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BANACH-TARSKI Paradox

“A ball can be decomposed into a finite number of point sets and reassembled into two balls identical to the original.”

It can be difficult to comprehend a world where one can chop up a solid ball and reconstruct it to form two solid spheres of precisely the same size, but such is the world of mathematics.

A couple of Polish mathematicians, Stefan Banach and Alfred Tarski, tell us exactly this:  The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. The reason the Banach–Tarski theorem is called a paradox is because it contradicts basic geometric intuition. ‘Doubling the ball’ by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible.

What makes the Banach-Tarski paradox particularly remarkable in geometry is that it relies on the once-controversial Axiom of Choice. This axiom states that for a group of non-empty sets, there exists a function called the “choice function” f such that f(s) for any one of those sets is an element of s. This means, given a collection of sets we can selectively choose exactly one element out of each of those sets. This may seem bland and innocuous, but the controversy is that there is no specified rule for which element is chosen from each set. In this way, the choice function is uniquely and intriguingly human and leads to answers that can be difficult to grasp, such as the Banach-Tarski paradox. [The-Banach-Tarski-Paradox]

How the Axiom of Choice can lead to this paradox regarding our notions of volume is that the axiom allows for the construction of non-measurable sets. This decomposition and reconstruction cannot be done in the physical world, because if you take and apple and cut it up into pieces with a knife, each of these pieces is “measurable.” That is, they have boundaries and a sense of volume, whereas the sets constructed using the Axiom of Choice do not obey these properties. Unmeasurable sets do not have a volume or size in any ordinary sense and such sets have been used historically to shed light on the true meaning of length, area, and volume in set theory.

A strong form of the Banach–Tarski paradox:

Given any two bounded subsets A and B of an Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of A and B into a finite number of disjoint subsets, A = A1 ∪ … ∪ Ak, B = B1 ∪ … ∪ Bk, such that for each i between 1 and k, the sets Ai and Bi are congruent.Now let A be the original ball and B be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball A into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set B, which contains two copies of A.

This theorem depends on the Axiom of Choice and include another interesting consequence of non-measurable sets .

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What a crock of shit. Your list is bullshit.

Gender is an arbitrary societal construct, but Hillary must be elected because we need a female president.

Gay marriage and abortion are in the constitution, while the right to bear arms is not.

My opinion as a woman only matters if I’m a Democrat.

A man can never understand a woman’s perspective on abortion, but he can become a woman.

If two people want to get married, it’s none of your business. However, if two consenting adults agree to a wage for services, it is your business how much said wage should be.

You’re too old and too white to run for office … unless you’re a Democrat.

Death penalty for babies is part of women’s rights. Death penalty for sadistic murders is abhorrent.

Global warming is “settled science,” but a fetus is not alive.

Tolerance is a one-way street with liberals deciding the direction.

Christians are out of line for saying homosexuality is a sin, but Muslims throwing homosexuals to their deaths off of buildings are just practicing their peaceful religion.

If you want it, you’re entitled to it.

If I am losing a debate strictly on facts, I can call my opponent a racist or a bigot and the debate is over because I win.

If Christians would just learn to coexist with the peaceful Muslims, they wouldn’t be beheaded… It’s the Christians’ fault because — Crusades!

Gay marriage will have zero impact on you … Now bake a gay wedding cake or get sued for all you have worked for.

The Bible is completely irrelevant and absolutely false … until someone hurts their feelings. Then, God is love, and, “Judge Not.”

Science is king … except for that one lesson in biology regarding identifying sex as male or female. That’s not accurate at all.

Koch brothers = evil corporate interests; George Soros = benevolent donor.

The religion of preference is Islam. The favorite special interest group is the LGBT community… It does not matter that support of the first would result in the eradication of the other.

Gender is fluid; climate is not.

Life begins when they say, and chromosomes don’t matter.

Molesting kids is okay — even poignant and funny — as long as it’s same-sex liberal molestation.

Atheists, who claim God does not exist, believe He will be the end of civilization if we worship Him.

Thanks to my friends on the left, everything is as clear as mud, and I suddenly see the error in my Christian conservative ways.

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Echoes of Light

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Sunday neurosis, that kind of depression which afflicts people who become aware of the lack of content in their lives when the rush of the busy week is over and the void within themselves becomes manifest.

Victor Frankl 

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ferreal

Experience the vivid sensation of your initiative.

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One of my first thoughts after coming across these illustrations was “how have I never seen this artist’s work before?”, because they are freaking amazing. Few too many artists today have the courage to tackle important social commentaries with their art. But Spanish artist Luis Quiles has undoubtedly broken this mold with his series of illustrations […]

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The Five Remembrances n

I am of the nature to grow old; there is no way to escape growing old I am of the nature to have ill health: there is no way to escape having ill health I am of the Nature to die: there is no way to escape death All that  is dear to me  and everyone I love are of the nature to change; there is no way to escape being separated from them My deeds are my closest companions; I am the beneficiary of my deeds. my deeds are the ground on which I stand.

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Just like a riddle in the sky... I could drown in your kerosene eyesss. Oh, where do my blue-bird fly?

The Tallest Man on Earth

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I love love," she said, closing her eyes. I promised her beautiful love. I gloated over her. Our stories were told; we subsided into silence and sweet anticipatory thoughts. It was as simple as that."

Jack Kerouac

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With slightly watery eyes

If I squint a certain way with my eyes half closed (i'm such a pessimist), and stare at multi-colored lights it's like a light show that only I can see.

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Truth and knowledge can, and will, destroy comfort and happiness.

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Actual happiness always looks pretty squalid in comparison with the overcompensations for misery.

Aldous Huxley

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You practice the song she gave to you, until the quick bleeds and your fingers are through...

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A mountain is the best medicine for a troubled mind. Seldom does man ponder his own insignificance. He thinks he is master of all things. He thinks the world is his without bonds. Nothing could be farther from the truth. Only when he tramps the mountains alone, communing with nature, observing other insignificant creatures about him, to come and go as he will, does he awaken to his own short-lived presence on earth.

Finis Mitchell (via observando)

Safe in the steep cliffs...

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The heart wants what it wants.

When you get home late and listen to GOOOD songs that rub the nostalgic scar tissue just behind your tired eyes. And you crack a beer instead of a smile. Your heart keeps beating slowly when you wish it would skip a bit like it used to. You want to show someone how you miss them and that noone else fits, but you can’t because just because… Everything’s changed and they don’t want you to anyways because just because… Everything’s goodbye. Find a song for that, dj. I believe in you.