehrsatzes, daB zwi-
schen je zwei Werthen, die em entgegengesetztes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege, Prag 1817, Ostwalds Klassiker Nr. 153, Leipzig 1905
[5] BOLZANO,B.: ReineZahlenlehre, 7. Abschnitt, in: Bernhard Bolzano — Gesamtausgabe,eds. E. Winter/i. Berg/F. Kambartel/I.
Rootselaar, Reihe II. A. Nachgelassene Schriften Bd. 8, Stuttgart/Bad Cannstatt 1976
[6] CARTAN, H.: Un théorème sur les groupes ordonnés, in: Bull. Sci.
Math. 63 1939, 201—205
[7] DEDEKIND, It.: Stetigkeit und lrrationalzahlen, Braunschweig 1872,
[81 DUGAC, P.: Elements d'analyse de Karl Weierstra8, in: Arch. hist.
ex. Sciences 10, 1973, 41—176
[9] EULER, L.: De progressionibus harmonicis observationes (1734/35), in: Op. omn. 1, 14, 73—86
[10] FRITZ, K.v.: The Discovery of Incommensurability by Hippasus of Metapontum, in: Annals of Mathematics 46, 1945, 242—264
[11] HELLER, S.: Die Entdeckung der stetigen Teilung, Abh. d. DL Ak.
Wiss. Berlin, Kiasse für Mathematik, Physik u. Technik 1958, Nr. 6, Berlin 1958
[121 HERMES, H.: Aufzählbarkeit, Eutscheidbarkeit, Berechenbarkeit, Einführung in
die Theorie der rekursiven Funktionen, Berlin/
Heidelberg/ New York 1971
[13] HILBERT, D.: Grundlagen der Geometrie, Leipzig 1899, ed. With supplements by P. Bernays, Stuttgard 111972
[13a] HOLDER, 0.: Die Axiome der Quantit.it und die Lehre vom Mafi.
Berichte Verh. Kg). Sächs. Ges. Wiss., Leipzig, Math. Phys. KI. 53,
1901, 1—64.
[14] IAMBLICHI: de communi mathematica scientia liber (ed. N. Festa), Leipzig 1891
[15] LkMBLICrn: de vita Pythagorica Liber (ed. L. Deubner), Leipzig 1937 [16] LANDAU, E.: Grundlagen der Analysis (Das Rechnen mit ganzen, rationalen, irrationalen, komplexen Zahien). Ergänzung zu den Lehr- büchern der Differential- und Integralrechnung, Leipzig 1930 (repr.
Frankfurt 1970)
[17] LIPSCHITZ, R.: Grundlagen der Analysis, Bonn 1877
[18] LORENZEN, P.: Differential und Integral. Eine konstruktive Einführung in die klassische Analysis, Frankfurt 1965
[19] MERAY, C.: Remarques sur Ia nature des quantités définies par Ia condition de servir de limites a des variables données, in: Revue des Sociétés savantes. Sciences mathém., phys. et naturelles, series, t.
IV, 1869
[201 NEUGEBAUER, 0. and A. SAcHs: Mathematical Cuneiform Texts, New Haven 1945
[21] STEVIN, S.: La practique d'arithmetique, Leiden 1685
[22] STIFEL, M.: Arithmetica integra, Nurnberg 1544, Buch II, Kap. I [23) TROPFKE, 3.: Geschichte der Elementarmathematik. Vol. 1 Arith-
metik und Algebra, Berlin/New York 1980
[24] WEIERSTRASS, K.: Einleitung in die Theorie der analytischen Funk- tionen. Vorlesung 1880/81. Nachschrift von A. Kneser
[25] VAN DER WAERDEN, B.L.: Die Pytbagoreer: religiöse Bruderschaft und Schule d. Wigs., Zurich 1979
[26] VAN DER WAERDEN, B.L.: Algebra II, Berlin/Heidelberg/New York 1967
FURTHER READING
[27] BURRILL, C.W.: Foundations of Rea' Numbers. New York: McGraw- Hill, 1967
[28] CoHEN, LW. and GERTRUDE EHRLIcH: The Structure of the Real Number System. Princeton: D. Van Nostrand, 1963
[29] DEDEKIND, R.: Essays on the Theory of Numbers, tr. by W.W. Be- man. Chicago: Open Court, 1901; New York: Dover, 1963
[30] DODGE, C.W.: Numbers and Mathematics. Boston: Prindle, Weber
& Schmidt, 1969
[31] LANDAU, EDMUND: Foundations of Analysis, tr. by F. Steinhardt, New York: Chelsea, 1951
[32] NIvEN, IVAN: Irrational Numbers, Carus Mathematical Monograph No. 11. New York: John Wiley, 1956