Cantor and Infinity

Numbers and Infinite Numbers

We encounter problems when we start asking questions like:− What is a number?− What is infinity?− Is infinity a number?− If it is, can there be many infinite numbers?

We can resolve some aspects of these questions with a few simple ideas, rigorously applied.

Some Definitions

A set is a collection of well-defined, well-distinguished objects. These objects are then called the elements of the set.

For a given set S, the number of elements of S, denoted by |S|, is called the cardinal number, or cardinality, of S.

A set is called finite if its cardinality is a finite nonnegative integer.

Otherwise, the set is said to be infinite.

Galileo’s Paradox of Equinumerosity

Consider the set of natural numbers Ν = {1, 2, 3, 4, …} and the set of perfect squares (i.e. the squares of the naturals) S = {1, 4, 9, 16, 25, …}.

Galileo produced the following contradictory statements regarding these two sets ...

A Contradiction?

1.While some natural numbers are perfect squares, some are clearly not. Hence the set N must be more numerous than the set S, or |N| > |S|.

2.Since for every perfect square there is exactly one natural that is its square root, and for every natural there is exactly one perfect square, it follows that S and N are equinumerous, or |N| = |S|.

Many Contradictions?

We could repeat this reasoning for N and:− The even numbers E = {2, 4, 6, …}− Triples− Cubes− Etc.

In each case we have systematically picked out an infinite proper subset of N.

A is a proper subset of B if A is contained in B, but A ≠ B.

One-to-One Correspondence

Galileo’s exact matching of the naturals with the perfect squares constitutes an early use of a one-to-one correspondence between sets – the conceptual basis for Cantor’s approach to infinity.

Galileo's 'Solution'

To resolve the paradox, Galileo concluded that the concepts of “less,” “equal,” and “greater” were inapplicable to the cardinalities of infinite sets such as S and N, and could only be applied to finite sets.

Cantor showed how these concepts could be applied consistently, in a theory of the properties of infinite sets.

The Basis of Cantor's Theory

A bijection is a function giving an exact pairing of the elements of two sets.

Two sets A and B are said to be in a one-to-one correspondence if and only if there exists a bijection between the two sets. We then write A~B.

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

Which sets have the same cardinality as N?

The cardinality of the natural numbers, |N|, is usually written as

Other well-known infinite sets have cardinality − The integers, I, { …, -3, -2, -1, 0, 1, 2, 3, …} (fairly

easy to show N~I).− The rationals, Q (a bit more difficult to show N~Q).

Showing N~I

Showing N~Q

We can establish a one-to-one correspondence between the naturals and the rationals.

We use an infinite 2-d arrangement of Q, and a systematic path through this arrangement.

Are there any infinite sets with greater cardinality than N?

We can show that |R| is greater |N|. We cannot establish a one-to-one

correspondence between N and R. |R| is a higher order of infinity, usually written

as Let's prove this ...

Cantor's Diagonal Argument For any

hypothesised enumeration of the real numbers, we can show that there is a real which is not in that enumeration.

We rely on forming a new real by the systematic alteration of the digits in the enumeration.

Transfinite Arithmetic

Cantor devised a new type of arithmetic for these infinite numbers.

For the infinite numbers we have rules such as these ...

Note: the cardinality of the set of real numbers is often written 'c' for 'continuum'.

The Continuum Hypothesis

We have seen that |R| is greater than |N|. But, are there any infinite numbers in between

these? The hypothesis that there is no infinite number

between |N| and |R| is called the continuum hypothesis.

It was shown to be formally undecidable by Gödel and Cohen.