Cantor and Infinity - Galileo paradoxically showed that the set of natural numbers and the set of squares both have the same infinite size, despite the squares being a proper subset. - Cantor resolved this by developing a theory of one-to-one correspondences between sets to formally define their cardinalities (sizes). - Using this, he proved that the set of real numbers is a higher level of infinite size than the natural numbers or integers, denoted aleph-null and aleph-one. However, whether other infinities exist between these is undecidable.

1. Cantor and Infinity

2. 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.

3. 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.

4. 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 ...

5. 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|.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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

11. 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.

12. 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 ...

13. 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.

14. 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'.

15. 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.