Abhijit Dasgupta

Mathematical Foundations of Randomness

Publisher Summary This chapter introduces the definitions of the notion of a random sequence using the three main ideas described that have dominated algorithmic randomness. There are three key approaches for defining randomness in sequences: unpredictability, typicality, and incompressibility. A notion is set up notation for strings and sequences, the Cantor Space, which forms the basic framework for carrying out further discussions, and contains a brief and informal introduction to Lebesgue measure on the unit interval and the Cantor space, which one treats as equivalent. Discussion of mathematical randomness defines a series of classical stochastic or frequency properties in an effort to distill out randomness. This chapter further forms the core material on algorithmic randomness: Von Mises randomness, Martin-Lof randomness, Kolmogorov complexity and randomness of finite strings, application to Godel incompleteness, and Schnorrs theorem. It forms an overview of other forms of related randomness notions that have been studied and indicates the current state of affairs.

The Dedekind–Peano Axioms

This chapter develops the theory of natural numbers based on Dedekind–Peano Axioms, also known as Peano Arithmetic Peano Arithmetic. The basic theory of ratios (positive rational numbers) is also developed. It concludes with a section on formal definition by primitive recursion.

Cantor–Bendixson Analysis of Countable Closed Sets

We devote this chapter to the Cantor, G. Bendixson, I. O.Cantor–Bendixson analysis of countable closed sets. We first prove the effective Cantor–Bendixson theorem which decomposes a closed set into an effectively countable set and a perfect set. We then obtain a full topological classification for the class of countable closed bounded subsets of R: The Cantor–Bendixson rank is shown to be a complete invariant for the relation of homeomorphism between these sets, and the countable ordinals ω ν n + 1 (ν < ω 1, n ∈ N) are shown to form an exhaustive enumeration, up to homeomorphism, of the countable closed bounded sets into ℵ 1 many pairwise non-homeomorphic representative sets.

Cardinal Arithmetic and the Cantor Set

We continue the basic theory of cardinals, covering the Cantor–Bernstein Theorem, arbitrary cardinal products and cardinal arithmetic, binary trees and the construction of the Cantor set, the identity \({2}^{\aleph _{0}} =\boldsymbol{ \mathfrak{c}}\) and effective bijections between familiar sets of cardinality \(\boldsymbol{\mathfrak{c}}\), Cantor’s theorem and Konig’s inequality, and the behavior of \({\kappa }^{\aleph _{0}}\) for various cardinals κ.

Paradoxes and Resolutions

Unless carefully restricted, the informal naive set theory that we have so far been using can produce certain contradictions, known as set theoretic paradoxes. These contradictions generally result from consideration of certain very large sets whose existence can be derived from the unrestricted comprehension principle. This chapter discusses three such classical paradoxes due to Burali-Forti, Cantor, G.Cantor, and Russell, B.Russell, which showed the untenability of naive set theory and the need for more careful formalizations. The two earliest responses to the paradoxes, namely Russell’s theory of types and Zermelo, E.Zermelo’s axiomatization of set theory, are discussed.

Preliminaries: Sets, Relations, and Functions

This preliminary chapter informally reviews the prerequisite material for the rest of the book. Here we set up our notational conventions, introduce basic set-theoretic notions including the power set, ordered pairs, Cartesian product, relations, functions, and their properties, sequences, strings and words, indexed and unindexed families, partitions and equivalence relations, and the basic definition of linear order. Much of the material of this chapter can be found in introductory discrete mathematics texts.

Postscript I: What Exactly Are the Natural Numbers?

This postscript to Part I consists of philosophical and historical remarks concerning the nature of the natural numbers. It contrasts the absolutism absolutist approach requiring absolute constructions of individual natural numbers such as those given by Frege, G.Frege, Russell, B.Russell, Zermelo, E.Zermelo, and von Neumann, with Dedekind, R.Dedekind’s structuralism structuralist approach in which the natural numbers can be taken as members of any Dedekind–Peano system.

Postscript IV: Landmarks of Modern Set Theory

This part contains brief informal discussions (with proofs and most details omitted) of some of the landmark results of set theory of the past 75 years. Topics discussed are constructibility, forcing and independence results, large cardinal axioms, infinite games and determinacy, projective determinacy, and the status of the Continuum Hypothesis.

Interval Trees and Generalized Cantor Sets

This elementary chapter applies the nested intervals theorem to obtain base expansion of real numbers via trees of uniformly subdivided nested closed intervals, with detailed illustrations for ternary expansions. The construction of the Cantor set is then generalized to Cantor systems (systems of nested intervals indexed by binary trees), to formally introduce generalized Cantor sets.

The Heine–Borel and Baire Category Theorems

This chapter starts with the Heine–Borel theorem and its characterization of complete orders, and then uses Borel’s theorem to give a measure-theoretic proof that \(\mathbf{R}\) is uncountable. Other topics focus on measure and category: Lebesgue measurable sets, Baire category, the perfect set property for \(\mathcal{G}_{\delta }\) sets, the Banach–Mazur game and Baire property, and the Vitali and Bernstein constructions.