Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory. Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.
She walks in beauty, like the night
Of cloudless climes and starry skies;
And all that ‘s best of dark and bright
Meet in her aspect and her eyes:
Thus mellow’d to that tender light
Which heaven to gaudy day denies.
One shade the more, one ray the less,
Had half impair’d the nameless grace
Which waves in every raven tress,
Or softly lightens o’er her face;
Where thoughts serenely sweet express
How pure, how dear their dwelling-place.
And on that cheek, and o’er that brow,
So soft, so calm, yet eloquent,
The smiles that win, the tints that glow,
But tell of days in goodness spent,
A mind at peace with all below,
A heart whose love is innocent!
“I like drinking coffee alone and reading alone. I like riding the bus alone and walking home alone. It gives me time to think and set my mind free. I like eating alone and listening to music alone. But when I see a mother with her child, a girl with her lover, or a friend laughing with their best friend, I realize that even though I like being alone, I don’t fancy being lonely. The sky is beautiful, but the people are sad. I just need someone who won’t run away.”—Unknown (via girlinlondon)
“You live like this, sheltered, in a delicate world, and you believe you are living. Then you read a book… or you take a trip… and you discover that you are not living, that you are hibernating. The symptoms of hibernating are easily detectable: first, restlessness. The second symptom (when hibernating becomes dangerous and might degenerate into death): absence of pleasure. That is all. It appears like an innocuous illness. Monotony, boredom, death. Millions live like this (or die like this) without knowing it. They work in offices. They drive a car. They picnic with their families. They raise children. And then some shock treatment takes place, a person, a book, a song, and it awakens them and saves them from death. Some never awaken.”—Anaïs Nin (via girlinlondon)
“I love no one but you, I have discovered, but you are far away and I am here alone. Then this is my life and maybe, however unlikely, I’ll find my way back there. Or maybe, one day, I’ll settle for second best. And on that same day, hell will freeze over, the sun will burn out and the stars will fall from the sky.”—