DS-1994-07:
Shavrukov, V. Yu.
(1994)
*Adventures in Diagonalizable Algebras.*
Doctoral thesis, University of Amsterdam.

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## Abstract

The topic of this dissertation is diagonalizable algebra's, in

particular those of formal theories such as Peano arithmetic.

Part one discusses algebras that can be embedded in a diagonalizable

algebra of a given formal theory. In the case of embeddings with a

recursively enumerable range, the set of such algebras is

characterized by a small number of simple conditions of an algebraic

and recursive nature. In the general case, without restrictions on the

range of the embedding, the same holds for theories that are not

$\Sigma_1$-correct. In this case, we use Solovay--functions travelling

through Kripke models changing with the passage of time.

Our constructions are supported by argumentation stemming from modal logic.

Amongst others, we prove a uniform variant of the interpolation theorem

for the modal logic {\bf L}.

Part two is concerned with differences between diagonalizable algebras

of Peano arithmetic and Zermelo--Fraenkel set theory. These algebras

turn out not to be isomorphic. The proof for this relies on estimates

of the length of proofs of specific sentences in these theories.

In part three we look at the first-order theories of diagonalizable

algebras. Methods from interpretability logic are applied in the

construction of an interpretation of the true first-order

arithmetic in the first order theory of diagonalizable algebras of

$\Sigma_1$-correct formal theories (with the use of results from part

one). The interpretation shows that these diagonalizable algebras have

an undecidable, even non-arithmetical first-order theory.

Furthermore, an alternative approach is developed, that makes use of

Post canonical systems, which enables proof of the undecidability of

diagonalizable algebras of a broader class of theories.

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-1994-07 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 1994 |

Subjects: | Logic |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/1970 |

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