Patrick Uftring
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View article: On inverse Goodstein sequences
On inverse Goodstein sequences Open
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…
View article: The uniform Kruskal theorem over RCA$_0$
The uniform Kruskal theorem over RCA$_0$ Open
Kruskal's theorem famously states that finite trees (ordered using an infima-preserving embeddability relation) form a well partial order. Freund, Rathjen, and Weiermann extended this result to general recursive data types with their unifo…
View article: More conservativity for weak Kőnig's lemma
More conservativity for weak Kőnig's lemma Open
We prove conservativity results for weak Kőnig's lemma that extend the celebrated result of Harrington (for $Π^1_1$-statements) and are somewhat orthogonal to the extension by Simpson, Tanaka and Yamazaki (for statements of the form $\fora…
View article: On inverse Goodstein sequences
On inverse Goodstein sequences Open
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…
View article: Weihrauch degrees without roots
Weihrauch degrees without roots Open
We answer the following question by Arno Pauly: "Is there a square-root operator on the Weihrauch degrees?". In fact, we show that there are uncountably many pairwise incomparable Weihrauch degrees without any roots. We also prove that the…
View article: The uniform Kruskal theorem: between finite combinatorics and strong set existence
The uniform Kruskal theorem: between finite combinatorics and strong set existence Open
The uniform Kruskal theorem extends the original result for trees to general recursive data types. As shown by A. Freund, M. Rathjen and A. Weiermann, it is equivalent to $Π^1_1$-comprehension, over $\mathsf{RCA_0}$ with the chain antichai…
View article: Weak and Strong Versions of Effective Transfinite Recursion
Weak and Strong Versions of Effective Transfinite Recursion Open
Working in the context of reverse mathematics, we give a fine-grained characterization result on the strength of two possible definitions for Effective Transfinite Recursion used in literature. Moreover, we show that $Π^0_2$-induction alon…
View article: THE CHARACTERIZATION OF WEIHRAUCH REDUCIBILITY IN SYSTEMS CONTAINING
THE CHARACTERIZATION OF WEIHRAUCH REDUCIBILITY IN SYSTEMS CONTAINING Open
We characterize Weihrauch reducibility in $ \operatorname {\mathrm {E-PA^{\omega }}} + \operatorname {\mathrm {QF-AC^{0,0}}}$ and all systems containing it by the provability in a linear variant of the same calculus using modifications of …