Large Numbers, Infinities & God (3/3)

Posted on 04/02/2012 by Neural Outlet..

Continuing on from my posts on Large Numbers and Infinities. For the sake of argument, this post implies the existance of a Christian God.

Georg Cantor & God
Cantor was a very religious man, ironically he began his work on infinity to disprove it – because nothing could be as infinite as God. After finding multitudes of infinites of different type and form, he refered to them as transfinite: more than any finite number but not what he would later call Absolute Infinity. Religion often gets a bad rep for limiting scientific growth through things like persecution, elitist education systems, lack of rationality and the entirity of the dark ages.. but many scientists were inspired by God. Indeed Cantor himself actually believed God was speaking to him, that he was God’s messenger for this glorious new mathematics. The mathematical infinity was last in a series of three infinities diverging from the absolute (God), the second was physical infinity here I assume the universe. Of the three, the second (physical infinity) seems so much more out there. I assume he’s refering to the universe because as physics stands – matter is quantised (early on: Atomism, later the Standard Model) and so the only infinity is outward bound. Personally I don’t see reason to believe in a physical infinity. The nearest would be the universe as a closed manifold in Eliptic (Non-Euclidean) geometry – if you kept going you’d never reach the end, but it’s because you’ve looped round to the begining (modulus NOT infinite).

Back to the absolute infinity, what would that entail? Philosophy states it as an unconditional reality which transcends limited, conditional, everyday existence. As in all trains of thought there are variations, the general attributes found in most are: infinity, indescribability, formlessness, transcendence and immanence. Infinity inside infinity? Exactly how Cantor described it (Absolute→Physical→Abstract). These are also strong beliefs in pantheism (God is everything) and panentheism (everything is God), Cantor’s belief was that God holds every aspect of every infinity and finity. I get the feeling that Cantor saw God as ‘everything and more’, a sort of transpanentheism incorporating Christian dogma.

“Can any hide himself in secret places that I shall not see him? saith the Lord. Do I not fill heaven and earth? saith the Lord.” – Jeremiah 23:24.

Donald Knuth & God
The book Things a Computer Scientist Rarely Talks About is my influence for these three posts, and in it was an idea that captivated me: Does God have to be infinite to fit biblical criteria? Well, in the King James Version of the Bible “infinite” only appears three times and only once pertaining to an attribute of God: “Great is our Lord, and of great power: his understanding is infinite.” – Psalms 147:4-6. Also note that the Hebrew in this text can be more accurately translated as the phrase “too big to count”.

Knuth invites us to invision the number $10\uparrow \uparrow \uparrow \uparrow 3$ which, as we remember from the first post, means $10\uparrow \uparrow \uparrow (10\uparrow \uparrow \uparrow 10).$ Ofcourse we need to further explain $(10\uparrow \uparrow \uparrow 10)$ and we shall call it $\boldsymbol{\mathcal{K}}:$

$\boldsymbol{\mathcal{K}} = 10\uparrow \uparrow (10\uparrow \uparrow (10\uparrow \uparrow (10\uparrow \uparrow (10\uparrow \uparrow (10\uparrow \uparrow (10\uparrow \uparrow (10\uparrow \uparrow (10\uparrow \uparrow 10))))))))$

Now Knuth’s K was much more fancy but here we see that $10\uparrow \uparrow \uparrow \uparrow 3 = 10\uparrow \uparrow \uparrow \boldsymbol{\mathcal{K}}.$ Hopefully you are beginning to see the magnitude of the number we are dealing with, if not take into acount to attempt to define it further, we must say:

$10\uparrow \uparrow \uparrow \uparrow 3 = 10\uparrow \uparrow \uparrow \boldsymbol{\mathcal{K}} = \underbrace{10\uparrow \uparrow (10\uparrow \uparrow (10\uparrow \uparrow \dots \uparrow \uparrow (10\uparrow \uparrow 10) \dots ))}_{ K \mbox{ times.}}$

From now on we’ll refer to 10↑↑↑↑3 as Special K (Knuth calls it Super K but I am cereal about my names), Special K is an unfathomably large number – too big to count. Not only is Special K massive, it is one of the smallest finite numbers around, almost every other finite number is larger than it…

To say that God is not infinite but limited by numbers such as Special K is not a comprehensible limitation at all. Knuth puts it well saying that this cannot contradict the Bible or any other sacred text because there are no words to explain such large magnitudes, because they are quite simply incomprehensible (which itself is often seen as an important attribute of God).

Side note: God as Beauty
Carl Sagan once said on a programme called God, the Universe & Everything Else that he saw God as the sum total of the physical laws which describe the universe, not as a religious figure or spiritual being and infact opposed the idea. Einstein thought much the same, as is apparent in these two extracts:

“It was, of course, a lie what you read about my religious convictions, a lie which is being systematically repeated. I do not believe in a personal God and I have never denied this but have expressed it clearly. If something is in me which can be called religious then it is the unbounded admiration for the structure of the world so far as our science can reveal it” – 24th March, “On A Personal God”

“It seems to me that the idea of a personal God is an anthropological concept which I cannot take seriously. I feel also not able to imagine some will or goal outside the human sphere. My views are near those of Spinoza: admiration for the beauty of and belief in the logical simplicity of the order which we can grasp humbly and only imperfectly.” – Albert Einstein, 1947

Which belief do you favour? Cantor’s or Knuth’s?

[EDIT]
I wanted to share something I just read:

“Whatever is in the heavens and the earth glorifies Allah. He is the Mighty, the Wise. His is the kingdom of the heavens and the earth; He bestows life and he causes death; and He has power to do all that he wills. He is the first and the last Manifest and theHidden, and He has full knowledge of all things.” – Qur’an 57:1.

To me this is extremely close to what Cantor believed, except he had the math to define more than is stated here.
[/EDIT]

Large Numbers, Infinities & God (2/3)

Posted on 31/01/2012 by Neural Outlet..

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, …, ω^ω, …, ω^{ω^ω}, …, ε₀, … }

ω_n = sup{ ω_n-1, ω_n-1 + 1, ω_n-1 + 2, …, ω_n-1·2, ω_n-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.

$\aleph * \infty * \omega$

Related links:

Post-Transcendental Numbers

Posted on 30/09/2011 by Neural Outlet..

I might be wrong off the starting bat here, but e can be defined in an infinite series 1/0! + 1/1! + 1/2! + … + 1/n!

So, i was wondering, Does a number exist that:
-> Cannot be expressed as a fraction (Irrational)
-> Cannot be expressed as a polynomial (Transcendental)
-> Cannot be expressed by an infinite series following a set rule (???)