# Large Numbers, Infinities & God (3/3)

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]

## 12 thoughts on “Large Numbers, Infinities & God (3/3)”

1. I’m siding with Cantor. God encompasses –
the biggest big, the smallest small,
the highest high, and no one goes lower,
not even the devil. The truth cuts both ways.

I’m not the sharpest knife in the woods, either.
So what? No big deal. Peace to you today! UT 🙂

2. Both are great, but I too am siding with Cantor. If God is infinite, then he must encompass everything.

3. Very interesting ideas you put forth in these posts. I dig your site! Since your post “implies the existance of a Christian God,” than another thing to note would be the trinity (father, son,holy spirit). How does that tie into this? Maybe that should be your next post? Since “the mathematical infinity was last in a series of three infinities diverging from the absolute (God),” and God encompasses the father,son and holy spirt- that would be great to ponder!!??
***Felicia @ My Voyage Through Time

• I like that, what an aesthetic idea!

• excellent idea! I love studying Christian theology as well and the Trinity is one of the most beautiful parts about God…even if it does make my head hurt!

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5. Great article. It reminds me a litle bit of Gregory Chaitin’s essay, “A Hundred Years of Controversy . . .” . where he speaks of infinite sets, paradoxical sets and metamathematics.

The interesting question for me is how does consciousness handle concepts like infinities and what does that say about a transcendant basis for conscious life? I’m convinced the answer points to a transcendant power.

• Everyone loves a bit of metamathematics!

I’ve been drafting a post on Artificial Intelligence for about a week now, trying to make all my bullet points into a readable post that makes sense! So hopefully i’ll get round to that sometime next week, it talks about consciousness vaigely.

6. Great blog and great post. I definitely will be a “frequent flyer”. Also, thank you for taking a look at my last post also. Take care.

7. Pingback: Complex Bases | Neural Outlet..

8. Just a quick thanks for your posts on these topics, and I will begin checking out your blog more frequently! Religion began to take on a new context for me when I was, of all places, in Geometry class in high school, considering “pure mathematics” more seriously and beginning to look at the world a little differently. And concerning the topics at hand in these posts, I was thankful to have read Rudy Rucker’s Infinity and the Mind at a young age, because it was mind-expanding in so many ways. As a result, my children have grown up talking about “different ‘sizes’ of infinity” and such ideas from their earliest childhood, and we’ve been enriched for it.

I was unaware of Knuth’s book and will have to check it out. Thanks again!