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Foundations of Mathematics Vol. 1

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Bernays Project: Text No. 12Foundations of Mathematics Vol. 1(1934)Paul Bernays(Grundlagen der Mathematik, Vol. 1)Translation by: Ian MuellerComments:none1. The Problem of consistency in axiomatics as a decision prob-lem.The situation in the field of foundations of mathematics, to which ourpresentation is related, is characterized by three kinds of investigations:1. the development of the axiomatic method, especially in foundations ofgeometry,2. the founding of analysis by today’s rigorous methods through the re-duction of mathematics [Gr¨ossenlehre] to the theory of numbers and setsof numbers,3. investigations in the foundations of number theory and set theory.A deeper set of problems, related to the situation reached through theseinvestigations, arises when methods are subjected to stricter demands; these1problems involve a new kind of analysis of the problem of the infinite. Wewill introduce these problems by considering axiomatics.The term ‘axiomatic’ is used in a wider and a narrower sense. We callthe development of a theory axiomatic in the widest sense of the word whenthe fundamental concepts and presuppositions are put at the beginning andmarked as such and the further content of the theory is logically derived fromthese with the help of definitions and proofs. In this sense the geometry ofEuclid, the mechanics of Newton, and the thermodynamics of Clausius wereaxiomatically founded.The axiomatic point of view was made more rigorous in Hilbert’s Founda-tions of Geometry; The greater rigor consists in the fact that in the axiomaticdevelopment of a theory one makes use of only that portion of the represen-tational subject matter [sachlichen Vorstellungsmaterial] from which the funda-mental concepts of the theory are formed which is formulated in the axioms;one abstracts from all remaining content. Another factor in axiomatics inthe narrowest sense is the existential form. This factor serves to distinguishthe axiomatic method from the constructive or genetic method of foundinga theory. (compare Hilbert’s “¨Uber den Zahlenbegriff”) In the constructivemethod the objects of a theory are introduced merely as a species of things(Brouwer and his school use the word “species” in this sense.); but in anaxiomatic theory one is concerned with a fixed system of things (or severalsuch systems) which constitutes a previously delimited domain of subjects forall predicates from which the assertions of the theory are constructed.Except in the trivial cases in which a theory has to do just with a fixedfinite totality of things, the presupposition of such a totality, the so-called2“domain of individuals”, involves an idealizing assumption over and abovethose formulated in the axioms.It is a characteristic of the more rigorous kind of axiomatics involvingabstraction from material content and also the existential form (“formal ax-iomatics” for short) that it requires a proof of consistency; but contentualaxiomatics introduces its fundamental concepts by reference to known expe-riences and its basic assertions either as obvious facts which a person canmake clear to himself or as extracts from complicated experiences;1express-ing the belief that man is on the track of laws of nature and at the same timeintending to support this belief through the success of the theory.However, formal axiomatics also needs a certain amount of evidence inthe performance of deductions as well as in the proof of consistency; there is,however, an essential difference: the evidence required does not depend onany special epistemological relation to the material being axiomatized, butrather it is one and the same for every axiomatization, and it is that prim-itive kind of knowledge which is the precondition of every exact theoreticalinvestigation whatsoever. We will consider this kind of evidence more closely.The following points of view are especially important for a correct eval-uation of the significance for epistemology of the relationship between con-tentual and formal axiomatics:Formal axiomatics requires contentual axiomatics as a supplement, be-cause only in terms of this supplement can one give instruction in the choice1Alternative translation: to known experiences and either presents its basic asser-tions as obvious facts which a person can make clear to himself or formulates them asextracts from complicated experiences,3of formalisms and, in the case of a particular formal theory, give an indicationof its applicability to some part of reality.On the other hand we cannot just stay at the level of contentual axiomat-ics, since in science we are almost always concerned with theories which gettheir significance from a simplifying idealization of an actual state of affairsrather than from a complete reproduction of it. A theory of this kind doesnot get a foundation through a reference to either the evident truth of itsaxioms or to experience; rather such a foundation is given only when theidealization involved, i.e. the extrapolation through which the concepts andfundamental assertions of the theory come to overstep the bounds of intuitiveevidence and of experience, is seen to be free of inconsistency. Reference tothe approximate correctness of the fundamental assertions is of no use forthe recognition of consistency; for an inconsistency could arise just becausea relationship which holds only in a restricted sense is taken to hold exactly.We must then investigate the consistency of theoretical systems withoutconsidering matters of fact and, therefore, from the point of view of formalaxiomatics.The treatment of this problem up until now, both in the case of geom-etry and of branches of physics, involved arithmetizing: one represents theobjects of a theory through numbers and systems of numbers and basic re-lations through equalities and inequalities thereby producing a translationof the axioms of the theory; under this translation the axioms become ei-ther arithmetic identities or provable arithmetic assertions (as in the case ofgeometry) or (as in physics) a system of conditions the simultaneous satis-fiability of which can be proved on the basis of certain existence assertions4of arithmetic. This procedure presupposes the correctness of arithmetic, i.e.the theory of real numbers (analysis); so we must ask what this correctnessamounts to.However, before we concern ourselves with this question we want to seewhether there isn’t a direct way of attacking the problem of consistency.


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