Ten Ways of Looking at Real Numbers Robert Mayans Department of Math/CSci/Physics Fairleigh Dickinson University Varieties of Mathematical Text • Books – Reference books – Text books – Lecture Notes – Handbooks and encyclopedias Varieties of Mathematical Text • Papers – Research papers – Survey articles – Collected works • Online Resources – MathSciNet – Online databases – E-journals The Mathematics Hypertext Project (MHP) • A Web-based hypertext of mathematics • A design paper describes goals, organization, technology issues, etc. http://jodi.ecs.soton.ac.uk/Articles/v05/i01/Mayans/jodi/ • First release in June, 2005 • This presentation discusses some of the pages on the real numbers Text on Real Numbers • We aim for a “comprehensive introduction" to the real numbers. • Real numbers are everywhere dense in mathematics. • Real numbers have different meanings in different contexts. Text on Real Numbers • What to include on a text on real numbers? – Foundations and construction of real numbers – Characterizations of the real numbers by structure. – Extensions and substructures of the real numbers – Real numbers classified in different ways. – Real numbers are a system related to other systems. #1. Foundations of the Real Number System • How to define real numbers and their basic operations from rationals or integers • Presentation of three methods – Dedekind cuts of rational numbers – Equivalence classes of Cauchy sequences of rational numbers – Base-10 digit sequences #1. Foundations of the Real Numbers • Discussion of foundations from John Conway, "On Numbers and Games" R R+ Q Q+ Z N #2. Real Numbers as a Linear Order • The real numbers form the unique linear order that is: – dense – without endpoints – Dedekind-complete – separable (countable dense subset) #2. Real Numbers as a Linear Order • Suslin Problem: Replace separability with the countable chain condition: – every collection of disjoint nontrivial closed intervals is at most countable. • Does this characterize the real numbers? • A counterexample is called a Suslin line • The existence of Suslin lines are independent of the axioms of ZFC set theory #2. Real Numbers as a Linear Order • The real numbers may be viewed as a space of branches of an infinite tree. • Trees are partial orders whose initial segments {x : X<p} are well-ordered. The branches are the maximal chains in the partial order. • Different infinite trees (Aronzsajn tree, Kurepa tree) give rise to different linear orders (Aronszajn line, Kurepa line). #3. Real Numbers as a Topological Space • A characterization of the usual topology of the real line: • If X is a regular, separable, connected, locally connected space, in which every point is a cut point, then X is homeomorphic to the real line. – A point p in a connected space X is a cutpoint if X\{p} is disconnected. • Another characterization: Replace “regular” with “metrizable”. #3. Real Numbers as a Topological Space • The real numbers form a complete separable metric space, a “Polish space”. Also, it is perfect. • Other examples of perfect Polish spaces: – Cantor space: all sequences of 0’s and 1’s – Baire space: all sequences of natural numbers – Finite/countable products of perfect Polish spaces • Every perfect Polish space is Borel-isomorphic to the real numbers. #3. Real Numbers as a Topological Space • The real line is a one-dimensional topological manifold. • Classification of connected Hausdorff onedimensional manifolds – the real line – the circle – the long line – the open long ray #4. Real Numbers as a Completion of the Rational Numbers • A valuation is a function from a field to the nonnegative real numbers with properties analogous to a norm or absolute value: a 0 iff a 0 a b a b a b C max a , b #4. Real Numbers as a Completion of the Rational Numbers • Two valuations are equivalent if one is a power of the other. • Every valuation is equivalent to one which satisfies: a b a b • Such a valuation defines a metric on a field: the distance between a and b is a b b a #4. Real Numbers as a Completion of the Rational Numbers • Ostrowski’s theorem: The inequivalent valuations on the rational numbers are absolute value, the trivial valuation, and the p-adic valuations for every prime p. #4. Real Numbers as a Completion of the Rational Numbers • The metric completions of the rationals defined by a valuation are: – discrete topology on Q (with the trivial valuation) – R (with absolute value) – Rp (the p-adic reals, with the p-adic valuation). #5. The Real Numbers as a Field • The real numbers form an ordered field. • Subfields of the real numbers: – rational numbers – real algebraic number fields – computable real numbers – constructible real numbers • The algebraic completion of the real numbers is the field of complex numbers #5. The Real Numbers as a Field • The field of real numbers is the prototypical realclosed field: its algebraic closure is a finite extension. • The Artin-Schreier theorem characterizes a realclosed field: – – – – – it has characteristic 0 algebraic closure by adjoining i, where i2 = -1 it has a linear order every positive number has a square root -1 is not a sum of squares #5. The Real Numbers as a Field • Any field is a vector space over a subfield. • The real numbers form a vector space over the rational numbers. • A basis for this vector space is called a Hamel space. #6. The Real Numbers as an Algebra • To what extent can the operations on the reals extend to finite-dimensional algebras over the reals? • Here we list a few results. #6. The Real Numbers as an Algebra • The finite-dimensional associative real division algebras are the real numbers, complex numbers, and the quaternions. (Frobenius) • The finite-dimensional real commutative division algebras with unit are the real numbers and the complex numbers. (Hopf) • The finite-dimensional real division algebras have dimension 1, 2, 4, or 8. (Kervaire, Milnor) #7. The Cardinal of the Real Numbers • Cantor showed that the real numbers are not equinumerous with the integers. • Write c as the cardinal of the set of real numbers, the cardinal of the continuum. • The Continuum Hypothesis: Does c 1 ? #7. The Cardinal of the Real Numbers • The continuum must satisfy: c 0 , cf ( c ) 0 • The second condition guarantees that: c • Not much else restricts the possible values of the continuum. #7. The Cardinal of the Real Numbers • Easton’s theorem: Let be any regular cardinal in the ground model of ZFC with cofinality 0 • Then there is a generic extension which preserves cardinalities, in which c • For example, the continuum could be 1 , 2 , , n , but not . #7. The Cardinal of the Real Numbers • A variety of “cardinal invariants” of the continuum: cardinals between 1 and c . • We give two examples: the bounding number b, and the dominating number d. • Let f, g: N N. We say f dominates g iff f(n)≥g(n) for sufficiently large n. #7. The Cardinal of the Real Numbers • The bounding number b: the minimum number of functions f such that no g dominates every f. • The dominating number d: the minimum number of functions f such that every g is dominated by a function f. • b d c 1 #8. Number Theoretic Classification of Real Numbers • Rational numbers, algebraic numbers, transcendental numbers. • Liouville’s theorem: numbers that can be very well approximated by rationals must be transcendental. • If, for infinitely many n, there is a rational such that p / q q n ,then α is transcendental. #8. Number Theoretic Classification of Real Numbers • Mahler's classification of real numbers – A: algebraic numbers – S, T, U: classes of transcendental numbers • Roughly speaking, it measures how well can a number be approximated by algebraic numbers. • If x, y are algebraically dependent, then x and y belong to the same Mahler class. • Most real numbers are S-numbers by measure, U-numbers by category. #9. The Real Numbers as a FirstOrder Theory • Tarski's decidability theorem: The first-order theory of real-closed fields is decidable. • There is an algorithmic procedure to determine if a first-order sentence about the real numbers in the language of ordered fields is true or false. #9. The Real Numbers as a FirstOrder Theory • Nonstandard real numbers extend the real number system with infinitesimal numbers. • One construction is with an ultrapower of an first-order model of the real numbers, with all possible constants, predicates, and functions. • Every nonstandard real number may be written uniquely as a sum of a standard real number and an infinitesimal. #10. Surreal Numbers • Surreal numbers are a subclass of a class of finitely-move two-person games. • One development: a surreal is an ordinal-length sequence of +’s and –’s. • Surreals are lexicographically ordered by -, (empty), +. • The surreal numbers, as a proper class, form an ordered field. • The real numbers are a subfield of the surreals of order . #10. Surreal Numbers Examples of surreal numbers in order: • --2 •-1 • -+ -1/2 • 0 • +-+ ¾ • ++++ 4 #10. Surreal Numbers Surreals of order : all dyadic fractions Surreals of order : all real numbers all dyadic fractions 1 Many more views of the real numbers • • • • Geometry axioms for the real line Real numbers as infinite continued fractions Numeration schemes for real numbers Alternative foundations: constructivism, intuitionism, nonstandard set theory • Computational approximations to real numbers: floating point numbers, interval arithmetic, and so on. Many more views of the real numbers • Complexity and randomness measures on real numbers (for example, Turing degrees) • Historical and philosophical perspectives: the real numbers as an idealization of a measurement, the meaning and use of infinitesimals, and so on. • Real numbers as a representation of an infinite sequence of Bernoulli trials • Real numbers generated by formal languages. Many more views of the real numbers • Digit patterns in real numbers, such as normal numbers. • Real numbers as set-theoretic codes. A real number may code: – a cardinal collapse – a Borel set – a countable model of set theory – a strategy for an infinite two-person game. Organizing Multiple Theories • How should the hypertext on real numbers be organized? • Less than a grand all-encompassing architecture • More that a simple listing of topics in unrelated slots. • The goal is a readable, searchable, general introduction to the real number system. Organizing Multiple Theories • It must also show relationships across categories. • It must lead to more in-depth text • It must be in a form that is easy to update and extend. • It must help readers searching for a topic. Hypertext Structures • Most text in this system is in one of two forms: “book text” and “core text”. Book Text • Book text gives an orderly development of mathematical ideas. • Shorter and more narrowly focused than most math books. • Theorems, proof, definitions, examples • Other books attached in a tree-like structure. Core Text • Short, highly-linked texts, organized around a concept or method • Discursive, condensed discussions of a mathematical topic • Previews, surveys, summaries, leading to other text. • Helps the user navigate to other topics. • The same topic may reappear in several core texts. Core Text Real Numbers main essay Core Text Sets of Real Numbers Real Numbers Vector Spaces Functions of a Real Variable Complex Numbers Core Text and Book Text Mathematics Hypertext Project • First step of a very long term project. • Need for contributors and collaboration. • Goal is the building of large-scale structures of mathematical text.

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