353 research outputs found
Well-orders in the transfinite Japaridze algebra
This paper studies the transfinite propositional provability logics
\glp_\Lambda and their corresponding algebras. These logics have for each
ordinal a modality \la \alpha \ra. We will focus on the closed
fragment of \glp_\Lambda (i.e., where no propositional variables occur) and
\emph{worms} therein. Worms are iterated consistency expressions of the form
\la \xi_n\ra \ldots \la \xi_1 \ra \top. Beklemishev has defined
well-orderings on worms whose modalities are all at least and
presented a calculus to compute the respective order-types.
In the current paper we present a generalization of the original
orderings and provide a calculus for the corresponding generalized order-types
. Our calculus is based on so-called {\em hyperations} which are
transfinite iterations of normal functions.
Finally, we give two different characterizations of those sequences of
ordinals which are of the form \la {\formerOmega}_\xi (A) \ra_{\xi \in \ord}
for some worm . One of these characterizations is in terms of a second kind
of transfinite iteration called {\em cohyperation.}Comment: Corrected a minor but confusing omission in the relation between
Veblen progressions and hyperation
The intuitionistic temporal logic of dynamical systems
A dynamical system is a pair , where is a topological space and
is continuous. Kremer observed that the language of
propositional linear temporal logic can be interpreted over the class of
dynamical systems, giving rise to a natural intuitionistic temporal logic. We
introduce a variant of Kremer's logic, which we denote , and show
that it is decidable. We also show that minimality and Poincar\'e recurrence
are both expressible in the language of , thus providing a
decidable logic expressive enough to reason about non-trivial asymptotic
behavior in dynamical systems
Models of transfinite provability logic
For any ordinal \Lambda, we can define a polymodal logic GLP(\Lambda), with a
modality [\xi] for each \xi<\Lambda. These represent provability predicates of
increasing strength. Although GLP(\Lambda) has no Kripke models, Ignatiev
showed that indeed one can construct a Kripke model of the variable-free
fragment with natural number modalities. Later, Icard defined a topological
model for the same fragment which is very closely related to Ignatiev's.
In this paper we show how to extend these constructions for arbitrary
\Lambda. More generally, for each \Theta,\Lambda we build a Kripke model
I(\Theta,\Lambda) and a topological model T(\Theta,\Lambda), and show that the
closed fragment of GLP(\Lambda) is sound for both of these structures, as well
as complete, provided \Theta is large enough
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