Daisuke Ikegami: Omega-logic and Boolean valued second order logic

Wednesday, September 12, from 15:30 to 17:00
Seminar Room, Department of Logic and History and Philosophy of Science, UB, Montalegre 6, 4th floor, Barcelona.

Speaker: Daisuke Ikegami (UC Berkeley).
Title: Omega-logic and Boolean valued second order logic

Abstract:
Woodin’s Omega-logic is a logic on generic absoluteness and Woodin’s Omega-conjecture states that all the possible results on generic absoluteness must be explained by looking at certain good sets of reals called “universally Baire sets”. The Omega-conjecture explains the phenomena of generic absoluteness obtained by large cardinals in the last decades.

Second order logic has two standard semantics, one is full semantics where one interprets second order quantifiers over all the subsets of a universe and the other is Henkin semantics where a second order structure satisfies full comprehension. Second order logic with full semantics can express many complicated things but does not enjoy good logical properties (e.g. completeness) while the one with Henkin semantics is very weak and enjoys basic logical properties as first order logic does.

We introduce the Boolean valued semantics for second order logic and investigate Boolean valued second order logic. We will compare this logic with Omega-logic and second order logic with full semantics in terms of basic logical properties such as validity, completeness, and compactness.

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