Large Numbers, Infinities & God (2/3)

This post is continuing on from my discussion on large numbers and notation. I am going to assume some knowledge of Set Theory for this post as there really is a lot to write about and I have found it hard to cut it down to a nice size.

2. Infinities
Infinity as a concept has scared mathemeticians for a long time, not only was it hard to get your head around, it wasn’t really usable. A recent use is in calculus as, what I would call, a nominal infinity (‘increases without bound’ as opposed to a value or position): \displaystyle{\lim_{x \to \infty}}. Leibniz was facinated by infinity and it’s counterpart infintesimal \frac{1}{\infty} = 0.000\dots 1.

The Isha Upanishad, a Hindu scripture, gives what I think is the polite version of infinity: “if you remove a part from infinity or add a part to infinity, still what remains is infinity” – this philosophical statement explains a mathematical proposition, cardinal arithmetic, and it comes from a branch of mathematics called set theory developed by Georg Cantor and Richard Dedekind in the 1870s.

Cardinal infinites (Aleph numbers):
Say we have a set that holds all the natural numbers: \boldsymbol{\mathbb{N}} = \{0,1,2,3,4,...\} what is the size of such a set? Well \boldsymbol{|\mathbb{N}| = \aleph_0} (aleph-null) the smallest possible infinity and the first of the Aleph numbers. We just delt with the cardinality of the natural numbers, now lets move on to the rational numbers \boldsymbol{\mathbb{Q}} , these are all the numbers that can be expressed as \frac{n}{m} :

It turns out | \boldsymbol{\mathbb{Q}} | = \aleph_0 but how?
By definition a set S is countable if there exists an injective function: f: S \to \mathbb{N} from S to the natural numbers. The fractions would seem uncountable in a linear sense, but Georg Cantor came up with a method for one-to-one pairing with the natural numbers. (See right)

The final task is the real numbers , \boldsymbol{\mathbb{R}} , this contains every possible finite number. So it has 0.000001, √2, e, π, 55, 73896737483 and everything else! This set has no possible one-to-one pairing with the natural numbers. Cantor put forward something called the Continuum Hypothesis that stated no such set exists with a cardinality between that of \boldsymbol{\mathbb{N}} \text{ and } \boldsymbol{\mathbb{R}} . Although the cardinality of the real numbers is stated as: | \boldsymbol{\mathbb{R}} | = \mathfrak c = 2^{\aleph_0} > \aleph_0 \, the Continuum Hypothesis implies: | \boldsymbol{\mathbb{R}} | = \aleph_1 . Gödel and Cohen later showed that the hypothesis can neither be disproved nor be proved in ZF set theory.

In cardinal arithmetic  \aleph_0 + n = \aleph_0, n \aleph_0 = \aleph_0, {\aleph_0}^n = \aleph_0,   just like in the Hindu scripture.

Beth numbers:
The Beth numbers run parrallel to the Aleph numbers and are the successive cardinalities to power sets of the natural numbers. For an example of a power set, see: S=\{x,y,z\} it’s power set is: P(S) = \left\{\{\}, \{x\}, \{y\}, \{z\}, \{x, y\}, \{x, z\}, \{y, z\}, \{x, y, z\}\right\}\,\!.

\beth_0 = \aleph_0
\beth_{n+1} = 2^{\beth_n}
| P(\mathbb{N}) | = \beth_1
| P(P(\mathbb{N})) | = \beth_2 etc..

These numbers are related to the Generalised Continuum Hypothesis.

Ordinal infinities:
In set theory a set which is well-ordered (a is less than b which is less than c, etc) has an ordinal number, the smallest ordinal infinity is omega:

\omega = \{0<1<2<3<4<...\} and | \omega | = \aleph_0

To get the cardinality of an ordinal infinity we ignore the order, this will become important when we assess ordinal arithmetic. For example here are three ordinal additions:

\omega + \boldsymbol{1} = \{0_0<1_0<2_0<3_0<4_0<...\boldsymbol{0_1}\}

\boldsymbol{1} + \omega = \{\boldsymbol{0_0} < 0_1<1_1<2_1<3_1<4_1<...\} = \omega \text{ (after relabeling the elements)}

\omega + \boldsymbol{\omega} = \{0_0<1_0<2_0<3_0<4_0<...< \boldsymbol{0_1<1_1<2_1<3_1<4_1<...}\}

The difference between ω+1 and 1+ω is the placing of the dots, as they imply ad infinitum. In ω+1 there is no direct predecessor before the second 0, just infinity dots.
Now although ω, ω+1 and ω+ω are three seperate ordinal values – they all have the cardinality \aleph_0 and this is because when we get the size, we ignore the order. It also stands that n·ω = ω ≠ ω·n. The first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω is the supremum of all countable ordinals. The elements of ω1 are the countable ordinals, of which there are uncountably many:

ω1 = sup{ ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε0, … }

ωn = sup{ ωn-1, ωn-1 + 1, ωn-1 + 2, …, ωn-1·2, ωn-1·2 + 1, … } and | \omega_n | = \aleph_n.

Surreal numbers:
The surreal number system was thought up by John Conway, it contains the real numbers as well as infinite and infinitesimal ordinal numbers. They are also known as the extraordinal numbers. They have fathoms of interesting properties but if I carry on i’ll never stop. For more on these amazing numbers I recommend the book ‘Surreal Numbers‘ by Donald Knuth.

\aleph * \infty * \omega

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5 thoughts on “Large Numbers, Infinities & God (2/3)

  1. > This set has no possible one-to-one pairing with the natural numbers so it is known as uncountable: | \boldsymbol{\mathbb{R}} | = \aleph_1

    Sorry for nitpicking, but |\mathbb{R}| = 2^{\aleph_0} = \mathcal{c} = \aleph_1 only if you accept Continuum hypothesis

  2. Pingback: Large Numbers, Infinities & God (3/3) | Neural Outlet..

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