78 research outputs found
First-order limits, an analytical perspective
In this paper we present a novel approach to graph (and structural) limits
based on model theory and analysis. The role of Stone and Gelfand dualities is
displayed prominently and leads to a general theory, which we believe is
naturally emerging. This approach covers all the particular examples of
structural convergence and it put the whole in new context. As an application,
it leads to new intermediate examples of structural convergence and to a "grand
conjecture" dealing with sparse graphs. We survey the recent developments
Evitable iterates of the consistency operator
Let's fix a reasonable subsystem of arithmetic; why are natural
extensions of pre-well-ordered by consistency strength? In previous work,
an approach to this question was proposed. The goal of this work was to
classify the recursive functions that are monotone with respect to the
Lindenabum algebra of . According to an optimistic conjecture, roughly,
every such function must be equivalent to an iterate of
the consistency operator in the limit.
In previous work the author established the first case of this optimistic
conjecture; roughly, every recursive monotone function is either as weak as the
identity operator in the limit or as strong as in the limit.
Yet in this note we prove that this optimistic conjecture fails already at the
next step; there are recursive monotone functions that are neither as weak as
in the limit nor as strong as in the limit.
In fact, for every , we produce a function that is cofinally as strong
as yet cofinally as weak as
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