19 citations · 101 across the 35 of their papers we have counts for
40 papers
Fraïssé's conjecture, partial impredicativity and well-ordering principles, part II
Anton Freund, Katarzyna W. Kowalik, Davide Manca
We exhibit a well-ordering principle that is equivalent to a theory of partial impredicativity. The latter goes back to Towsner and relates to recent work of Suzuki and Yokoyama. T…
Induction on Dilators and Bachmann-Howard Fixed Points
Juan P. Aguilera, Anton Freund, Andreas Weiermann
One of the most important principles of J.-Y. Girard's -logic is induction on dilators. In particular, Girard used this principle to construct his famous functor . He cla…
Fraïssé's conjecture, partial impredicativity and well-ordering principles, part I
Anton Freund
Fraïssé's conjecture (proved by Laver) is implied by the -comprehension axiom of reverse mathematics, as shown by Montalbán. The implication must be strict for reasons of qu…
Dilators and the reverse mathematics zoo
Anton Freund
A predilator is a particularly uniform transformation of linear orders. We have a dilator when the transformation preserves well-foundedness. Over the theory from…
An Introduction to Mathematical Logic
Anton Freund
This introduction begins with a section on fundamental notions of mathematical logic, including propositional logic, predicate or first-order logic, completeness, compactness, the…
Provable better quasi orders
Anton Freund, Alberto Marcone, Fedor Pakhomov +1
It has recently been shown that fairly strong axiom systems such as cannot prove that the antichain with three elements is a better quasi order (). I…
Weak well orders and Fraïssé's conjecture
Anton Freund, Davide Manca
The notion of well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical st…
The logical strength of minimal bad arrays
Anton Freund, Fedor Pakhomov, Giovanni Soldà
This paper studies logical aspects of the notion of better quasi order, which has been introduced by C. Nash-Williams (Mathematical Proceedings of the Cambridge Philosophical Socie…
Normal functions and maximal order types
Anton Freund, Davide Manca
Transformations of well partial orders induce functions on the ordinals, via the notion of maximal order type. In most examples from the literature, these functions are not normal,…
The uniform Kruskal theorem: between finite combinatorics and strong set existence
Anton Freund, Patrick Uftring
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 -c…
On the logical strength of the better quasi order with three elements
Anton Freund
The notion of better quasi order (), due to Nash-Williams, is very fruitful mathematically and intriguing from the standpoint of logic, due to several long-standing o…
Higman's lemma is stronger for better quasi orders
Anton Freund
We prove that Higman's lemma is strictly stronger for better quasi orders than for well quasi orders, within the framework of reverse mathematics. In fact, we show a stronger resul…
Impredicativity and Trees with Gap Condition: A Second Course on Ordinal Analysis
Anton Freund
These lecture notes introduce central notions of impredicative ordinal analysis, such as the Bachmann-Howard ordinal and the method of collapsing, which transforms uncountable proo…
R.E. Bruck, proof mining and a rate of asymptotic regularity for ergodic averages in Banach spaces
Anton Freund, Ulrich Kohlenbach
We analyze a proof of Bruck to obtain an explicit rate of asymptotic regularity for Cesàro means in uniformly convex Banach spaces. Our rate will only depend on a norm bound and a…
Well ordering principles for iterated -comprehension
Anton Freund, Michael Rathjen
We introduce ordinal collapsing principles that are inspired by proof theory but have a set theoretic flavor. These principles are shown to be equivalent to iterated -compre…
Reverse mathematics of a uniform Kruskal-Friedman theorem
Anton Freund
The Kruskal-Friedman theorem asserts: in any infinite sequence of finite trees with ordinal labels, some tree can be embedded into a later one, by an embedding that respects a cert…
Unprovability in Mathematics: A First Course on Ordinal Analysis
Anton Freund
These are the lecture notes of an introductory course on ordinal analysis. Our selection of topics is guided by the aim to give a complete and direct proof of a mathematical indepe…
Bounds for a nonlinear ergodic theorem for Banach spaces
Anton Freund, Ulrich Kohlenbach
We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong con…
Bachmann-Howard Derivatives
Anton Freund
It is generally accepted that H. Friedman's gap condition is closely related to iterated collapsing functions from ordinal analysis. But what precisely is the connection? We offer…
Patterns of resemblance and Bachmann-Howard fixed points
Anton Freund
Timothy Carlson's patterns of resemblance employ the notion of -elementarity to describe large computable ordinals. It has been conjectured that a relativization of these patt…
Ackermann and Goodstein go functorial
Juan P. Aguilera, Anton Freund, Michael Rathjen +1
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…
Well ordering principles and -statements: a pilot study
Anton Freund
In previous work, the author has shown that -induction along is equivalent to a suitable formalization of the statement that every normal function on the ordinal…
What is effective transfinite recursion in reverse mathematics?
Anton Freund
In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders,…
A mathematical commitment without computational strength
Anton Freund
We present a new manifestation of Gödel's second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert's program. Specifically, we…