How incomputable is the separable Hahn-Banach theorem?

Hahn-Banach定理可计算性再探讨

Computability of the Hahn-Banach Theorem Revisited

本文研究了Hahn-Banach定理的计算性质，证明其在Banach空间ℓ¹中已达到最大复杂性，并将其与中间值定理及弱Kőnig引理等数学原理建立等价关系。

Multiple Choices imply the Ingleton and Krein-Milman axioms

In set theory without the Axiom of Choice, we consider Ingleton's axiom which is the counterpart in ultrametric analysis of the Hahn-Banach axiom. We show that in $ZFA$, set theory without the Axiom of Choice weakened to allow "atoms", Ingleton's axiom does not imply the Axiom of Choice (this solves in $ZFA$ a question raised by van Rooij (1992). We also prove that in $ZFA$, the "multiple Choice" axiom implies the Krein-Milman axiom. We deduce that, in $ZFA$, the conjunction of the Hahn-Banach, Ingleton and Krein-Milman axioms does not imply the Axiom of Choice.

The algebra of conditional sets and the concepts of conditional topology and compactness

The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel-Lebesgue theorem, the Hahn-Banach theorem, the Banach-Alaoglu theorem an…

Effective Choice and Boundedness Principles in Computable Analysis

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles which are cornerstones among Weihr…

构造分析中的McShane-Whitney扩张

McShane-Whitney extensions in constructive analysis

本文在Bishop构造数学中研究McShane-Whitney扩张定理，证明Lipschitz函数在完全有界空间上的均匀稠密性，并引入McShane-Whitney对描述扩张的构造内容，同时推广到Hölder和ν-连续函数。

Weihrauch Degrees, Omniscience Principles and Weak Computability

In this paper we study Weihrauch reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice with the disjoint union of multi-valued functions as greatest lower bound operation. We prove that parallelization is a closure operator for this semi-lattice and the parallelized Weihrauch degrees even form a lattice with the product of multi-valued functions as greatest lower bound operation. We show that the Medvedev lattice and hence Turing degrees can be …