Notions of Infinity

My children have asked questions such as “what is infinity minus one?”. That got me thinking about such things again. Well, it depends which sort of infinity you mean.

Signed Limit Infinity

In statements such as

$\lim_{n \rightarrow \infty} \left( 1 + \dfrac{1}{n} \right)^{n} = \rm{e}$

the symbol ∞, or if we are not to be ambiguous, the symbols −∞ and +∞, are used in the representation of a process, and are not really in themselves numbers.

We could supplement the real number system with these symbols to form R# = R ∪ {−∞,+∞}. With the usual ordering, this set has the topology of a closed interval (whereas R has the topology of an open interval).

Similarly, we can supplement the complex numbers with directional limits: C# = C ∪ {∞z | z∈C and |z|=1}. This has the topology of a closed disk.

One problem with the signed infinity is as follows:

$\lim_{x \rightarrow 0-} \dfrac{1}{x} = -\infty$, but

$\lim_{x \rightarrow 0+} \dfrac{1}{x} = +\infty$;

so, in this sense, we can’t really extend arithmetic with a rule

$\dfrac{1}{0} = \infty$.

Unsigned Limit Infinity

On the other hand, if we extend the reals with an unsigned infinity ∞ or ±∞ to form R* = R ∪ {∞}, then the rule

$\dfrac{1}{0} = \infty$

makes more sense. This system has the topology of a circle. Note that this set can not be totally ordered.

There are at least two distinct ways to extend this to the complex case. The first is C* = C ∪ {∞}. This has the topology of a sphere (or, more precisely, a spherical shell).

The second is C* = C ∪ {±∞z | z∈C and |z|=1}, where we consider that −∞z = +∞(−z), identifying diametrically opposite points at infinity. This has a more complex topology: that of a projective plane; this is essentially a spherical shell with diametrically opposite points identified, so the ‘points’ of this space are pairs of points on the sphere.

Infinite Ordinals

Now we come to numbers. The ordinals are positive (or more accurately, non-negative) ordering integers more familiar as 1st, 2nd, 3rd, 4th,… They are really so-called well-ordered sets.

The finite ordinals are represented by numbers: 0, 1, 2, 3, … . The first (smallest) infinite ordinal is given the symbol ω; this is then followed by further infinite ordinals. There is no largest ordinal; for any ordinal, there is always a bigger one. Ordinal arithmetic is non-commutative. Some example sums include

$1+2=3$

$1+\omega = \omega < \omega + 1 < \omega + 2 < \omega + \omega = \omega \times 2 < \omega \times \omega = \omega^2 < \omega^{\omega}$

$2 \times \omega = \omega$

Note here, in particular, that $1+\omega \ne \omega + 1$ and $2 \times \omega \ne \omega \times 2$.

The value ω−1 is meaningless, as is 1−2, in the world of ordinals; in fact, subtraction is not defined.

Infinite Cardinals

The cardinals represent the sizes of sets, and are more familiar as 1, 2, 3, 4…

The finite cardinals are also represented by numbers: 0, 1, 2, 3, … . Zero is essentially the same as the empty set. The first (smallest) infinite cardinal is given the symbol ℵ0; this is then followed by further bigger and bigger cardinals. There is no largest cardinal; for any cardinal, there is always a bigger one.

Cardinal arithmetic is commutative.

$1+2=3$

$1+\aleph_0 = \aleph_0 + 1 = \aleph_0 + \aleph_0 = \aleph_0 \times 2 = 2 \times \aleph_0 = \aleph_0 < 2^{\aleph_0}$.

Infinite Surreal Numbers

This is where the fun really begins. The mathematician J. H. Conway combined the constructions of ordinal with Dedekind cuts, and formed the Surreal numbers (but the name came from D. E. Knuth).

These numbers are akin to the ordinal numbers, in that they include the finite ordinals 0, 1, 2, 3,… and the infinite ordinals such as ω. However, the surreals also include such numbers as the negatives −1, −2, …, the reals, the infinitesimals (infinitely small, but non-zero) 1/ω, and also for example −ω, ω+1, ω−1, ω/2, $\sqrt{\omega}$, and many others….

Other?

There may be other notions of infinity.