# The structure & [and] limits of finitism

## Abstract/Contents

- Abstract
- In the years since Hilbert announced his now famous proof theory and the years following Gödel's Incompleteness Theorems much important work has been done in the subject that has cemented its fate as one of the pillars of metamathematics. To be sure, many of the directions that proof theory has taken since the 1920s can be found as germs in Hilbert and the Hilbert School's writings from the period - Kreisel's unwinding program a classic example of the metamathematical analysis of a proof to extract its mathematical content, Kohlenbach's program to extract the computational content of a proof using proof-theoretic techniques, Feferman's conservativity results, a spectacular example that shows the proof-theoretic relationships between formal theories, and so many more. But the one direction that precipitated the whole of proof theory - namely, to provide an epistemic foundation for a mathematical theory by providing it with a consistency proof from the point of view of a theory strictly weaker than it, a finitistic theory in the Hilbert School's language - has remained mostly unearthed. Our first chapter studies Hilbert's early years and shows that there are more resources for the pursuit of Hilbert's Program than the tradition typically recognizes. We then turn to the structure and limits of finitist reasoning, where it is argued that by progressing through the recursive functions systematically, finitists are able to reason about higher recursions and that such reasoning provides a platform that might make the execution of Hilbert's Program's consistency proof possible, albeit for weaker formal systems than originally envisaged. In the final chapter we study some possibilities with respect to the consistency problem for finitists and show that there are some ways in which the question is left open. Hence, the goal of our dissertation is to excavate the old stones and to provide some insight into the question of limits for finitists and their associated consistency problem.

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2014 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Angelides, Alexei E |
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Associated with | Stanford University, Department of Philosophy. |

Primary advisor | Lawlor, Krista |

Primary advisor | Ryckman, Thomas |

Thesis advisor | Lawlor, Krista |

Thesis advisor | Ryckman, Thomas |

Thesis advisor | Burgess, Alexis, 1980- |

Thesis advisor | Sieg, Wilfried, 1945- |

Advisor | Burgess, Alexis, 1980- |

Advisor | Sieg, Wilfried, 1945- |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Alexei E. Angelides. |
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Note | Submitted to the Department of Philosophy. |

Thesis | Thesis (Ph.D.)--Stanford University, 2014. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2014 by Alexei E Angelides
- License
- This work is licensed under a Creative Commons Attribution 3.0 Unported license (CC BY).

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