1,189 research outputs found
De Jongh's Theorem for Intuitionistic Zermelo-Fraenkel Set Theory
We prove that the propositional logic of intuitionistic set theory IZF is
intuitionistic propositional logic IPC. More generally, we show that IZF has
the de Jongh property with respect to every intermediate logic that is complete
with respect to a class of finite trees. The same results follow for CZF.Comment: 12 page
The modal logic of set-theoretic potentialism and the potentialist maximality principles
We analyze the precise modal commitments of several natural varieties of
set-theoretic potentialism, using tools we develop for a general
model-theoretic account of potentialism, building on those of Hamkins, Leibman
and L\"owe, including the use of buttons, switches, dials and ratchets. Among
the potentialist conceptions we consider are: rank potentialism (true in all
larger ); Grothendieck-Zermelo potentialism (true in all larger
for inaccessible cardinals ); transitive-set potentialism
(true in all larger transitive sets); forcing potentialism (true in all forcing
extensions); countable-transitive-model potentialism (true in all larger
countable transitive models of ZFC); countable-model potentialism (true in all
larger countable models of ZFC); and others. In each case, we identify lower
bounds for the modal validities, which are generally either S4.2 or S4.3, and
an upper bound of S5, proving in each case that these bounds are optimal. The
validity of S5 in a world is a potentialist maximality principle, an
interesting set-theoretic principle of its own. The results can be viewed as
providing an analysis of the modal commitments of the various set-theoretic
multiverse conceptions corresponding to each potentialist account.Comment: 36 pages. Commentary can be made about this article at
http://jdh.hamkins.org/set-theoretic-potentialism. Minor revisions in v2;
further minor revisions in v
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
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