17 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
Hyperations, Veblen progressions and transfinite iterations of ordinal functions
In this paper we introduce hyperations and cohyperations, which are forms of
transfinite iteration of ordinal functions.
Hyperations are iterations of normal functions. Unlike iteration by pointwise
convergence, hyperation preserves normality. The hyperation of a normal
function f is a sequence of normal functions so that f^0= id, f^1 = f and for
all ordinals \alpha, \beta we have that f^(\alpha + \beta) = f^\alpha f^\beta.
These conditions do not determine f^\alpha uniquely; in addition, we require
that the functions be minimal in an appropriate sense. We study hyperations
systematically and show that they are a natural refinement of Veblen
progressions.
Next, we define cohyperations, very similar to hyperations except that they
are left-additive: given \alpha, \beta, f^(\alpha + \beta)= f^\beta f^\alpha.
Cohyperations iterate initial functions which are functions that map initial
segments to initial segments. We systematically study cohyperations and see how
they can be employed to define left inverses to hyperations.
Hyperations provide an alternative presentation of Veblen progressions and
can be useful where a more fine-grained analysis of such sequences is called
for. They are very amenable to algebraic manipulation and hence are convenient
to work with. Cohyperations, meanwhile, give a novel way to describe slowly
increasing functions as often appear, for example, in proof theory
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
Turing-Taylor expansions for arithmetic theories
Turing progressions have been often used to measure the proof-theoretic
strength of mathematical theories. Turing progressions based on -provability
give rise to a proof-theoretic ordinal. As such, to each theory
we can assign the sequence of corresponding ordinals . We call this sequence a \emph{Turing-Taylor expansion} of
a theory.
In this paper, we relate Turing-Taylor expansions of sub-theories of Peano
Arithmetic to Ignatiev's universal model for the closed fragment of the
polymodal provability logic . In particular, in this
first draft we observe that each point in the Ignatiev model can be seen as
Turing-Taylor expansions of formal mathematical theories.
Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor
expression will define a unique point in Ignatiev's model.Comment: First draf