**Mathematics of the 19th Century**

**The Non-Denumberability of the Continuum**

-no single preeminent mathematician

-century of abstraction and generalization

-deeper analysis of logical foundations of math

-increasing disconnection with the "real" world

-philosophical thrust paralleled with that of the art movement of the time

-criticism from the modern math community

-logical foundations of calculus questioned

**Underpinnings of Calculus: the Limit**

-need for logically precise terminology for "infinitely large" and "infinitely small" quantities

-Newton: ultimate ratio of vanishing quantitities

"...is to be understood the ratio of the quantities, not before they vanish, nor after, but that with which they vanish."

-wanted to consider the ratio at the exact moment when both numerator and denominator became zero: 0/0 has NO MEANING

-Leibniz' contributions: mathematical gibberish

"It will be sufficient if, when we speak of...infinitely small quantitities (i.e., the very least of those within our knowledge), it is understood that we mean quantities that are...indefinitely small....If anyone wishes to understand these [the infinitely small] as the ultimate things..., it can be done..., ay even though he think that such things are utterly impossible; it will be sufficient simply to make use of them as a tool that has advantages for the purpose of calculation, just as the algebraists retain imaginary roots with great profit."

Bishop George Berkeley

-caustic essay: "The Analyst, or a Discourse addressed to an Infidel Mathematician

-"...he who can digest a second or third fluxion, a second or third differential, need not, methinks, be squeamish about any point in divinity."

"ghosts of departed quantities"

Internal concerns & Berkeley:

led to work on foundational questions of calculus

1821: Augustin-Louis Cauchy

"When the values successively attributed to a particular variable approach indefinitely a fixed value, so as to end by differing from it by as little as one wishes, this latter is called the limit of all the others."

Went on to prove the major theorems of calculus using this definition

"limit avoidance" definition

Problems with Cauchy's Definition

Vague ideas about points "moving around" and "approaching" one another...too intuitive

Word "indefinitely"...too indefinite

Definition...too wordy

Needed to have words and phrases replaced with clear, unambiguous symbols

He needed...

KARL WEIERSTRASS

L is the limit of the function f(x) as x approaches a if,

for any ε > 0, there exists a δ > 0 so that, if 0 < |x - a| < δ, then |f(x) –L| < ε.

-symbolic

-static

-solid

foundation for "the edifice of calculus that remains to the present day"

**Cantor and the Challenge of the Infinite**

dealt with some "highly peculiar and unsettling discoveries"

rationals/irrationals

nature of the real number system

profound properties of sets

"sometimes maligned, sometimes paranoid genius"

Russia --> Germany

Judaism, Protestantism, Roman Catholicism

Artistic (drawing, violin)

Married, 6 children

doctorate, University of Berlin

studied w/ Weierstrass

University of Halle

personal/professional affronts: depression

died alone in a sanatorium in 1918

1845-1918

Georg Ferdinand Ludwig Philip Cantor

**More on the Infinite**

Equivalent Sets

previous thought: counting -> equinumerosity

cantor: equinumerosity -> counting

method: one-to-one correspondence

examples:

fingers on hands

people and seats in auditorium

Cantor's definition:

Two sets M and N are EQUIVALENT...if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other.

"same power" "same cardinality"

applies to finite OR infinite sets

uncharted territory

Objections

"a concept better let alone"

only the "potential infinite" recognized

"completed infinite" ?!?!

"...I protest above all against the use of an infinite quantity as a completed one, which in mathematics is never allowed. The Infinite is only a manner of speaking..." ~Gauss

In summary...

two fundamental premises:

equal cardinality of sets determined by one-to-one matchings

completed infinite - a valid concept

**N, E, Z**

any set in 1-to-1 correspondence with N: "denumerable" or "countably infinite"

new number and symbol:

Denumerability of Q

was every infinite set denumerable?!

1874, "On a Property of the Collection of all Algebraic Numbers"

a set not denumerably infinite: all real numbers!

"showed that no interval of real numbers, regardless of how small its length, could be put into a one-to-one correspondence with N.

Great Theorem:

The Non-Denumerability of the Continuum

"continuum"- an interval of real numbers

(a,b) = the set of all real numbers x such that a<x<b

(0,1) - the unit interval

real numbers in this interval can be expressed as infinite decimals.

*technical note: 0s and 9s

Theorem: The interval of all real numbers between 0 and 1 is not denumerable.

Assume (0,1) can be matched in a 1-to-1 fashion with N and derive a contradiction.

b

1) b is a real number since it is an infinite decimal. it cannot be 0 or 1. it falls strictly between 0 and 1.

2) b cannot appear anywhere among our numbers in the right column.

by contradiction, a 1-to-1 correspondence cannot exist between N and (0,1), so (0,1) is not denumerable.

*technical note 2

the skeptics must become believers

"More" Infinite

points in (0,1) are so abundant that they outnumber the positive integers!

from (0,1) to (a,b)...any finite interval

same cardinality:

(0,1) and set of ALL real numbers

N: the standard for transfinite cardinal

(0,1): the standard for new, larger infinite cardinal

c

Application: difference between rationals and irrationals

Theorem U: If B and C are denumerable sets and A is the set of all elements belonging either to B or to C (or to both), then A itself is denumerable. (A is the union of B and C, so A = B U C.)

B and C are denumerable and thus 1-to-1 with N:

Generate 1-to-1 matching between N and A=B U C

A is denumberable: shows union of two denumberable sets is denumerable.

Rationals are denumerable; irrationals are not

Proof? Suppose irrationals are denumerable.

Then union of rationals and irrationals would be denumerable (Thm U).

But this union is the set of all real numbers, which is non-denumerable.

By contradiction, irrationals are non-denumerable.

So the irrationals far outnumber the rationals!

Epilogue...

Cantor applied the non-denumerability of intervals to the existence of transcendental numbers

Real numbers can be divided into algebraic and transcendental numbers

Algebraic numbers: vast set

Transcendentals: very few on the scene

(1st example only 30 years prior)

Proved set of algebraic numbers is denumerable

By Theorem, (a,b) is not denumerable.

By Theorem U, transcendental numbers must be non-denumerable and must outnumber algebraic numbers in any interval

"genuinely provocative theorem"

transcendentals in the vast majority,

did not ever exhibit a single concrete example of one

"The algebraic numbers are spotted over the plane like stars against a black sky; the dense blackness is the firmament of the transcendentals." ~Eric Temple Bell

And just for fun...