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**Kreisel’s Church.**
*(English)*
Zbl 0897.03005

Odifreddi, Piergiorgio (ed.), Kreiseliana: about and around Georg Kreisel. Wellesley, MA: A K Peters. 389-415 (1996).

This is a comprehensive presentation of Georg Kreisel’s various approaches, discussions and applications of Church’s thesis, using as far as possible Kreisel’s original words.

In a first section with “General remarks” (pp. 390-398) Church’s thesis is named as a candidate for “informal rigour.” Several variants of Church’s thesis are given, and Kreisel’s opinion is discussed that the equivalence of different characterizations of recursiveness does not give empirical support for Church’s thesis. Kreisel’s stronger version of Church’s thesis (what he called “Church’s superthesis”) is presented, as well as its relations to Turing’s theory of universal computers, to perfect fluids and perfect machines. As a prototype for applications of recursiveness and Church’s thesis in areas outside computability serves Hyman’s application to group theory. A last topic is the infinitistic character of recursiveness.

In the second section (pp. 398-404) Church’s thesis for constructive mathematics is treated, especially the distinction between mechanical and constructive rules, Kreisel’s suggestions for formal versions of Church’s thesis with their consistency proofs, and a discussion of their validity. Further topics are Church’s thesis as a reducibility axiom and Church’s rule.

Topic in section 3 (pp. 404-408) is the relation between Church’s thesis and mathematical reasoning, in particular individual and collective reasoning and the philosophy of mind, focusing on Gödel’s attempts to justify that minds are no Turing machines.

In the final section on “Physical theories” (pp. 408-413) the question whether physically realizable functions are recursive is treated, in particular in classical mechanics and biological processes.

For the entire collection see [Zbl 0894.03002].

In a first section with “General remarks” (pp. 390-398) Church’s thesis is named as a candidate for “informal rigour.” Several variants of Church’s thesis are given, and Kreisel’s opinion is discussed that the equivalence of different characterizations of recursiveness does not give empirical support for Church’s thesis. Kreisel’s stronger version of Church’s thesis (what he called “Church’s superthesis”) is presented, as well as its relations to Turing’s theory of universal computers, to perfect fluids and perfect machines. As a prototype for applications of recursiveness and Church’s thesis in areas outside computability serves Hyman’s application to group theory. A last topic is the infinitistic character of recursiveness.

In the second section (pp. 398-404) Church’s thesis for constructive mathematics is treated, especially the distinction between mechanical and constructive rules, Kreisel’s suggestions for formal versions of Church’s thesis with their consistency proofs, and a discussion of their validity. Further topics are Church’s thesis as a reducibility axiom and Church’s rule.

Topic in section 3 (pp. 404-408) is the relation between Church’s thesis and mathematical reasoning, in particular individual and collective reasoning and the philosophy of mind, focusing on Gödel’s attempts to justify that minds are no Turing machines.

In the final section on “Physical theories” (pp. 408-413) the question whether physically realizable functions are recursive is treated, in particular in classical mechanics and biological processes.

For the entire collection see [Zbl 0894.03002].

Reviewer: V.Peckhaus (Erlangen)

### MSC:

03-03 | History of mathematical logic and foundations |

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

03D20 | Recursive functions and relations, subrecursive hierarchies |

03D10 | Turing machines and related notions |