On Godel's Way In: The Influence of Rudolf Carnap The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Goldfarb, Warren. 2005. On Godel's way in: The influence of Rudolf Carnap. Bulletin of Symbolic Logic 11, no. 2: 185-193. Published Version http://dx.doi.org/10.2178/bsl/1120231629 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3153305 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA On Godel's Way in: The Influence of Rudolf Carnap Author(s): Warren Goldfarb Source: The Bulletin of Symbolic Logic, Vol. 11, No. 2 (Jun., 2005), pp. 185-193 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/1556748 Accessed: 24/06/2009 18:20 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Bulletin of Symbolic Logic. http://www.jstor.org THE BULLETINOF SYMBOLICLOGIC Volume 11, Number 2, June 2005 ON GODEL'SWAY IN: THE INFLUENCEOF RUDOLF CARNAP WARREN GOLDFARB The philosopherRudolf Carnap (1891-1970), although not himself an originatorof mathematicaladvances in logic, was muchinvolved in the de- velopmentof the subject.He was the most importantand deepestphiloso- pher of the ViennaCircle of logical positivists,or, to use the label Carnap laterpreferred, logical empiricists.It was Carnapwho gavethe most fully developedand sophisticatedform to the linguisticdoctrine of logical and mathematicaltruth: the view that the truthsof mathematicsand logic do not describesome Platonisticrealm, but ratherare artifactsof the way we establisha languagein whichto speakof the factual,empirical world, fall- outs of the representationalcapacity of language.(This view has its roots in Wittgenstein'sTractatus, but Wittgenstein's remarks on mathematicsbeyond first-orderlogic are notoriouslysparse and cryptic.) Carnapwas also the thinkerwho, afterRussell, most emphasizedthe importanceof modernlogic, and the distinctiveadvances it enablesin the foundationsof mathematics, to contemporaryphilosophy. It was throughCarnap's urgings, abetted by Hans Hahn, once Carnaparrived in Viennaas Privatdozentin philosophy in 1926, that the ViennaCircle began to take logic seriouslyand that posi- tivist philosophybegan to grapplewith the questionof how an accountof mathematicscompatible with empiricismcan be given (see Goldfarb1996). A particularfacet of Carnap'sinfluence is not widelyappreciated: it was Carnapwho introducedKurt Godel to logic, in the serioussense. Although Godel seemsto haveattended a courseof Schlick'son philosophyof mathe- maticsin 1925-26,his secondyear at the University,he did not at that time pursuelogic further,nor did the seminarleave much of a traceon him. In the earlysummer of 1928,however, Carnap gave two lecturesto the Circlewhich Godel attended,or so I surmise.At theseoccasions, Carnap presented mate- rial from his manuscript treatise, Untersuchungenzur allgemeinenAxiomatik, that is, "Investigationsinto generalaxiomatics", which dealt with questions of consistency,completeness and categoricity.Carnap later circulatedthis materialto variouspeople including Godel. Received February 2, 2005; accepted February 9, 2005. I am grateful to Steven Awodey, John Baldwin, Solomon Feferman, Akihiro Kanamori, and Stephen Simpson for helpful suggestions, and to Enzo De Pellegrin for research and editorial assistance. ? 2005, Association for Symbolic Logic 1079-8986/05/1102-0006/$1.90 185 186 WARREN GOLDFARB Godel's serious interest in logic dates from that time. Subsequently he began the systematic reading in logic that brought him to the frontiers of what was then known, as his libraryrequest-slips show (Dawson 1997, p. 53). So, for example, in a letter to his fellow student Herbert Feigl, later a distinguished philosopher of science in America, he reports on what he did with his 1928 summer vacation: I myself was in Briinn the whole time and among other things read a part of Principia mathematica,about which, however, I was less enthusiastic [begeistert]than I had expected from its reputation. (letter to Feigl, 24 September 1928; Godel2003, p. 403) That he was put onto Principiais no doubt also Carnap'sinfluence; Carnap's concern with details of Principia far outstripped what would have been com- mon among mathematiciansby that time. It is, of course, a simplifiedversion of the system of Principiathat appears in Godel's 1931 incompletenesspaper. Hilbert and Ackermann's Grundziigeder theoretischenLogik, published in 1928, was another of the first books Godel looked at in logic. He found his dissertation problem in it: the question of the completeness of first-order logic is explicitly formulated for the first time in the book, on p. 68. In 1975, the sociologist Burke Grandjean sent a questionnaire to Godel, in which he asked, "When did you become interested in the problem... of the complete- ness of logic and mathematics?" and "Are there any influences you would single out as especially important in this regard?" Godel's answers were: "1928"; and "Hilbert Ackermann: Introduction to math Logic, Carnap: Lectures of math Logic" (Godel 2003, p. 447). Now, Carnap's logical work did not influence Godel in any mathematical or technical way: there is in Carnap's material no mathematical idea that could be exploited for serious results. What G6del learned from Carnap concerned the concepts that needed to be investigated. The peculiarities of the way Carnap framed those concepts motivated Godel concerning both the problems he set himself to investigate and the correct formulation of the concepts. Carnap'sfocus in the Untersuchungenwas on properties of axiom systems such as consistency and completeness.' The subtitle of the first volume of the manuscript, and the title of Carnap'slectures, was Metalogik. But in fact Carnap's work was not at all metalogical, in our sense. For Carnap worked in a Russellian tradition, that is, with a conception of logic that can be called "universalist"(the term stems from vanHeijenoort 1967). All notions were to be defined withinthe logical system. Consider the conjunction F of axioms of a system (Carnap's treatment is limited to finitely axiomatized theories). F contains various nonlogical vocabulary; let us call F(R) what we obtain from F by replacing these with variables (of the appropriate types). Then 'Carnap never published this manuscript;it appeared only recently as Carnap2000. In my account of this work in the next few paragraphs, I draw heavily on Awodeyand Carus2001. ON GODEL'SWAY IN: THE INFLUENCEOF RUDOLFCARNAP 187 the assertionthat the axioms have a model (are satisfiable)is expressedin the systemby (3R)F (R). The assertionthat an assertionG followsfrom the axiomsis expressedas (VR)(F(R) D G(R)), that is, in the terminologyof Russell1903, F(R) formallyimplies G(R). The assertionthat the axioms are inconsistentis then the assertionthat a contradictionfollows from them: (3G)(V2R)(F(R)D G(R).-G(R)). The quantification"(3G)" is to be construedhere not as overformulas, but ratherover propositional functions (i.e., properties)of a higherorder. Note how simpleit is to show,once the conceptsare definedin this way, thatif the axiomsare consistent then theyhave a model. For assumethey do not havea model: that is, (VR)^F(R). Then (VR)(F(R) D H) for any H, and in particular,for some H of the form G(R).G (R). Hencethe axioms are inconsistent. So herewe havea trivialproof of a claimthat verballysounds exactly like a completenesstheorem: an axiom system has a model if it is consistent. Of course, it is not really such a theorem;in particularthe definitionof consequencethat is used to framethat of consistencydoes not accordwith whatis ordinarilymeant by a deductiveconsequence of the axiom system. The opening of Godel's doctoral dissertation,completed in July 1929, reactsto this pseudo-completenesstheorem. He startsby being insistently expliciton whatnotion of consequenceis at issue in the dissertation.In the second sentence,he frameshis theoremthus: everyvalid formulaof first- orderlogic "canbe derivedfrom the axiomsby meansof a finitesequence of formalinferences." Two sentenceslater, after reformulating the theoremas aboutconsistent axiom systems,