Notre Dame Journal of Formal Logic

Boolean-Valued Second-Order Logic

Daisuke Ikegami and Jouko Väänänen

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Abstract

In so-called full second-order logic, the second-order variables range over all subsets and relations of the domain in question. In so-called Henkin second-order logic, every model is endowed with a set of subsets and relations which will serve as the range of the second-order variables. In our Boolean-valued second-order logic, the second-order variables range over all Boolean-valued subsets and relations on the domain. We show that under large cardinal assumptions Boolean-valued second-order logic is more robust than full second-order logic. Its validity is absolute under forcing, and its Hanf and Löwenheim numbers are smaller than those of full second-order logic.

Article information

Source
Notre Dame J. Formal Logic, Volume 56, Number 1 (2015), 167-190.

Dates
First available in Project Euclid: 24 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1427202979

Digital Object Identifier
doi:10.1215/00294527-2835065

Mathematical Reviews number (MathSciNet)
MR3326594

Zentralblatt MATH identifier
1372.03090

Subjects
Primary: 03-06: Proceedings, conferences, collections, etc.
Secondary: 03C95: Abstract model theory 03E40: Other aspects of forcing and Boolean-valued models 03E57: Generic absoluteness and forcing axioms [See also 03E50]

Keywords
Boolean-valued second-order logic full second-order logic $\Omega$-logic Boolean validity

Citation

Ikegami, Daisuke; Väänänen, Jouko. Boolean-Valued Second-Order Logic. Notre Dame J. Formal Logic 56 (2015), no. 1, 167--190. doi:10.1215/00294527-2835065. https://projecteuclid.org/euclid.ndjfl/1427202979


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