Ten Ways of Looking at Real Numbers

Page
4

#7. The Cardinal of the Real Numbers

The continuum must satisfy:

The second condition guarantees that:

Not much else restricts the possible values of the continuum.

Slide 27

#7. The Cardinal of the Real Numbers

Easton’s theorem: Let be any regular cardinal in the ground model of ZFC with cofinality

Then there is a generic extension which preserves cardinalities, in which

For example, the continuum could be

Slide 28

#7. The Cardinal of the Real Numbers

A variety of “cardinal invariants” of the continuum: cardinals between .

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.

Slide 29

#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.

Slide 30

#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 ,then α is transcendental.

Slide 31

#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.

Slide 32

#9. The Real Numbers as a First-Order Theory

Tarski's decidability theorem: The first-order theory of real-closed fields is decidable.

There is an algorithic procedure to determine if a first-order sentence about the real numbers in the language of ordered fields is true or false.

Slide 33

#9. The Real Numbers as a First-Order 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.