##
**Kurt Gödel. Truth and provability. Vol. 2: Compendium on the work.
(Kurt Gödel. Wahrheit und Beweisbarkeit. Band 2: Kompendium zum Werk.)**
*(German)*
Zbl 1028.03002

Wien: öbv & hpt. 445 S. (2002).

The second volume of W. DePauli-Schimanovich-Göttig and P. Weibel’s long-term joint project contains a collection of 39 contributions and documents related to K. Gödel’s work, whereas the first volume gives an exposition of biographical research, documents, interviews, and photographs. Most of the contributions of the second volume are research papers accompanied by short extracts from Gödel’s work and pieces from various correspondences providing material treated in the research papers.

The papers are distributed over five sections: formal logic, computer science and logic, set theory, cosmology, and philosophy.

Section on formal logic: M. Baaz and R. Zach survey in “Das Vollständigkeitsproblem und Gödels Vollständigkeitsbeweis” (The completeness problem and Gödel’s completeness proof, pp. 21-27) the emergence of the completeness problem and the significance of Gödel’s completeness proof. B. Buldt discusses in “Kompaktheit und Endlichkeit in der formalen Logik” (Compactness and finiteness in formal logic, pp. 31-49) the context and the significance of the compactness theorem in formal logic. G. H. Moore interprets in “Die Kontroverse zwischen Gödel und Zermelo” (The controversy between Gödel and Zermelo, pp. 55-64) the quarrel between E. Zermelo and K. Gödel about Gödel’s incompleteness results. It started at the 1931 DMV Bad Elster meeting. As their correspondence shows, it was due to misunderstandings on both sides. The respective correspondence between Zermelo and Gödel is also provided (pp. 51-54). J. Hintikka interprets in “Die Dialektik in Gödel’s Dialectica-Interpretation” (The dialectics in Gödel’s Dialectica interpretation, pp. 67-90) the Dialectica interpretation in terms of game-theoretical semantics and independence friendly logic. In his attachment to the previous paper (pp. 91-93) G. Sandu gives a technical result concerning the translation from first- to second-order logic.

Section on computer science and logic: K.-D. Schulz tells in “Wie allgemein sind die allgemein-rekursiven Funktionen?” (How general are the general recursive functions?, pp. 103-122) in a detailed way the story of the theory of computability and general recursive functions. Special emphasis is given to J. Herbrand’s role. The correspondence between Herbrand and Gödel is also provided (pp. 99-102). A. Leitsch interprets in “Über nicht-rekursive Beweisverkürzungen” (On non-recursive shortenings of proofs, pp. 127-134) Gödel’s contributions on the length of proofs in terms of the theory of complexity, especially the shortening of proofs by extending the theory. C. Smoryński gives in “Gödels Unvollständigkeitssätze” (Gödel’s incompleteness theorems, pp. 147-159) a description of Gödel’s incompleteness theorems, placing them into the context of Hilbert’s Programme and proof theory, and discussing their effects.

Section on set theory: U. Felgner surveys in “Zur Geschichte des Mengenbegriffs” (On the history of the concept of set, pp. 169-185) the history of the notion of set as routed in the notions of the infinite and of number from ancient times up to W. Ackermann’s axiomatization of 1956. N. Brunner and U. Felgner discuss in “Gödels Universum der konstruktiblen Mengen” (Gödel’s universe of constructible sets, pp. 189-198) Gödel’s hierarchy of constructible sets, which was used to prove the consistency of AC and GCH with ZF. U. Felgner deals in “Ein Brief Gödels zum Fundierungsaxiom” (A letter of Gödel’s on the axiom of Fundierung, pp. 205-216) with a correspondence of Gödel with P. Bernays in 1931 on the axiom of Fundierung.

Section on cosmology: H. Rupertsberger surveys in “Das Gödelsche Universum” (Gödel’s universe, pp. 219-229) Gödel’s contributions (and their reception) to the global structure of space and time as proposed in three papers dealing with A. Einstein’s general theory of relativity and his field equations. A. Bartels focusses in “Das Gödel-Universum und die Philosophie der Zeit” (Gödel’s universe and the philosophy of time, pp. 231-250) on the aspect of time in the Gödel universe and the possibility of time travels. The author relates Gödel’s contributions to the discussions on time travels in the 1960s, which were ignorant of Gödel’s cosmology. He then presents Gödel’s theory in the light of scientific realism. G. Hon discusses in “Gödel, Einstein, Mach: Die Vollständigkeit physikalischer Theorien” (Gödel, Einstein, Mach: The completeness of the theories of physics, pp. 251-267) the violation of Mach’s principle in Gödel’s cosmology and its effects on the general theory of relativity. P. Yourgrau deals in “Philosophische Betrachtungen zu Gödels Kosmologie” (Philosophical considerations on Gödel’s cosmology, pp. 269-298) with implications of Gödel’s cosmology on the philosophical notion of time. M. Stöltzner stresses in “Zeitreisen, Singularitäten und die Unvollständigkeit physikalischer Axiomatik” (Time travels and the incompleteness of the axiomatics of physics, pp. 289-304) Gödel’s significance for the study of space-time singularities. He furthermore relates Gödel’s rotating universes to Hilbert’s programme of axiomatizing relativistic field physics.

Section on philosophy: J. Czermak presents in “Abriss des ontologischen Argumentes” (Sketch of the ontological argument, pp. 309-324) Gödel’s ontological proof of the existence of God in a comprehensive way. A reprint of the manuscript of two pages is provided (pp. 307-308). P. Hájek discusses in “Der Mathematiker und die Frage der Existenz Gottes (betreffend Gödels ontologischen Beweis)” (The mathematician and the problem of the existence of God – in respect to Gödel’s ontological proof, pp. 325-336) different interpretations of Gödel’s proof of the existence of God. E. Kähler in his “Gödels Platonismus” (Gödel’s Platonism, pp. 341-386) gives an extensive interpretation of Gödel’s conceptual Platonism and its main results according to which mathematics is non-empirical or objective or it has both features. Gödel’s “weak Platonism”, regarding mathematics as independent from the human mind and objective, assumes that on each level of the mind there are mathematical theorems into which the mind has no insight. The author supplements it by a strong version of Platonism demanding that the reality of mathematics is perceived by a mind on a higher level, not only, however, through its senses (p. 358). L. E. J. Brouwer is seen as a weak Platonist. Among other aspects the relation between Platonism and theology is discussed. C. Thiel reports in “Gödels Anteil am Streit über Behmanns Behandlung der Antinomien” (Gödel’s part in the controversy on Behmann’s treatment of the paradoxes, pp. 387-394) on Gödel’s role in a quarrel H. Behmann had with W. Dubislav on his attempt to solve Russell’s Paradox and to save type-free logic. The volume is closed by B. Buldt’s comprehensive paper “Philosophische Implikationen der Gödelschen Sätze? Ein Bericht” (Philosophical implications of Gödel’s theorems? A report, pp. 395-438). He criticizes some implications of Gödel’s myth which can be seen in applying Gödel’s results outside the domain of mathematical logic. He especially discusses the relation of Gödel’s theorems to Hilbert’s programme, logicism, and the notions of truth and provability.

Despite these final words, critical even against Gödel’s self-assessment on the philosophical implications of his results, this impressive volume makes clear the reasons for Gödel’s myth: his massive and fruitful influence on contemporary and subsequent researchers in various fields of knowledge. Although the volume represents more than 20 years of research on Gödel, authors and editors succeed in drawing a vivid and up-to-date picture of Gödel and his influence. Potential readers from outside the German-speaking world will regret the decision to publish this volume completely in German and to translate even texts originally written in English.

The papers are distributed over five sections: formal logic, computer science and logic, set theory, cosmology, and philosophy.

Section on formal logic: M. Baaz and R. Zach survey in “Das Vollständigkeitsproblem und Gödels Vollständigkeitsbeweis” (The completeness problem and Gödel’s completeness proof, pp. 21-27) the emergence of the completeness problem and the significance of Gödel’s completeness proof. B. Buldt discusses in “Kompaktheit und Endlichkeit in der formalen Logik” (Compactness and finiteness in formal logic, pp. 31-49) the context and the significance of the compactness theorem in formal logic. G. H. Moore interprets in “Die Kontroverse zwischen Gödel und Zermelo” (The controversy between Gödel and Zermelo, pp. 55-64) the quarrel between E. Zermelo and K. Gödel about Gödel’s incompleteness results. It started at the 1931 DMV Bad Elster meeting. As their correspondence shows, it was due to misunderstandings on both sides. The respective correspondence between Zermelo and Gödel is also provided (pp. 51-54). J. Hintikka interprets in “Die Dialektik in Gödel’s Dialectica-Interpretation” (The dialectics in Gödel’s Dialectica interpretation, pp. 67-90) the Dialectica interpretation in terms of game-theoretical semantics and independence friendly logic. In his attachment to the previous paper (pp. 91-93) G. Sandu gives a technical result concerning the translation from first- to second-order logic.

Section on computer science and logic: K.-D. Schulz tells in “Wie allgemein sind die allgemein-rekursiven Funktionen?” (How general are the general recursive functions?, pp. 103-122) in a detailed way the story of the theory of computability and general recursive functions. Special emphasis is given to J. Herbrand’s role. The correspondence between Herbrand and Gödel is also provided (pp. 99-102). A. Leitsch interprets in “Über nicht-rekursive Beweisverkürzungen” (On non-recursive shortenings of proofs, pp. 127-134) Gödel’s contributions on the length of proofs in terms of the theory of complexity, especially the shortening of proofs by extending the theory. C. Smoryński gives in “Gödels Unvollständigkeitssätze” (Gödel’s incompleteness theorems, pp. 147-159) a description of Gödel’s incompleteness theorems, placing them into the context of Hilbert’s Programme and proof theory, and discussing their effects.

Section on set theory: U. Felgner surveys in “Zur Geschichte des Mengenbegriffs” (On the history of the concept of set, pp. 169-185) the history of the notion of set as routed in the notions of the infinite and of number from ancient times up to W. Ackermann’s axiomatization of 1956. N. Brunner and U. Felgner discuss in “Gödels Universum der konstruktiblen Mengen” (Gödel’s universe of constructible sets, pp. 189-198) Gödel’s hierarchy of constructible sets, which was used to prove the consistency of AC and GCH with ZF. U. Felgner deals in “Ein Brief Gödels zum Fundierungsaxiom” (A letter of Gödel’s on the axiom of Fundierung, pp. 205-216) with a correspondence of Gödel with P. Bernays in 1931 on the axiom of Fundierung.

Section on cosmology: H. Rupertsberger surveys in “Das Gödelsche Universum” (Gödel’s universe, pp. 219-229) Gödel’s contributions (and their reception) to the global structure of space and time as proposed in three papers dealing with A. Einstein’s general theory of relativity and his field equations. A. Bartels focusses in “Das Gödel-Universum und die Philosophie der Zeit” (Gödel’s universe and the philosophy of time, pp. 231-250) on the aspect of time in the Gödel universe and the possibility of time travels. The author relates Gödel’s contributions to the discussions on time travels in the 1960s, which were ignorant of Gödel’s cosmology. He then presents Gödel’s theory in the light of scientific realism. G. Hon discusses in “Gödel, Einstein, Mach: Die Vollständigkeit physikalischer Theorien” (Gödel, Einstein, Mach: The completeness of the theories of physics, pp. 251-267) the violation of Mach’s principle in Gödel’s cosmology and its effects on the general theory of relativity. P. Yourgrau deals in “Philosophische Betrachtungen zu Gödels Kosmologie” (Philosophical considerations on Gödel’s cosmology, pp. 269-298) with implications of Gödel’s cosmology on the philosophical notion of time. M. Stöltzner stresses in “Zeitreisen, Singularitäten und die Unvollständigkeit physikalischer Axiomatik” (Time travels and the incompleteness of the axiomatics of physics, pp. 289-304) Gödel’s significance for the study of space-time singularities. He furthermore relates Gödel’s rotating universes to Hilbert’s programme of axiomatizing relativistic field physics.

Section on philosophy: J. Czermak presents in “Abriss des ontologischen Argumentes” (Sketch of the ontological argument, pp. 309-324) Gödel’s ontological proof of the existence of God in a comprehensive way. A reprint of the manuscript of two pages is provided (pp. 307-308). P. Hájek discusses in “Der Mathematiker und die Frage der Existenz Gottes (betreffend Gödels ontologischen Beweis)” (The mathematician and the problem of the existence of God – in respect to Gödel’s ontological proof, pp. 325-336) different interpretations of Gödel’s proof of the existence of God. E. Kähler in his “Gödels Platonismus” (Gödel’s Platonism, pp. 341-386) gives an extensive interpretation of Gödel’s conceptual Platonism and its main results according to which mathematics is non-empirical or objective or it has both features. Gödel’s “weak Platonism”, regarding mathematics as independent from the human mind and objective, assumes that on each level of the mind there are mathematical theorems into which the mind has no insight. The author supplements it by a strong version of Platonism demanding that the reality of mathematics is perceived by a mind on a higher level, not only, however, through its senses (p. 358). L. E. J. Brouwer is seen as a weak Platonist. Among other aspects the relation between Platonism and theology is discussed. C. Thiel reports in “Gödels Anteil am Streit über Behmanns Behandlung der Antinomien” (Gödel’s part in the controversy on Behmann’s treatment of the paradoxes, pp. 387-394) on Gödel’s role in a quarrel H. Behmann had with W. Dubislav on his attempt to solve Russell’s Paradox and to save type-free logic. The volume is closed by B. Buldt’s comprehensive paper “Philosophische Implikationen der Gödelschen Sätze? Ein Bericht” (Philosophical implications of Gödel’s theorems? A report, pp. 395-438). He criticizes some implications of Gödel’s myth which can be seen in applying Gödel’s results outside the domain of mathematical logic. He especially discusses the relation of Gödel’s theorems to Hilbert’s programme, logicism, and the notions of truth and provability.

Despite these final words, critical even against Gödel’s self-assessment on the philosophical implications of his results, this impressive volume makes clear the reasons for Gödel’s myth: his massive and fruitful influence on contemporary and subsequent researchers in various fields of knowledge. Although the volume represents more than 20 years of research on Gödel, authors and editors succeed in drawing a vivid and up-to-date picture of Gödel and his influence. Potential readers from outside the German-speaking world will regret the decision to publish this volume completely in German and to translate even texts originally written in English.

Reviewer: Volker Peckhaus (Paderborn)

### MSC:

03-06 | Proceedings, conferences, collections, etc. pertaining to mathematical logic and foundations |

03A05 | Philosophical and critical aspects of logic and foundations |

03-03 | History of mathematical logic and foundations |

01A60 | History of mathematics in the 20th century |

83F05 | Relativistic cosmology |