PP-2012-07:
Väänänen, Jouko and Wang, Tong
(2012)
*Internal Categoricity in Arithmetic and Set Theory.*
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## Abstract

Second order logic was originally considered as an innocuous variant

of first order logic in the works of Hilbert. Later study reveals that

the analogy with first order logic does not do full justice to second

order logic. Quine famously referred to second order logic as "set

theory in disguise". Second order logic truly transcends first order

logic in terms of strength, and is more appropriate to be compared to

(first order) set theory. In second order logic, a large part of set

theory becomes essentially logical truth. There is the debate between

the "set theory view" and the "second order view" in the foundation of

mathematics . The set theory view holds that mathematics is best

formalized using first order set theory. The second order view holds

that mathematics is best formalized in second order logic.

Two important issues in this debate are completeness and

categoricity. It is usually conceived that one merit of the set theory

view is that first order logic has a complete proof calculus, while

second order logic has not. One merit of the second order view is that

second order theories of classical structures (e.g. N, R) are

categorical, while first order theories allow for non-standard models.

The aim of this paper is to synthesize completeness and categoricity

in the second order, while working within the framework of normal

second order logic instead of full second order logic.

Item Type: | Report |
---|---|

Report Nr: | PP-2012-07 |

Series Name: | Prepublication (PP) Series |

Year: | 2012 |

Uncontrolled Keywords: | second order logic; arithmetic; categoricity; set theory |

Subjects: | Logic |

Depositing User: | Jouko |

Date Deposited: | 12 Oct 2016 14:37 |

Last Modified: | 12 Oct 2016 14:37 |

URI: | https://eprints.illc.uva.nl/id/eprint/449 |

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