How big (really) is infinity?

A definite answer through transfinite arithmetic

This workshop will be an informal introduction to transfinite arithmetic and the questions related to the quantification of infinities. Basically, it will try to provide a satisfactory answer to all those nagging questions we’ve all had - likely before the age of six - about the size of infinity. Queries like “What is infinity plus one?” or “What is infinity plus infinity?” or even “What is infinity times infinity?” It turns out Georg Cantor, the founder of set theory, had all the answers! Even so, he commented on one of his seminal proofs that “although I see it, I do not believe it.” Prepare to see things that you will hardly believe.. and - by the same token – learn a quality lesson in parenting.

Motivational Questions

If all humans are inherently equipped with the ability to count and to repeat the process of “adding one more”ad infinitum – starting with the number 1, add 1 to 1 to get 2, then add 1 to 2 to get 3, and so forth – then why are we seemingly incapable of…

1)…providing a satisfactory answer (or justification) to our kids when they ask: “What is infinity plus one?” or “What is infinity plus infinity?” or even “What is infinity times infinity?”

2)…explaining such an absurdity as Zeno’s paradox, which basically states that motion through space is impossible or, equivalently, that motion is an illusion?

3)…deciding whether there are more real numbers between 0 and 1 than there are rational numbers (i.e. fractions) altogether?

I - Historical Perspective

Ancient GreeksThe Greek philosopher Zeno of Elea proposes his

(5th century B.C.) famous paradoxes of motion. These are rooted in deep questions about the nature of time and space and, also, in some misconceptions about infinity:

- Achilles and the Tortoise (you can never catch up!)

-The dichotomy paradox (you can’t even start!)

-The arrow paradox (you can’t even move!)

Ancient Romans Use the symbol ∞to represent 1000, a “big” number to them.

Galileo of PisaGalileo states his famous “paradox"and begins the

(Early 17th century)first modern line of inquiry towards the infinite by writing: “infinity should obey a different arithmetic than finite numbers.”

John WallisThe English mathematician proposes that ∞ stand (1665) for infinity.

Newton / LeibnitzBoth mathematicians independently develop the

(Late 17th century)calculus,which systematically assumes the existence of “infinitesimal” processes.

Analysis - one of the pillars of modern mathematics - is born!

Georg CantorAn ambitious Russian/German mathematician completely

(1870’s)upsets the mathematical order by founding the theory of sets and by putting forth his (heretic) transfinite arithmetic.

His tremendous accomplishment is twofold: he has single-handedly invented the language of contemporary mathematics - set theory –and, in addition to that, he has groundedthe concept of infinity on a firm logical foundation.

He was rewarded for all this with a lifetime of controversy, a stifled career and, ultimately, a sad history of mental health.

Now, however, he justly holds a very special place in the pantheon of mathematics.

Kurt GödelThe brilliant Austrian logician Kurt Gödel shows

(1930’s)that the continuum hypothesis cannot be disproved from the axioms of set theory…

Paul Cohen…and then the American mathematician Paul Cohen

(1960’s)shows that the continuum hypothesis cannot be proved!

As it stands today, we have still not settled the question of the continuum!

II - Mathematical Perspective – Set Theory and Transfinite Arithmetic

As humans we have a fundamental ability to perceive the items that surround us as collections and to categorize them into abstract groups. This explains why we also seem to have an inherent drive to collect things.

Definitions

A set is any collection of well-defined, well-distinguished objects. These are called the elements, or members, of the set. For a given set S, the number of elements of S is called the cardinality of S. (Think of the cardinality as the “size” of the set.) If S has a finite number of elements, it is called a finite set. Otherwise, the set is called an infinite set.

Examples

1 - A set of 12 crystal champagne glasses (card = 12)

2 -A complete set of golf clubs (card = 13):

{PW, SW, Putter, 1W, 3W, 5W, 3, 4, 5, 6, 7, 8, 9}

3 - The set of prime numberssmaller than 100 (card = 25)

{2, 3, 5,7, … ,89, 97}

4 – The set of counting numbers

N = {1, 2, 3, …}

Question: What about the cardinality of infinite sets, like N in example 4?

This is where Cantor comes in to provide all the answers…

Cantor’s Definition of an Infinite Set:

A collection is infinite, if some of its parts are as big as the whole.

Alternative Definition:

A set is infinite if it can be put in a one-to-one correspondence with a proper subset of itself.

Cantor proposed that, since we can’t count the number of elements for any two infinite sets, we should instead compare whether they have the same “size” (i.e. are equinumerous) by seeking a one-to-one match-up between the elements of the two sets.

Example:

Thus, the example above illustrates that the set of even numbers is equinumerous to the set of counting numbers, contrary to what our intuition would otherwise suggest (one should have twice as many elements as the other, right? No, that’s wrong!)

The cardinalityof the set of counting numbers (or even numbers) is called aleph-nulland it actually constitutes the smallest type of infinity.

Now, what can you say about the cardinalities of…

1)…the set of integers?

2)…the set of rationals?

3)…the set of reals?

Are they all equal to aleph-null? How can you show if theyhave a different cardinality?

Cantor answered this fundamental question and – in the process – posed one of the most important queries in the development of mathematics:

the continuum hypothesis.

CANTOR’S DIAGONALIZATION PROOF

See hand-out.

CONTINUUM HYPOTHESIS

Theorem (Cantor)

If X is any set, then there exists at least one set, the power set of X, which is cardinally larger than X.

Set of “cardinals” = {0, 1, 2, 3, … , aleph-null, aleph-one, aleph-two, … }

Let c denote the cardinality of the continuum (i.e. all real numbers on a segment of the real line). Then, is it true that c = aleph-null? The affirmative answer is what is called the continuum hypothesis.