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So we fix our eyes not on what is seen, but on what is unseen, since what is seen is temporary, but what is unseen is eternal.


The Profile


Zanzibar
Age. 39
Gender. Female
Ethnicity. that of my father and his father before him
Location Altadena, CA
School. Other
» More info.
The World









The Link To Zanzibar's Past
This is my page in the beloved art community that my sister got me into:

Samarinda

Extra points for people who know what Samarinda is.
The Phases of the Moon Module
CURRENT MOON
Croc Hunter/Combat Wombat
My hero(s)
Only My Favorite Baseball Player EVER


Aw, Larry Walker, how I loved thee.
The Schedule
M: Science and Exploration
T: Cook a nice dinner
W: PARKOUR!
Th: Parties, movies, dinners
F: Picnics, the Louvre
S: Read books, go for walks, PARKOUR
Su: Philosophy, Religion
The Reading List
This list starts Summer 2006
A Crocodile on the Sandbank
Looking Backwards
Wild Swans
Exodus
1984
Tales of the Alhambra (in progress)
Dark Lord of Derkholm
Do Androids Dream of Electric Sheep?
The Lost Years of Merlin
Harry Potter a l'ecole des sorciers (in progress)
Atlas Shrugged (in progress)
Uglies
Pretties
Specials
A Long Way Gone (story of a boy soldier in Sierra Leone- met the author! w00t!)
The Eye of the World: Book One of the Wheel of Time
From Magma to Tephra (in progress)
Lady Chatterley's Lover
Harry Potter 7
The No. 1 Lady's Detective Agency
Introduction to Planetary Volcanism
A Child Called "It"
Pompeii
Is Multi-Culturalism Bad for Women?
Americans in Southeast Asia: Roots of Commitment (in progress)
What's So Great About Christianity?
Aeolian Geomorphology
Aeolian Dust and Dust Deposits
The City of Ember
The People of Sparks
Cube Route
When I was in Cuba, I was a German Shepard
Bound
The Golden Compass
Clan of the Cave Bear
The 9/11 Commission Report (2nd time through, graphic novel format this time, ip)
The Incredible Shrinking Man
Twilight
Eclipse
New Moon
Breaking Dawn
Armageddon's Children
The Elves of Cintra
The Gypsy Morph
Animorphs #23: The Pretender
Animorphs #25: The Extreme
Animorphs #26: The Attack
Crucial Conversations
A Journey to the Center of the Earth
A Great and Terrible Beauty
The Absolutely True Diary of a Part-Time Indian
Dandelion Wine
To Sir, With Love
London Calling
Watership Down
The Invisible
Alice in Wonderland
Through the Looking Glass
20,000 Leagues Under the Sea
The Host
The Hunger Games
Catching Fire
Shadows and Strongholds
The Jungle Book
Beatrice and Virgil
Infidel
Neuromancer
The Help
Flip
Zion Andrews
The Unit
Princess
Quantum Brain
The Immortal Life of Henrietta Lacks
No One Ever Told Us We Were Defeated
Delirium
Memento Nora
Robopocalypse
The Name of the Wind
The Terror
Sister
Tao Te Ching
What Paul Meant
Lao Tzu and Taoism
Libyan Sands
Sand and Sandstones
Lost Christianites: The Battles for Scripture and the Faiths We Never Knew
The Science of God
Calculating God
Great Contemporaries, by Winston Churchill
City of Bones
Around the World in 80 Days, by Jules Verne
Divergent
Stranger in a Strange Land
The Old Man and the Sea
Flowers for Algernon
Au Bonheur des Ogres
The Martian
The Road to Serfdom
De La Terre � la Lune (ip)
In the Light of What We Know
Devil in the White City
2312
The First Fifteen Lives of Harry August
Red Mars
How to Be a Good Wife
A Mote in God's Eye
A Gentleman in Russia
The Fatal Conceit: The Errors of Socialism
Seneca: Letters from a Stoic
The Juanes Module


Juanes just needed his own mod. Who can disagree.
If Death Were a Number, I Would Be Aleph-Zero
Sunday. 9.20.09 9:39 pm
"Yes"
"No"
"Yes"
"No"
"Yes yes yes!"
"No No No!"
"Yes times infinity!"
"No times infinity to the infinity!"

Have you ever had this argument? Probably most of us has. It is possible that at the time we sat and pondered exactly how much infinity times infinity was, and if it were really possible to add infinities together or take infinities to the power of other infinities. What does infinity really mean, anyway, and can we ever hope to wrap our minds around it?

It is also possible that some hitting and kicking and punching and pulling of hair ensued, though whether this was an outgrowth of the fundamental frustration of the human mind to conceive of infinity, I cannot say.

So there are several different kinds of infinity.

Imagine the natural numbers, let's say, "1, 2, 3, 4, 5...etc." They go on into infinity, non?

But let's imagine the rational numbers, that is, numbers that can be represented using fractions, like 1/2, 1/3, or 7/22. How many of those are there? Also an infinite number!

But despite the fact that the rationals and the naturals are infinite in number, we say that they are also "countable". How can something be infinite and yet countable? Well that depends on your definition of "countable". Sure, you could never count to infinity in your lifetime or an infinite number of lifetimes. But on the other hand, everyone can start counting to infinity in the natural numbers and be absolutely certain that they aren't missing any. 1, 2, 3, 4... there aren't any natural numbers between 1 2 3 or 4 that you would be missing. In the same way you could count the rationals, making a grid where you had 1/1, 2/1, 3/1, etc. going horizontally and 1/1, 1/2, 1/3, etc. going vertically and you could ostensibly get all of the rational numbers in existence, because after all, the rationals are just combinations of the naturals. You would probaby get an infinity on the order of infinity times infinity. It would still be countable: you could start counting up rationals and be sure you aren't missing any.

However, if you made it your goal to count the real numbers, you would immediately run into a big problem. What problem is this? Well imagine we wanted to begin by counting up all the real numbers between 0 and 1. We could start with 0.01, and then go up to 0.02. But if we did that, we'd be missing all the numbers like 0.001 or 0.0000001 or 0.0000023343434 that lie between 0.0 and 0.01 and 0.02. In fact, between any two real numbers, there exists an infinite number of real numbers, because you could just extend those decimal places out as far as you like, in whatever combinations you can imagine, making however many numbers as you like between 0 and 1. This, to be sure, is a new kind of infinity. This is an uncountable kind of infinity, and as you may have guessed, its magnitude is far greater than the normal kind of infinity that you just imagined when we were counting natural numbers or rational numbers.

Mathematicians, thinking of this idea for the first time, decided to differentiate between different magnitudes of infinity. They called infinity after the Hebrew letter "aleph". For the regular old type of countable infinity, they chose the term Aleph-Zero. Most types of infinity that one encounters fit into this term. Even if someone made some kind of proclamation like, "Infinity times infinity" or "infinity to the infinity", they would all still fall under the countable kind of infinity of Aleph-Zero.
Even when we discuss the number of real numbers, we can say that the number of digits available to use for each number is Aleph-Zero, because we can say that there is one digit, there are two digits, three, four, and not be missing any in between. But each one of an infinite number of numbers can have Aleph-Zero digits, so we know we are dealing with a much larger form of infinity.

The number of real numbers fits under the term Aleph-One.

6 Comments.


Oh my gosh.
I was definitely arguing about this with my math teacher at the beginning of the year. He said that the "infinity" of a number line and the "infinity" between 0 and 1 were the same, and I begged to differ because the number line infinity would obvously have to include all the infinities between the inifinite number of numbers. Thank you.
» middaymoon on 2009-09-20 10:19:53

Tell that to my math teacher. Please.
» middaymoon on 2009-09-20 11:49:52

Wooooh this is fascinating. Halfway through I started thinking about all of the infinities between the real numbers (decimals) and then you started talking about it. It was beautiful.
» The-Muffin-Man on 2009-09-21 12:44:43

That read like an LSAT reading comprehension passage...
» undisputed on 2009-09-21 08:17:58

yep, now my head hurts.

i remember having conversations with one of my professors about infinity. my head hurt just from thinking about infinity in general (mostly because it's almost a concept i can't grasp). and now, this? goodness!
» thaitanic on 2009-09-21 12:20:42

Calm down!
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» Chauncey (41.190.16.17) on 2011-06-08 01:28:10

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