And the Axiom of Choice implies the Banach-Tarski parodox.
It's anti-intuitive:
Disassemble a pea into a finite number of pieces. Then reassemble those pieces to create the sun.
It's anti-intuitive:
Disassemble a pea into a finite number of pieces. Then reassemble those pieces to create the sun.
"Finite number of pieces" is a tricky thing. It's a finite number of pieces, sure, but each of those pieces is made of an uncountably infinite number of pieces. Banach-Tarski is a clever way of sweeping an infinite number of dust bunnies under the rug until just the right moment, when it reveals all infinitely many of them in the shape of a pea and claims to have just created them, when in reality, they were actually there all along, hiding under the rug.
For me, Banach-Tarski is just an another version of Hilbert's Hotel, but with uncountable infinities instead of countable ones. Once you accept, in your heart, that infinity is real and is equally deserving of your love and respect as finite natural numbers, then the paradoxicalness of Banach-Tarski melts away.
Such sets, of cause, cannot be constructed or even described as their description will contain infinite number of steps.
> However, the pieces themselves are not "solids" in the traditional sense, but infinite scatterings of points.