30 citations · 86 across the 24 of their papers we have counts for
24 papers
On the computational properties of ambivalent sets and functions
Dag Normann, Sam Sanders
Examples of discontinuous functions already appear in the work of Euler, Abel, Dirichlet, Fourier, and Bolzano. A ground-breaking discovery due to Baire was that many discontinuous…
The uncountability of the reals and the Axiom of Choice
Dag Normann, Sam Sanders
The uncountability of the reals was first established by Cantor in what was later heralded as the first paper on set theory. Since the latter constitutes the official foundations o…
On two recent extensions of the Big Five of Reverse Mathematics
Dag Normann, Sam Sanders
The program Reverse Mathematics in the foundations of mathematics seeks to identify the minimal axioms required to prove theorems of ordinary mathematics. One always assumes the ba…
On sequential theorems in Reverse Mathematics
Dag Normann, Sam Sanders
Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then…
On some computational properties of open sets
Dag Normann, Sam Sanders
Open sets are central to mathematics, especially analysis and topology, in ways few notions are. In most, if not all, computational approaches to mathematics, open sets are only st…
The Biggest Five of Reverse Mathematics
Dag Normann, Sam Sanders
The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent…
On the computational properties of basic mathematical notions
Dag Normann, Sam Sanders
We investigate the computational properties of basic mathematical notions pertaining to -functions and subsets of , like finiteness, c…
Betwixt Turing and Kleene
Dag Normann, Sam Sanders
Turing's famous 'machine' model constitutes the first intuitively convincing framework for computing with real numbers. Kleene's computation schemes S1-S9 extend Turing's approach…
On robust theorems due to Bolzano, Weierstrass, Cantor, and Jordan
Dag Normann, Sam Sanders
Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e…
On the uncountability of
Dag Normann, Sam Sanders
Cantor's first set theory paper (1874) establishes the uncountability of . We study this most basic mathematical fact formulated in the language of higher-order arithme…
Computability and Non-monotone induction
Dag Normann
Non-monotone inductive definitions were studied in the late 1960's and early 1970's with the aim of understanding connections between the complexity of the formulas defining the in…
The Axiom of Choice in Computability Theory and Reverse Mathematics, with a cameo for the Continuum Hypothesis
Dag Normann, Sam Sanders
The Axiom of Choice (AC for short) is the most (in)famous axiom of the usual foundations of mathematics, ZFC set theory. The (non-)essential use of AC in mathematics has been well-…
Open sets in computability theory and Reverse Mathematics
Dag Normann, Sam Sanders
To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reaso…
On the logical and computational properties of the Vitali covering theorem
Dag Normann, Sam Sanders
We study a version of the Vitali covering theorem, which we call and which is a direct weakening of the Heine-Borel theorem for uncountable coverings, called $\text…
Measure-theoretic Uniformity and the Suslin Functional
Dag Normann
We generalise results by Sacks and Tanaka concerning measure-theoretic uniformity for hyperarithmetical sets and a basis theorem for -sets of positive measure to computabili…
Pincherle's theorem in Reverse Mathematics and computability theory
Dag Normann, Sam Sanders
We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a loc…
The strength of compactness in Computability Theory and Nonstandard Analysis
Dag Normann, Sam Sanders
Compactness is one of the core notions of analysis: it connects local properties to global ones and makes limits well-behaved. We study the computational properties of the compactn…
On the mathematical and foundational significance of the uncountable
Dag Normann, Sam Sanders
We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, th…
Computability Theory, Nonstandard Analysis, and their connections
Dag Normann, Sam Sanders
We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately relat…
The sequential functionals of type form a dcpo for all
Dag Normann
We prove that the sequential functionals of some fixed types at type level 2, taking finite sequences of unary functions as arguments, do form a directed complete partial ordering.…
The extensional realizability model of continuous functionals and three weakly non-constructive classical theorems
Dag Normann
We investigate wether three statements in analysis, that can be proved classically, are realizable in the realizability model of extensional continuous functionals induced by Kleen…
Banach Spaces as Data Types
Dag Normann
We introduce the operators "modified limit" and "accumulation" on a Banach space, and we use this to define what we mean by being internally computable over the space. We prove tha…
A rich hierarchy of functionals of finite types
Dag Normann
We are considering typed hierarchies of total, continuous functionals using complete, separable metric spaces at the base types. We pay special attention to the so called Urysohn s…
Comparing hierarchies of total functionals
Dag Normann
In this paper we consider two hierarchies of hereditarily total and continuous functionals over the reals based on one extensional and one intensional representation of real number…