## Large numbers
There is one final number.Using ^ to represent exponentiation, it is 10^10^10^10^10^1.1. Note that exponents are resolved right-to-left. This is the maximum Poincare recurrence time for a certain type of universe, as calculated by Don Page. It doesn't matter whether you measure it in years, Planck times, or universe ages, because it''s virtually identical either way. Going back down for a moment, a googol is 10 Donald Knuth defined a notation for describing large numbers. a^^b = a^a^...a with b as. For example, 3^^3 = 3^3^3 = 3^27 which is roughly 7.6 trillion. Also, 10^^100 = 10^10^10...10 with 100 10s. This is called a giggol and was defined by Jonathan Bowers. Also, a^^^b = a^^a^^a...a with b as. Thus 3^^^3 = 3^^3^^3 = 3^3^3^3^...3 with about 7.6 trillion 3s. This number is too big to comprehend. It is called tritri. Also, 10^^^100 is known as gaggol. Similarly, a^^^^b = a^^^a^^^...a with b as. 3^^^^3 = 3^^^3^^^3 = 3^3^3...3 where the number of 3s is 3^3^3...3 where the number of 3s is 3^3^3...3 where... where the number of 3s is 3. The number of sections separated by the words "where the number of 3s is" is tritri, which is already unfathomable. This is G1. Ronald Graham defined a truly massive number known as Graham's number. I will use the notation a{c}b = a^^...^b with c up-arrows. This notation is due to Jonathan Bowers. G1 = 3{4}3. The next step is unbelievable. G2 = 3{G1}3 = 3^^...^3 with G1 up-arrows. Likewise, G3 = 3{G2}3. Graham's number is G64. It is an upper bound to a problem in Ramsey theory. This has been reduced to less than 9^^^4. The lower bound is currently 13. We can go much further. Define a{1,2}b = a{a{a...{a}...a}a}a with b nestings. This notation is due to me, but Jonathan Bowers prefers a{{1}}b. Now, a{2,2}b = a{1,2}a{1,2}...a with b as. Likewise, a{3,2}b = a{2,2}a{2,2}...a with b as. We can go beyond that in one huge step: a{1,3}b = a{a{a...a{a,2}a,2}a,2}...a,2}a with b nestings. Likewise, we can also have a{2,3}b and a{3,3}b. This allowed Jonathan Bowers to define tetratri, which is 3{3,3}3. Of course, there will always be bigger and bigger numbers. Sometimes
people hold "large number contests" where the aim is to name
the largest number. Of course, entries like gazillion are forbidden as
they do not represent a definite number - unless you define them. Also
forbidden are infinity and any infinite number, and entries like "one
plus the opponent's number". One such contest between Agustin Rayo
and Adam Elga ended at Rayo's number, which is so large that you need
over a googol symbols of first-order set theory to define it. It is defined
as being the smallest number greater than any definable in a googol or
less symbols of first-order set theory. For any computable number you
can name that can be written (using any computable notation whatsoever)
within the observable universe, it will take less than a googol symbols
of ZFC set theory to define it. Rayo's number will always be greater,
unless you write an uncomputable number. ## Related entries • Fast growing hierarchy | ||

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