MoL-2021-24:
Osinski, Jonathan
(2021)
*Symbiosis and Compactness Properties.*
[Pre-print]

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## Abstract

We investigate connections between model-theoretic properties of extensions of first-order logic and set-theoretic principles. We build on work of Bagaria and Väänänen, and of Galeotti, Khomskii and Väänänen, which used the notions of symbiosis and bounded symbiosis between a logic L and a predicate of set theory R, respectively, to show that if L and R are (boundedly) symbiotic, (upwards) Löwenheim-Skolem properties of L are equivalent to certain (upwards) reflection principles involving R. Similarly, we consider whether under the assumption of symbiosis compactness properties of L are related to some set-theoretic principle involving R.

For this purpose, we give a thorough introduction to symbiosis and the concepts from abstract model theory and set theory needed in its study. We further give a proof of a characterization of compactness properties of L in terms of extensions of specific partial orders stated by Väänänen. We use this and the novel concept of (R, κ)-extensions to formulate a set-theoretic principle which describes that in classes which are definable under the usage of R there exist (R, κ)-extensions with upper bounds for such partial orders. We show that this principle is related to compactness properties of a logic L symbiotic to R.

Item Type: | Pre-print |
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Report Nr: | MoL-2021-24 |

Series Name: | Master of Logic Thesis (MoL) Series |

Year: | 2021 |

Subjects: | Logic Mathematics |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 25 Oct 2021 13:13 |

Last Modified: | 25 Oct 2021 13:14 |

URI: | https://eprints.illc.uva.nl/id/eprint/1818 |

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