The Ladder

of the

Infinite

By Ernesto Pérez

“He told me his book was called the Book of Sand, because neither the book nor the sand has any beginning or end.”

Jorge Luis Borges

No matter the loose or

imprecise usage we give

to the notion of

infinity, we all have an

intuitive idea that the

infinity is the

unending and the

limitless.

Is it that the

idea of infinity

is inbred in our

minds?

The infinity is often

treated from three

distinct aspects:

● from the

philosophical way,

● from the mystical,

● and from the

mathematical point

of view.

However, this symbol

is mathematically

imprecise.

The infinite is usually

denoted by this symbol.

It tells nothing about the

type of infinite we are

referring to.

In mathematics, the

usual symbol for the

infinite is this one.

It is commonly called

the aleph.

Why the need for a new

symbol?

Because in mathematics

there are many types of

infinities, not just

one.

Some infinites are

“countable.”

Other infinites are

“uncountable.”

The amount of grains

of sand in the whole

planet Earth is finite

and countable.

The grains of sand in

the planet can be

theoretically possible

placed one next to

another.

Then we begin counting

the grains as “first

grain”, “second grain”,

etc.

The amount of fractions

on the number line is

infinite and countable.

See at right that the

fractions can be

counted following the

line “first fraction”,

“second fraction”, etc.

The grains of sand is

an example of a finite

and countable set.

The fractions is an

example of a

countably infinite

set.

The cardinality of a

set is the amount of

elements this set has.

The set of all natural

numbers is (countably)

infinite. This type of

infinitude is the

“smallest” of all

infinities.

The subscript added

to the aleph symbol

denotes how “big” is

one infinite respect

to another.

ℵ0, ℵ1, ℵ2, ...

The ladder of the

infinites go like this ...

What’s the meaning of the

next infinite ℵ1?

The next infinite symbol

ℵ1 is for the set of all

possible decimals.

The decimals are called

infinitely uncountable.

Is there an end to this

ladder of infinities?

No, for there is also

the set of all types of

infinite sets. and

many other “types.”

Is there an end to

this madness?

You guess, its your

turn.

Three people have

influenced me in the

understanding of the

infinite.

Archimedes, Georg

Cantor, and Jorge Luis

Borges.

This is the classic book

that began to count the

grains of sand. Click the

image to get the free EBook

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