Related papers: Fundamental sequences based on localization
We prove that Buchholz's system of fundamental sequences for the $\vartheta$ function enjoys various regularity conditions, including the Bachmann property. We partially extend these results to variants of the $\vartheta$ function,…
In the late 1980s, Abrusci, Girard and van de Wiele defined a variant of Goodstein sequences: the so-called inverse Goodstein sequence. In their work, they show that it terminates precisely at the Bachmann-Howard ordinal. This reveals that…
We generalize Goodstein's theorem (Goodstein 1944) and Cichon's independence proof (Cichon 1983) to $\Pi^1_1-\mathrm{CA}_0$ using results from (Wilken 2026). The method is generalizable to stronger notation systems that provide unique terms…
Caucal hierarchy is a well-known class of graphs with decidable monadic theories. It were proved by L. Braud and A. Carayol that well-orderings in the hierarchy are the well-orderings with order types less than $\varepsilon_0$. Naturally,…
The aim of this article is to refine a weak invariance principle for stationary sequences given by Doukhan & Louhichi (1999). Since our conditions are not causal our assumptions need to be stronger than the mixing and causal $\theta$-weak…
This paper serves to define an extension, which we call dimensional Veblen, of Oswald Veblen's system of ordinal functions below the large Veblen ordinal. This is facilitated by iterating derivatives of ordinal functions along…
We present variants of Goodstein's theorem that are equivalent to arithmetical comprehension and to arithmetical transfinite recursion, respectively, over a weak base theory. These variants differ from the usual Goodstein theorem in that…
In arXiv:2508.14768, a variant of Goodstein's original process was recently introduced which, given a set $B\subseteq \mathbb{N}$ of bases, writes each $n\in\mathbb{N}$ in $B$-normal form, namely $n=b^ea+r$, where $b\in B$ the greatest base…
Goodstein's principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to grow very quickly, but eventually decrease to zero.…
Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence $\mathbb{M}=(M_p)_{p\in\mathbb{N}_0}$, have been put forward by A. Lastra, S. Malek and the second author (Summability in general…
Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence $\mathbb{M}=(M_p)_{p\in\mathbb{N}_0}$, have been put forward by A. Lastra, S. Malek and the second author [1], and their validity…
Inspired by Leivant's work on absolute predicativism, Bellantoni and Cook in 1992 introduced a structurally restricted form of recursion called predicative recursion. Using this recursion scheme on the inductive structures of natural…
Timothy Carlson's patterns of resemblance employ the notion of $\Sigma_1$-elementarity to describe large computable ordinals. It has been conjectured that a relativization of these patterns to dilators leads to an equivalence with…
The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in \cite{bc}) that has been used to define several new concepts…
Goodstein sequences are numerical sequences in which a natural number m, expressed as the complete normal form to a given base a, is modified by increasing the value of the base a by one unit and subtracting one unit from the resulting…
In this paper, we propose a new framework to study the generalization property of classifier chains trained over observations associated with multiple and interdependent class labels. The results are based on large deviation inequalities…
These lectures present some basic ideas and techniques in the spectral analysis of lattice Schrodinger operators with disordered potentials. In contrast to the classical Anderson tight binding model, the randomness is also allowed to…
Ordinal time series analysis is based on the idea to map time series to ordinal patterns, i.e., order relations between the values of a time series and not the values themselves, as introduced in 2002 by C. Bandt and B. Pompe. Despite a…
Starting from Wigner's theory of elementary systems and following a recent approach of Schroer we define certain subspaces of localized wave functions in the underlying Hilbert space with the help of the theory of modular von-Neumann…
Using the relation established by Johnson--Zumbrun between Hill's method of aproximating spectra of periodic-coefficient ordinary differential operators and a generalized periodic Evans function given by the $2$-modified characteristic…