Law of Truly Large Numbers – 2^{À}^{0} = À_{1}_{ } is undecidable

http://mathworld.wolfram.com/LawofTrulyLargeNumbers.html

http://en.wikipedia.org/wiki/Law_of_Truly_Large_Numbers

http://skepdic.com/lawofnumbers.html

The
“Law of Truly Large Numbers” is summarized nicely in the following
limerick. I am not sure whether
Eddington or Pound should receive credit.

There
once was a *breathy* baboon

Who
always breathed down a bassoon,

For
he said, "It appears

That
in billions of years

I
shall certainly hit on a tune."

Sir Arthur Eddington (1882 –1944)

There
once was a *brainy* baboon

Who
always breathed down a bassoon

For
he said, "It appears

That
in billions of years

I
shall certainly hit on a tune."

Ezra Pound (1885 –1972)

That
2^{À}^{0} = À_{1} is the continuum hypothesis,
where À_{0} is aleph null and À_{1} is aleph one.

http://mathworld.wolfram.com/Aleph-0.html

http://mathworld.wolfram.com/Aleph-1.html

http://en.wikipedia.org/wiki/Continuum_hypothesis

http://mathworld.wolfram.com/ContinuumHypothesis.html

Kurt Gödel proved that assuming
the continuum hypothesis true does not contradict Zermelo-Fraenkel set
theory. Paul Cohen proved that assuming
the continuum hypothesis false does not contradict Zermelo-Fraenkel set theory. Therefore, the validity of the continuum
hypothesis is *undecidable* within
Zermelo-Fraenkel set theory.

The online essay “You
Can’t Get There From Here” on infinity at Platonic Realms (another
site I highly recommend) is nicely done and definitely worth a look!