Bhnl,[:amLV, LRil, [ :a,~l l [ .+=mmm'mu],ndu--~i D i r k H u y l e b r o u c k , E d i t o r ]

GOdel's Vienna JOHN W . DAWSON JR., AND KARL SIGMUND

Does y o u r h o m e t o w n h a v e a n y

mathemat ica l tourist at tractions such

as statues, plaques, graves, the car#

w h e r e the f a m o u s conjecture w a s

made, the desk w h e r e the f a m o u s

initials are scratched, birthplaces,

houses, or memorials? Have y o u

e n c o u n t e r e d a mathemat i ca l sight on

y o u r travels? I f so, we invite y o u to

submi t to this co lumn a picture, a

description o f its ma themat i ca l

significance, and either a m a p or

directions so that others m a y fo l low

in y o u r tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende. Belgium e-maih huylebrouck@gmail.com

B y August 17, 1939, a European war was imminent. Two weeks before Hitler invaded Poland, Dr.

Kurt G6del received a letter from his tailor: 'Sending repaired trousers. As I heard, you will journey to America again. You will certainly need a suit . . . . With German greetings, Decker.'

G6del ordered the suit. His journey back to Princeton seemed to offer no problems. On August 30, 1939, a few days after the Stalin-Hitler pact, GOdel blissfully announced to his friend Karl Menger his intention of returning to Princeton forthwith, in a letter which, in Menger's eyes, 'may well represent a record for unconcern on the threshold of world-shaking events' (Menger 1994).

Two days later, Hitler informed a wildly cheering German Reichstag that 'since 5:45, the fire has been returned.' G6del's outlook changed drastically. He had to write Oswald Veblen in No- vember: 'It now seems likely that I will not be able to come to Princeton this academic year, because it will probably be impossible to obtain a German visa during the war-time.' G6del was trapped in Vienna. He would spend the next few months in desperate attempts to leave for the US. Against all odds, he finally succeeded. But after the war, G6del would never return to Vienna again. He was through with it.

Vienna: A Logical Choice In 1924, when he arrived in Vienna as an eighteen-year-old from provincial Brno to study at the university, things had looked very different. Vienna had overcome years of hunger and misery, and the economy was picking up. The intellectual and cultural life underwent an amazing flowering. Very soon, Kurt GOdel would contribute to it. His work may one day well be viewed as the most lasting achievement of that epoch.

Today's tourists to Vienna follow the traces of Habsburg, visit the imperial

museums, and are shown the many dwellings of Beethoven and Mozart, or the churches where Haydn and Schu- bert performed. Increasingly, tours in- clude aspects of Vienna between the two world wars, most notably the re- cent Leopold museum, with its paint- ings by Klimt, Schiele, and Kokoschka, or the architectural monuments of Red Vienna, or the art ddco villas built by Hoffmann and Loos. If, as a tourist, you relax in a coffee-house between visits to these sights, you will already be very close to GOdel. Let us pick up his trail, a map of which appears at the end of this article. (For more details, see Daw- son 1997.)

Kurt G6del's parents were well o f f ~ his father was manager, and part owner, of a textile firm in Brt~nn, a charming little town which used to be called 'the Czech Manchester', less than two hours by train north from Vienna. In 1919, the treaty of St. Germain had established a border between Austria and Czechoslo- vakia, but for the large German-speak- ing segment of what was by then Brno, Vienna as the former capital was still the focus, and obviously the place to go to study. At that time, young GOdel could probably not have chosen a site more tailor-made to his talents any- where in the world.

Settling Down Kurt GOdel moved in with his brother Rudolf, four years his elder, who stud- ied medicine under the illustrious fac- ulty to which Freud had often dreamt of belonging. Kurt first enrolled for physics, but switched to mathematics under the spell of superb introductory lectures on calculus by Furtw5ngler and a 'survey of the major problems in phi- losophy' by Gomperz (Sigmund 2006).

In his fifteen years in Vienna, GOdel lived in seven different apartments. Tourists will be reminded of Beethoven or Mozart, who also moved a lot, The GOdel brothers obviously had a well-

THE MATHEMATICAL INTELLIGENCER �9 2006 Spnnger Science+Business Media, Inc.

Letter from G6del's tailor, August 1939.

defined image in mind in their apart- ment hunts: the seven houses look remarkably alike. All are massive four- sto W buildings erected at the turn of the century, staid and stately. If you have seen one, you have seen them all. Only one of the houses has a plaque com- memorating Kurt G6del, but this may change with the 2006 centenary--G6del was born on April 28, 1906, and Vienna is set to celebrate.

Most of the houses are close to the university, and especially close to the building on Boltzmanngasse where the institutes of physics and (in G6del's time) the Mathematische Seminar were located. On one occasion G6del lived just across the street, two floors above the Josephinum, one of the most dec- orative cafes favoured by Viennese aca- Care Josephinum.

demics and students (but brutally dis- figured today).

G6del was a very quiet young man, but not always the hermit he later be- came. He studied diligently, and was soon invited to join the Vienna Circle, a brilliant group of positivists gathered around the mathematician Hans Hahn (famed for his work on functional analy- sis) and the philosopher Moritz Schlick. Both were professors at the university. The young German philosopher Rudolf Camap had also just moved to Vienna and joined the Circle. G0del's closest stu- dent friends were Marcel Natkin and Her- bert Feigl, who studied mathematics and philosophy and were disciples of Schlick. They all met, every second Thursday, in a small lecture room of the Mathematis- che Seminar (Stadler 2002). Informally, most of them also met at theJosephinum, or the Cafe Reichsrat, Cafe Central, and Cafe Arkadenhof, among other places filled with journals and tobacco smoke. These cafes were crowded with intellec- tuals and world-reformers nurturing delusions of grandeur and talking phi- losophy, literature, psychoanalysis, eco- nomics, or politics late into the night.

A Nervous Splendor Austrian politics was dominated by a fierce struggle between the Social Dem- ocrats and the Conservatives. The for- mer had the majority in Red Vienna, and were engaged in a sweeping pro- gram of social reforms. The Conserva-

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Plaque on G6del's apartment house in Himmelsstrasse (Himmel means "heaven"). A divorced woman working for a living, Adele Nimbursky.

tives held the Catholic countryside. The strict fiscal policy of their chancellor, the prelate Seipel, had stabilised the cur- rency but exacerbated social unrest. The tensions exploded in a wild riot in 1927. After its bloody suppression, which claimed 80 deaths, the little republic seemed doomed to civil war.

In 1929, G6del struck up an ac- quaintance with a young woman living across the street, in Lange Gasse. Adele Nimbursky, n~e Porkert, had divorced and returned to live with her parents. A trained ballet dancer, she had worked for a spell at the Nachtfalter (literally, 'nocturnal moth'), a night-club in down- town Vienna. Thirty years old, she worked as a masseuse. According to Kurt's brother Rudolf (by then a young MD), Kurt had during his school-boy years experienced an 'escapade' with a mature woman, until his parents put a stop to it (cf. K6hler 2002). This time, he kept his new friendship under wraps. It was meant to last: Ten years later, Adele would become Mrs. GOdel.

In February 1929, G6del's father died

unexpectedly. In the following weeks GOdel obtained Austrian citizenship, and shortly afterward his PhD thesis was approved by Hans Hahn. The dis-

sertation was a remarkable piece of work: he had proved that the axioms for first-order logic given in Russell and Whitehead's Principia Mathematica

Authors on the Strudelhof- steige

JOHN W. DAWSON, JR., is Professor of Mathematics, Emeritus, at Penn State York He received his S.B. in mathematics from M.I.T. and his Ph.D. from the University of Michigan. Since 1979 his work has focused on G6del's life and work During the years 1982-1984 he catalogued G6del's Nachlass at the Insitute for Advanced Study in Princeton. His biography of G6del, Logical Dilemmas, was published in 1997.

Information Sciences and Technology Center Penn State York 1031 Edgecomb Ave. York PA 17403 USA e-mail: jwd7@psu.edu

46 THE MATHEMATICAL INTELLIGENCER

Kurt GOdel, (1906-2000).

were independent and complete.1 This solved two problems that Hilbert and Ackermann had posed in their recent book on mathematical logic. (In that same year, incidentally, G6del's sixty- year-old professor, Philipp Furtw/ingler, succeeded in proving Hilbert's Haup- tidealsatz [principal ideal theorem]).

After he received his doctorate, life for GOdel did not change much. His mother had moved to Vienna after her husband's death, and she now lived with her two sons and an aged relative in two spa-

GOdel studied at the Mathematische Seminar (a small institute with only three professors).

cious, adjacent apartments in the Josef- stadt. The GOdels often went to plays and concerts but otherwise kept pretty well to themselves. Kurt used to work at night, sleep late, and hang around the mathe- matics library, occasionally helping Hahn prepare a seminar or correct exams.

~The names "completeness" and "incompleteness" might wrongly be thought to imply the incompatiblity of those two theorems. The confusion arises from the two different senses in which the English term "com- plete" is used in logic. In the former sense it means "capable of proving every statement that holds in all

models of the axioms"; in the latter sense it means "capable of proving or refuting every sentence of the language." The corresponding German terms are vollstbndig and entscheidungsdefinit. The former is prop- erly glossed as "complete," but the latter is better rendered by "decisive," which captures the meaning of

the German root entscheid- and suggests that the theory makes a definite determination (that is, yields a proof or refutation) for each sentence.

On weekends and holidays, he made many excursions with his mother and brother to the scenic mountainside at Semmering or Mariazell, close to Vi- enna. Later, in letters to his mother, he would often hark back to those times, 'Our excursion on the Leopoldsberg must have been in 1932 because 1 re- member thinking all the time of my coming lecture in Hahn's seminar.' Or: 'Of course I remember the sausages in

KARL SIGMUND is professor at the faculty of mathematics at the University of Vienna for longer than he cares to think and lives mostly in the Thirties of the previous century. Somewhat after that time, he worked in ergodic theory and dynamical systems, and even later in biomathemathics and game theory. Together with John Daw- son and Kurt M0hlberger, he wrote: Kurt G6deI--The Album" (Vieweg, Wiesbaden, 2006)

Faculty for Mathematics University of Vienna Nordbergstrasse 15 1090 Vienna, Austria and Institute for Applied Systems Analysis Laxenburg, Austria e-mail: kad.sigmund@u nivie.ac.at Adele Nimbursky, 1932.

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Kurt G6del outside his workplace on the Strudlhofgasse.

Josefstadterstrasse 43, complete with in-house movie theatre.

Annaberg but if I am not mistaken the digestive tonic from the apothecary soon made me feel better.'

GOdel had become friends with Karl Menger, who in 1927 had returned from postdoctoral years in Amsterdam to take up a position as associate professor in geometry (Golland and Sigmund, 2002).

Although Menger was barely four years older than G6del, he acted as his men- tor. Together, the two gradually drifted away from the positions held by the Vienna Circle. They shared neither Schlick's infatuation with Wittgenstein nor Hahn's political activism, and they had little interest in the Circle's attempts

to 'go public' by founding the Ernst Mach Verein. But they kept attending the sessions, and they often met with individual members.

Incompleteness "No Small Matter" In the summer of 1930, while meeting in the Cafe Reichsrat (today's Konditorei Sluka) with Carnap and others to plan a joint journey to KOnigsberg, G6del first dropped the news that he had succeeded in proving, not the completeness, as ex- pected, but--rather startlingly--the in._= completeness of formalized arithmetic.

G6del on the road. How to drop a time-bomb.

THE MATHEMATICAL INTELLIGENCER

Ticket to fame. GOdel's standing-room ticket to Karl Menger's lecture on 'The New Logic'.

The axioms in Principia Mathematica did not suffice to derive all true state- ments about natural numbers.

Apparent ly the members of the Vi- enna Circle did not immediate ly grasp what this momentous news meant to Hilbert 's p rogram for proving the con- sistency of mathematics. They had p lanned to hold a workshop in K6nigs- berg, a satellite meet ing to the ritual Jahrestreffen [annual gathering] of the German Mathematical Society. The pur- pose of their meet ing was to discuss the foundat ions of mathematics. On the first day of that meeting, Carnap and Hahn gave their lectures as if nothing had happened , and GOdel presented his comple teness theorem from the year before. It was only during the fi- nal discussion that he ment ioned, al- most casually, his incompleteness re- suit. The protocol of the meet ing bears witness to a double take: at first, the discussion f lowed peaceful ly on, but then G0del was invited to add an ap- pendix to the proceedings, and explain what he meant. John von Neumann, of course, had unders tood immediately. A

few weeks later, he wrote to G6del say- ing that the p roof of the incomplete- ness theorem showed that Hilbert 's p rogram must fail: consistency cannot

be formally establ ished within the sys- tem. But by that time, GOdel had al- ready submit ted the same result to the Monatshefie.

Sanatorium Purkersdorf, designed by the architect Hoffmann.

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The impact was tremendous. Hahn hailed the work as an 'achievement of the first o r d e r . . , that will take its place in the history of mathematics'. And Mar- cel Natkin wrote to G0del: 'So you have proved that Hilbert's system of axioms contains unsolvable problems---why, this is no small matter.'

But Kurt G0del kept on, as before, with his quiet life, making himself useful at the Mathematische Seminar in many ways. Young Olga Taussky would later write about his kindness in helping students and colleagues. With some malicious amusement she

also noticed his encounters with mem- bers of the opposite sex. G6del was discreet, to be sure, but not unwilling to be seen with a good-looking girl (see Taussky 1987).

Standing up for The New Logic Some more bright young people had collected around Menger: N6beling, Wald, and Alt. Their informal group, the Kolloquium, became a hotbed of results on topology, mathematical economy, and of course, logic. G6del gave a num- ber of lectures, over the years, and most of them were published in the yearly

proceedings, the Ergebnisse eines math- ematischen Kolloquiums, of which he was co-editor. The sessions also at- tracted philosophers from beyond the Vienna Circle, such as Karl Popper, and economists such as Oskar Morgenstern. The latter, at the time, attempted to prove that predictions in economics were in principle impossible, in loose analogy to G6del's incompleteness re- suits in logic.

O O O

G6del applied to become Privatdozent (which would give him the right to lec- ture at the University, but implied no appointment there). In due course, he obtained his Habilitation, and thence the title. Fifty-year old Eduard Helly was also Privatdozent at the time; so was the topologist Waiter Mayer, before Ein- stein called him to his side in Berlin (and right afterwards took him along into exile at Princeton). Privatdozenten got pitifully small fees for their lectures (in one term, G6del would earn enough to have bought three beers). But hap- pily G6del did not need a salary from the university. He still had his own means, and more importantly, he was invited to the newly founded Institute for Advanced Study in Princeton! John von Neumann and Karl Menger had pointed him out to Oswald Veblen, who was talent-scouting for the IAS. Veblen listened to a Kolloquium lecture by G6del and was suitably impressed.

By that time, G6del was obviously the star of the institute in Vienna. Al- fred Tarski, John yon Neurnann, and Willard Quine visited him. Karl Menger, especially, sang his praises wherever he went. Menger and Hahn had launched a very well-attended series of public lec- tures, and they used the substantial ad- mission fees to pay the salary of young Olga Taussky, and to erect a funeral monument for Boltzmann. When Menger lectured on 'The New Logic', G6del's discoveries were for the first time pre- sented to a wider audience. G6del at- tended, but he only purchased a stand- ing-room ticket; after all, he was well acquainted with the topic!

G6del was often forced to cancel his lectures. The fees were modest, to say the least.

Private Rest Homes and Public Bedlam GOdel paid a heavy price for his genius. From his childhood, when he suffered a bout of rheumatic fever, he had been a

50 THE MATHEMATICAL INTELLIGENCER

Vienna's new Mayor inspects the University of Vienna in 1938.

A Nazi leader informs: G6del had studied with a Jewish professor. On the other hand, all of mathematics had formerly been 'verjudet'.

hypochondriac. The stress of his work led to episodes of nervous breakdown, and eventually, paranoia. His second visit to Princeton had to be aborted af- ter two months, and Rudolf was called upon to escort his brother from Paris back to Vienna. During the next few years, Kurt G6del spent much of his time in sanatoria and rest homes around Vienna, in Rekawinkel, Purkersdorf, and Aflenz.

During his time in Rekawinkel, he was so plagued by fears of poisoning that he would only eat what Adele pre- pared for him, in his presence. She had to eat from the same plate, with the same spoon. Later, she recorded that G6del's mother was so frightened by her son's psychosis that she locked her- self in her bedroom at night. Eventu- ally, G6del would recover (for a time) and return to the mathematics institute. But altogether, between his three visits to Princeton and his stays in sanatoria, he lectured for only two semesters at the University of Vienna, once on logic and once on set theory.

The political situation in Austria in the 1930s deteriorated rapidly. Under Hitler's greedy eyes, the Austrian chancellor Dollfuss had established a fascist regime in an ill-guided attempt to obtain Mus- solini's protection. In 1934, a short burst of civil war was followed by an abortive Nazi putsch. Dollfuss was murdered. His successor Schuschnigg was unable to build up a national consensus. In March 1938, Hitler's troops entered Austria, greeted by a jubilant mob.

The Nazis lost no time in "purifying" the university. GOdel had to prove that he was Aryan. His rank as Privatdozent was too low to demand a loyalty oath to the Eihrer, but all professors had to take the oath. Non-Aryans were literally sent packing, and Adolph Eichmann set up offices in Vienna to organise the exodus.

Intellectually, G6del was very iso- lated: Hahn had died in 1934; Schlick had been murdered in 1936 by a para- noiac former student; Menger had em- igrated in 1937 to the USA; Feigl and Natkin were gone, as were Carnap, Mayer, and Taussky; Helly, Alt, and Wald had all managed, with great pains, to flee to the United States.

But G6del was not alone. Adele was at his side. She had helped him while he was in the sanatorium, and she had helped him to get out of it. Now she helped him on the stairs of the Strudl-

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The dean writes that GOdel has hardly any inner feeling for National Socialism. He will hardly be up to the difficulties awaiting him abroad. On the other hand, his manners are good.

hofstiege next to the Seminar, when he was stopped by SA-rowdies who took him to be Jewish and knocked his glasses off. She chased them away, with her handbag or with her umbrel la-- in any case with great spirit.

It was time to get married, even if his b r ide- -by now almost forty, a di- vorcee, a former dancer - -was not ex- actly up to the expectations of his fam- ily. Forty years later, Rudolf GOdel would write icily 'I do not presume to form an opinion on my brother's mar- riage' (KOhler 2002). GOdel's mother re- turned to her villa in Brno, and Kurt and Adele left the Josefstadt, moving to the outskirts of Vienna.

The civil ceremony took place in September 1938, at the height of the Czechoslovakia crisis. Two weeks later, G6del left for another stay in the United States--without Adele. When he boarded the steamer, the Munich treaty was just a week old.

GOdel had achieved another break- through by proving the consistency of the cont inuum hypothesis. One half of Hilbert 's famous first problem was thus solved. During his residency at IAS, news of the pogroms of the Kristallnacht shook public opinion in the US. GOdel spent the following spring with Karl Menger in South Bend, Indiana, and during that term, Hitler d i smembered what was left of Czechoslovakia. In spite of these dire developments and the urgent warn- ings of his former mentor, GOdel in- sisted on returning to Europe the fol- lowing summer.

Trapped in Vienna GOdel returned to Vienna at the climax of a political nightmare, which soon be- came a personal nightmare as well. He had hoped to return to the Institute for Advanced Study in the fall, this time with Adele, but now he encountered a

fiercely adverse bureaucracy. The min- istry of finance, for example, looked askance at his earnings in America. A confused exchange of letters between the ministry of education and the au- thorities at the University of Vienna re- suited in the loss of his title of Privat- dozent, when it emerged that the previous fall he had left without proper permission. Worst of all, GOdel was ex- amined for the Wehrmacht and found fit for garrison duty. In addition, the American Consulate, flooded as it was with demands for immigration visas, took a very dilatory attitude towards his application. Then, Hitler's troops rushed into Poland, and war was declared in the west. German troops on one side, and French and British on the other, dug themselves in, although the west- ern front, in this early stage, saw action only on the waters of the Atlantic.

Although there seemed to be no hope that GOdel would get back to

52 THE MATHEMATICAL INTELLIGENCER

Towards new shores. G6del and his wife, Adele, on shipboard, 1940. A receipt that never was signed, and a University of Vienna

diploma that was never used.

Princeton, the IAS persevered. In a mas- terful analysis, John yon Neumann ex- plained to the director exactly how and where to intervene with the State De- partment. It worked. Shortly before Christmas 1939, GOdel went to Berlin to pick up his visa. Since the German authorities could not provide a job for him, they decided to let him earn dol- lars in the States. Some worthies still de- bated whether he was man enough to

master the difficult situations he was bound to encounter in the States as a representative of the New Germany, but they let him go. His field of mathemat- ical logic, obviously 'infested by Jewish thinking' and aloof from Deutsche Mathematik, surely meant nothing for the German war effort, and he had not yet been called to serve garrison duty. In early Janua W 1940, G6del received his emigration permit, and shortly af-

terwards the necessary transit visa; this time he travelled to America with his wife. They had to take the long route, via the Soviet Union, Japan, and the Pa- cific. After a gruelling winter journey through Siberia and a three-week delay in Yokohama, G6del and Adele reached New York in mid-March. Nine months earlier, he had started from there for his return to Vienna and what would be- come a nerve-racking world tour. He

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The German consulate wanted G6del to return to Germany,

but the Fatherland could offer him no job. And as G6del pointed

out, his health was delicate.

In a letter to his mother, G0del reports his nightmares about

returning to Vienna.

"Decker's suit looks as good as new."

would never travel far again. And when his Viennese friend Oskar Morgenstern, also exi led in Princeton, asked him about Vienna, he remarked pithily, 'The coffee is wretched! '

Meanwhile, Back in Vienna The bureaucratic machinery GOdel had left behind cont inued milling for a time. He had appl ied to become Dozent

neuer Ordnung (the new German form of Habilitation), and al though he was considered politically lukewarm, even suspect, he eventually received an em- b lazoned diploma, c /o University of Vi- enna. That is where it still waits to be collected, together with a receipt GOdel was meant to sign.

For years, there would be official in- quiries about the whereabouts of Dozent GOdel. His brother Rudolf repl ied testily that as long as Germany

THE MATHEMATICAL INTELLIGENCER

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~, 5,11,29-16,11.}7 Josefst~idterstrasse 43/1/2/12a, Ado8:

7. Universitb, t Wien, A-1010

8, Mathematisches Institut, Strudelhofgasse, Aqo9o

~. 9.1t,39-M,2.48 Hegelgasse 51215a, A-Ioio

The GOdel trail. GOdel heard his philosophy lectures at the main building of the university (7), and his lectures on mathemat-

ics and physics in the complex building (8) between Boltzmanngasse, Strudlhofgasse, and Wfihringerstrasse, which had opened in 1915. The Strudlhofstiege is close by.

could not offer him a job, he was forced to remain in Princeton. Moreover, as Rudolf could honest ly point out, the German consul in New York had ex- plicitly advised against an Atlantic cross- ing; and the opt ion of a return journey via Siberia was not open for long.

After the war, every other Sunday GOdel wro te to his mothe r (who had m o v e d back to Vienna in 1944, and thus avo ided the expu l s ion of the Ge rman- speak ing popu la t ion from Czechoslovakia) . Al though Adele re- tu rned to Vienna a number of t imes, G6de l never jo ined her. In one letter, he confes sed to his mother that he had had n ightmares about be ing t r apped in Vienna, unable to leave. Fi- nally, his mother , by then almost eighty, f lew to New York to visit her famous son in Princeton. Her visit was such a success that it was r e p e a t e d every s econd year until her dea th in 1966.

Both Morgenstern and Menger re-

sumed their contacts with Austria after the war, but not GOdel. He adamant ly ref\~sed all honors conferred upon him by the University of Vienna and the Aus- trian Academy of Sciences; and when he wrote, 'It seems that Vienna is chang- ing only very slowly, ' he did not mean it as a compliment.

And, you may ask, what about the suit GOdel had ordered from his tailor Decker? In 1952 G6del wrote to his mother: 'Decker 's suit looks as good as new. ' Made of pre-war quality cloth, that was to be expected.

REFERENCES

AIt, Franz (1998) Afterword to Karl Menger, Ergebnisse eines mathematischen Kolloqui-

ums, ed. by E. Dierker and K. Sigmund, Springer-Verlag, Wien

Dawson, John (1997) Logical Dilemmas: The

Life and Work of Kurt GOdel, A. K. Peters, Wellesley, MA

Menger, Karl (1994) Reminiscences of the Vi-

enna Circle and the Mathematical Collo-

quium, Kluwer, Dordrecht Stadler, Friedrich (2001) The Vienna Circle,

Springer-Verlag, Wien, New York

Dawson John (2002), Max Dehn, Kurt Gddel, and the Trans-Siberian Escape Route, No- tices AMS 49, 1068-1075

Sigmund, Karl (2006) Pictures at an exhibition, Notices AMS 53, 426-430

Taussky-Todd, Olga (1987) Remembrances of Kurt G6del, in G6del remembered: Salzburg, 10-12 July 1983 (ed. P. Weingartner and L. Schmetterer), Bibliopolis, Naples, 29-41

Golland, Louise and Sigmund Karl (2000) Exact Thought in a Demented Time--Karl Menger and his Viennese Mathematical Col- Ioqium, Math Intelligencer 22(3), 34-45

Sigmund, Karl A (1995) Philosopher's Mathe- matician- Hans Hahn and the Vienna Circle, Math Intelligencer, 17(4), 16-29

KOhler, Eckehard, et al. (eds) (2002) Wahrheit

und Beweisbarkeit, HOlder-Pichler-Tempsky, Vienna

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