Numbers in the physical world (orders of magnitude)

Number living humans 7 * 10^9 (billions)

Number of stars in the Milky Way 10^11 (100 billions)

US national debt 10^13 (tens of trillions)

Number of ants on the Earth 10^15

Seconds since the big bang 10^17

Avogadro's number 10^23

Number of atoms in the universe 10^80

Number of grains of sand to fill the universe 10^90

A google 10^100

Number of protons to fill the universe 10^122

Number of planck volumes to fill the universe 10^185

A googleplex

google = 10^100

googleplex = 10 ^ (10 ^ 100)

How long to write down a googleplex?

Greater than the number of quantum states occupied by a human

$\displaystyle{\begin{matrix}a\times b&=&\underbrace {a+a+\dots +a} \\&&b{\mbox{ copies of }}\end{matrix}}$

$\displaystyle{\begin{matrix}a\uparrow b=a^{b}=& \underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix}}$

$\displaystyle{\begin{matrix}a\uparrow \uparrow b&={\ ^{b}a}=&\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} &= &\underbrace {a\uparrow (a\uparrow (\dots \uparrow a))} \\ &&b{\mbox{ copies of }}a&&b{\mbox{ copies of }}a\end{matrix}}$

And a “quadruple arrow” operator

$\displaystyle{\begin{matrix}a\uparrow \uparrow \uparrow b&= &\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} \\&&&{\mbox{b-1 towers of N high}}\end{matrix}}$

$a\uparrow ^{n}b$ Things are getting out of hand

$3\uparrow \uparrow 2=3^{3}=27$

${\displaystyle 3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987}$

$3\uparrow\uparrow 4=3^{3^{3^3}}=3^{3^{27}}=3^{7625597484987}\approx 1.2580143\times 10^{3638334640024}$

$3\uparrow\uparrow 5=3^{3^{3^{3^3}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}}$

From before: ${\displaystyle 3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987}$

Now: ${\displaystyle 3\uparrow \uparrow \uparrow 3 = 3\uparrow \uparrow (3 \uparrow \uparrow 3) = 3\uparrow \uparrow 7,625,597,484,987}$

A tower of 3's 7,625,597,484,987 high

That's a big number, it has about 3.6 trillion digits.

${\displaystyle 3\uparrow \uparrow \uparrow \uparrow 3 = 3\uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow 3)}$

A number with a 3.6 trillion digit exponent.

Connect each pair of geometric vertices of an n-dimensional hypercube
to obtain a complete graph on 2n vertices. Colour each of the edges of
this graph either red or blue. What is the smallest value of n for
which every such colouring contains at least one single-coloured
complete subgraph on four coplanar vertices?

Tree(3): https://youtu.be/3P6DWAwwViU

Makes Graham's number look like zero.

The Busy Beaver Function

100 print 1 200 goto 100This will never halt

100 let n = 2 200 n = n + 2 300 print 1 400 if n /= sum of two primes goto 200 500 print "Goldbach conjecture is false"How many ones will this print?

Consider all the programs that can be written with N symbols. For
each N, choose the program that writes out the largest number of ones
and halts. This describes a function from N->N.

This is the Busy Beaver Function.

It grows faster than any function we've seen so far.

There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.