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 $\xi< \Lambda$ 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 $<_\xi$ on worms whose modalities are all at least $\xi$ and presented a calculus to compute the respective order-types. In the current paper we present a generalization of the original $<_\xi$ orderings and provide a calculus for the corresponding generalized order-types $o_\xi$. 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 $A$. 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

Polimodalna logika dokazivosti GLP

U ovom diplomskom radu glavni cilj je dokazati topološku potpunost zatvorenog fragmenta polimodalne logike GLP. Definiramo logiku dokazivosti GL i takozvanu Kripkeovu semantiku za modalne logike koja uspostavlja vezu između relacijskih struktura i modalnog jezika. Ujedno definiramo aritmetičku interpretaciji logike dokazivosti te izričemo Solovayev prvi teorem aritmetičke potpunosti čiji se dokaz temelji na Kripkeovoj semantici za GL. Definiramo modalni sistem s beskonačno mnogo modalnih operatora, GLP, te dokazujemo nepotpunosti sistema GLP u odnosu na vlastitu Kripkeovu semantiku. Zatvoreni fragment sistema GLP, koji označavamo sa GLP$_0$, ima odgovarajuću Kripkeovu semantiku u obliku takozvanog Ignatievog univerzalnog okvira. Taj okvir onda proširujemo tako da bude izomorfan kanonskom okviru i taj rezultat iskorištavamo kako bi definirali jednostavan topološki model nad ordinalnim brojem $\epsilon_0$, na kojem je promatrana logika potpuna.The main goal of this thesis is to show the closed fragment of polymodal logic GLP is complete with respect to a certain topological model. We define provability logic GL and so-called Kripke semantics for modal logics, which establishes a connection between relational structures and modal language. Also, we define the arithmetical interpretation of logic of provability and state Solovay’s first arithmetical completeness theorem, whose proof relies on Kripke semantics for GL. We define a model system with infinitely many modalities, GLP, and show that it is incomplete with respect to its class of Kripke frames. Closed fragment of system GLP, which we denote by GLP$_0$, has appropriate semantics in the form of Ignatiev’s universal frame. That frame is then expanded to one that is isomorphic to the canonical frame and we use that result to define a simple topological model on the ordinal $\epsilon_0$ for which we have completeness result

Polimodalna logika dokazivosti GLP

U ovom diplomskom radu glavni cilj je dokazati topološku potpunost zatvorenog fragmenta polimodalne logike GLP. Definiramo logiku dokazivosti GL i takozvanu Kripkeovu semantiku za modalne logike koja uspostavlja vezu između relacijskih struktura i modalnog jezika. Ujedno definiramo aritmetičku interpretaciji logike dokazivosti te izričemo Solovayev prvi teorem aritmetičke potpunosti čiji se dokaz temelji na Kripkeovoj semantici za GL. Definiramo modalni sistem s beskonačno mnogo modalnih operatora, GLP, te dokazujemo nepotpunosti sistema GLP u odnosu na vlastitu Kripkeovu semantiku. Zatvoreni fragment sistema GLP, koji označavamo sa GLP$_0$, ima odgovarajuću Kripkeovu semantiku u obliku takozvanog Ignatievog univerzalnog okvira. Taj okvir onda proširujemo tako da bude izomorfan kanonskom okviru i taj rezultat iskorištavamo kako bi definirali jednostavan topološki model nad ordinalnim brojem $\epsilon_0$, na kojem je promatrana logika potpuna.The main goal of this thesis is to show the closed fragment of polymodal logic GLP is complete with respect to a certain topological model. We define provability logic GL and so-called Kripke semantics for modal logics, which establishes a connection between relational structures and modal language. Also, we define the arithmetical interpretation of logic of provability and state Solovay’s first arithmetical completeness theorem, whose proof relies on Kripke semantics for GL. We define a model system with infinitely many modalities, GLP, and show that it is incomplete with respect to its class of Kripke frames. Closed fragment of system GLP, which we denote by GLP$_0$, has appropriate semantics in the form of Ignatiev’s universal frame. That frame is then expanded to one that is isomorphic to the canonical frame and we use that result to define a simple topological model on the ordinal $\epsilon_0$ for which we have completeness result