Bernays Project: Text No. 14Hilbert’s investigations on the foundations ofarithmetic(1935)Paul Bernays(Hilberts Untersuchungen ¨uber die Grundlagen der Arithmetik, 1935.)Translation by: Dirk SchlimmComments:noneThe first investigations of Hilbert on the foundations of arithmetic followtemporally as well as theoretically [gedanklich] his investigations on the foun-dations of geometry. Hilbert begins the paper “On the concept of number”[¨Uber den Zahlbegriff”]1by bringing to bear [zur Geltung bringt] the axiomaticmethod for arithmetic, accordingly as [entsprechend wie] for geometry, whichhe contrasts [gegen¨uberstellt] to the otherwise usually applied [sonst gew¨ohnlichangewandten] “genetic” method.“At first let us bring to mind [Vergegenw¨artigen wir uns zun¨achst] the mannerof introducing [die Art und Weise der Einf¨uhrung] the concept of number. Start-ing from the concept of the number 1, usually one thinks at first the further1Jber. dtsch. Math.-Ver. Bd. 8 (1900); reprinted in Hilbert’s “Foundations of Geome-try”, 3rd-7th ed., as appendix VI.1rational positive numbers 2, 3, 4,. . . as arisen [entstanden] from the process ofcounting and developed their laws of calculation [Rechnungsgesetze entwickelt];then one arrives at the negative number by the requirement of the generalexecution [allgemeinen Ausf¨uhrung] of subtraction; one further defines the ra-tional [gebrochene] number perhaps [etwa] as a pair of numbers — then everylinear function has a zero [Nullstelle] —, and finally the real number as a cutor a fundamental sequence — thereby obtaining that every whole rationalindefinite [indefinite], and generally [¨uberhaupt] every continuous indefinitefunction has a zero [Nullstelle]. We can call this method of introducing theconcept of number the genetic, because the most general concept of real num-ber is generated [erzeugt] by successive expansion [Erweiterung] of the simpleconcept of number.One proceeds fundamentally differently with the structure [? Aufbau]of geometry. Here one tends [pflegt] to begin with the assumption of theexistence of all elements, i.e., one presupposes at the outset three systemsof things, namely the points, the lines, and the planes, and then bringsthese elements — essentially after the example of Euclid — in relation witheach other [miteinander in Beziehung] by certain axioms, namely the axiomsof concatenation [? Verkn¨upfung], of ordering [Anordnung], of congruency,and of continuity. Then the necessary task arises to show the consistency[Widerspruchslosigkeit] and completeness of these axioms, i.e., it must be proventhat the application of the compiled [aufgestellte] axioms can never lead tocontradictions, and moreover that the system of axioms suffices [ausreicht]to prove all geometric theorems. We want to call this chosen [eingeschlagene]procedure of investigation [Untersuchungsverfahren] the axiomatic method.2We raise the question, whether the genetic method is really the onlypertinent for the study of the concept of number and the axiomatic methodfor the foundations of geometry. It also appears to be of interest to contrastboth methods and to investigate which method is the most advantageousif one is concerned with [es sich ... handelt] the logical investigation of thefoundations of mechanics or other physical disciplines.My opinion is this: Despite the great pedagogical and heuristic value ofthe genetic method, the axiomatic method deserves the preference [Vorzug]for the final representation and complete logical security [? Sicherung] of thecontent of our knowledge [Erkenntnis].”Already Peano developed number theory axiomatically2Hilbert now setsup [stellt. . . auf ] an axiom system for analysis, by which the system of real2Peano, G. “Arithmetices principia nova methodo exposita”. (Torino 1889.) The intro-duction of recursive definitions is here not immaculate [einwandfrei]; the proof [Nachweis]of the solvability of the recursive equations is missing. Such a proof was provided alreadyby Dedekind in his essay “Was sind und was sollen die Zahlen” (Braunschweig 1887). Tointroduce the recursive functions beginning with Peano’s axioms it is best to proceed byfirst proving the solvability of the recursive equations for the sum after L. Kalm´ar by aninduction conclusion [? Induktionsschluß] after the argument of the parameter [? Param-eterargument], then defining the concept “less” with the help of the sum, and after thisusing Dedekind’s consideration [¨Uberlegung] for the general recursive definition. Thisprocedure is displayed [? dargestellt] in Landau’s textbook “Grundlagen der Analysis”(Leipzig 1930). Admittedly [allerdings] here the concept of function is used. If one wantsto avoid it, the recursive equations of the sum and product have to be introduced as ax-ioms. The proof of the general solvability of recursive equations follows then by a method[Verfahren] by K. G¨odel (cf. “¨Uber formal unentscheidbare S¨atze . . . ” [Mh. Math. Physik,Bd. 38 Heft 1 (1931)], and also Hilbert-Bernays Grundlagen der Mathematik Bd. 1 (Berlin1934) p. 412 ff.)3number is characterized as a real Archimedean field which cannot be extendedto a more extensive [umfassenderen] field of the same kind.A few exemplary [beispielsweise] remarks about dependencies follow theenumeration [Aufz¨ahlung] of the axioms. In particular it is mentioned that thelaw of commutativity of multiplication can be deduced from the remainingproperties of a field [K¨orpereigenschaften] and the order properties [Ordnung-seigenschaften] with the help of the Archimedean axiom, but not without it.The requirement [Forderung] of the non-extendibility [Nichterweiterbarkeit]is formulated by the “axiom of completeness”. This axiom has the advantageof conciseness [Pr¨agnanz]; however, its logical structure is complicated. Inaddition it is not immediately apparent [unmittelbar ersichtlich] from it [an ihm]that it expresses a demand of continuity [Stetigkeitsforderung]. If one wants,instead of this axiom, such a one that clearly has the character of a demandof continuity [Stetigkeitsforderung] and on the other hand does not include therequirement of the Archimedean axiom, it is recommended to take Cantor’saxiom of continuity, which says that if there is a series [Folge] of intervals suchthat every interval includes [umschließt] the following one, then there is a pointwhich belongs [angeh¨ort] to every interval. (The