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**Investigations on general axiomatics. Edited by Thomas Bond and Jesus Mosterin.
(Untersuchungen zur allgemeinen Axiomatik. Herausgegeben von Thomas Bonk und Jesus Mosterin.)**
*(German)*
Zbl 0959.03004

Darmstadt: Wissenschaftliche Buchgesellschaft. viii, 166 S. (2000).

Carnap never completed his investigations on general axiomatics which grew to a typescript of 105 pages now stored in the Archives for Scientific Philosophy, University of Pittsburgh Libraries. In the meritorious edition under review the typescript is published in the state as it was in the first half of 1929 when Carnap stopped working on it. The edition is supplemented by a valuable introduction by the editors (pp.1-52) providing a summary of Carnap’s text together with detailed information on its context. Readers will obtain a vivid picture of the state of research in logic and the foundations of mathematics in the pre-Gödelian period being heavily influenced by David Hilbert and Wilhelm Ackermann’s “Grundzüge der theoretischen Logik” [Berlin, Julius Springer (1928; JFM 54.0055.01)]. Logical research at that time aimed at formulating logical formal systems as suitable tools for proof theory which was itself needed for providing consistency proofs for axiomatic systems.

The edition covers a first part with three sections. Carnap’s plan for a second part with another three sections is documented by Carnap’s working agenda (pp.153-155).

Carnap proposed a strict logicism maintaining the identity of logic and mathematics which he applied to a Hilbert style theory of axiomatic systems. According to the agenda set by Hilbert and Ackermann the completeness of axiomatized theories stood in the focus of Carnap’s interest. He especially developed a theory of forking: an axiom system is forkable if there is a proposition \(S\), such that \(S\) or \(\neg S\) can be added to the axioms without contradiction, i.e. \(S\) and \(\neg S\) are independent from the axiom system. This forking theory is applied to model theory, especially to the isomorphism of models, and to the decidability problem.

The edition covers a first part with three sections. Carnap’s plan for a second part with another three sections is documented by Carnap’s working agenda (pp.153-155).

Carnap proposed a strict logicism maintaining the identity of logic and mathematics which he applied to a Hilbert style theory of axiomatic systems. According to the agenda set by Hilbert and Ackermann the completeness of axiomatized theories stood in the focus of Carnap’s interest. He especially developed a theory of forking: an axiom system is forkable if there is a proposition \(S\), such that \(S\) or \(\neg S\) can be added to the axioms without contradiction, i.e. \(S\) and \(\neg S\) are independent from the axiom system. This forking theory is applied to model theory, especially to the isomorphism of models, and to the decidability problem.

Reviewer: Volker Peckhaus (Erlangen)

### MSC:

03-03 | History of mathematical logic and foundations |

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

01A75 | Collected or selected works; reprintings or translations of classics |

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

00A30 | Philosophy of mathematics |