Notre Dame Journal of Formal Logic 56 (1):167-190 (2015)

Jouko A Vaananen
University of Helsinki
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
Keywords Boolean-valued second-order logic   full second-order logic   $\Omega$-logic   Boolean validity
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DOI 10.1215/00294527-2835065
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References found in this work BETA

Completeness in the Theory of Types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
Second-Order Logic and Foundations of Mathematics.Jouko Väänänen - 2001 - Bulletin of Symbolic Logic 7 (4):504-520.
Suitable Extender Models I.W. Hugh Woodin - 2010 - Journal of Mathematical Logic 10 (1):101-339.
Second Order Logic or Set Theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.
The Self-Iterability of L[E].Ralf Schindler & John Steel - 2009 - Journal of Symbolic Logic 74 (3):751-779.

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Second‐Order Logic and Set Theory.Jouko Väänänen - 2015 - Philosophy Compass 10 (7):463-478.

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