On the number horizon 7 - bigO and God's Number
Man is a beast of addition and multiplication, it's hard for him to imagine what comes after, what comes before and what's inbetween his two most familiar recursive operators.
| operators | examples |
|---|---|
|
0 = N(x) = x' N(-') = 0 |
N(0) = ' = 1 N(1) = '' = 2 N(n-1) = '''... = n |
| O(...) = O(...,0) |
O() = O(0) = 0 O(x) = O(x,0) = x |
|
O(x,n) = N(N(..N(x).)) = x + n O(x,-n) = -O(-x,n) = x - n |
0(2,3) = ''''' = 5 0(0,0) = 0 + 0 = 0 0(1,-1) = 1-1 = -0 |
|
O(x,n,1) = O(x,O(x,....O(x,x).)) = x * n O(x,n,-1) = O(x,O(n,-1,2),1) = x/n |
2*3 = 2+2+2 = 6 1000/10 = 100 |
|
O(x,n,2) = O(x,....O(x,x,1)...,1) = x ^ n O(x,n,-2) = log(x)/log(n) |
2^3 = 2*2*2 = 8 log(100)/log(10) = 2 |
|
O(x,n,3) = O(x,....O(x,x,2)...,2) = x ^^ n |
2^^3 = 2^(2^2) = 16 n^^-1 = 0 |
|
O(x,n,4) = O(x,....O(x,x,3)...,3) = x ^^^ n |
2^^^3 = 2^^(2^^2) = 65536 n^^^-1 = 0 n^^^-2 = -1 |
|
O(x,n,p+1) = O(x,....O(x,x,p)...,p) = x ^^^... n |
O(2,2,p) = 4 O(2,3,p) = O(2,O(2,3,p-1),p-2) O(x,0,p) = 1 {p>0} O(x,1,p) = x {p>0} |
|
B(0) = 1 B(i) = O(B(i-1),B(i-1)) B(n) = O(2,n,2) = O(n,n,3/2) |
B(3) = 2^3 = 8 B(4) = 2^3 + 2^3 = 2^4 = 16 B(-1) = 1/2 B(1/2) = 2^(1/2) |
|
B(n,1) = B(B(..B(1).)) = O(2,n,3) |
B(1,1) = B(1) = 2 B(4,1) = B(B(B(B(1)))) = 2^(2^(2^2)) = 2^^4 = 65536 |
To approximate the number that fits the universe you
take as unit the smallest possible length, the
Planck length of
L = 1.616252E-35 meter,
of which there are
L^-3 = 2.3685E104
in a cubic meter of space, and
9460730472580800*L^-3 = 2.2408E120
in a cubic light year.
Multiply that by the current
size of the universe,
which is a
sphere with a volume of
4/3*pi*(156E9/2)^3 = 2E33
cubic light-years,
then you get the current total of
5E153
Planck volume units in the universe.
Then there's the fourth dimension, which has a
Planck time
of
T = 5.391214E-44
second, or
1/T*60*60*24*365 = 5.85E50
Planck time units per year.
Multiply that by the current
age of the universe
and by its volume, and take roughly a quarter, then you have
5.85E50*13.7E9*5E153/4 = 1E214
possible Planck space-time events up to the present.
The hidden 7
curled up dimensions of String theory are according to a generous estimate at most
0.15 millimeter big, in total
(1.5E-3/L)^7 = 1E224
extra hidden Planck units.
The current string space with 11 dimensions would then spit out a universal number of at most
1E214*1E224 = 1E438 Planck units.
If every Planck unit needs to be represented separately as 0 or 1, or if indeed every Planck unit means a choice, a binary probability, instead of a fixed state, then up to the present the universe has received a number of choices in the range
4^^4 < 1E1E438 < 5^^4.
You can speculate about the end of space-time, but there are most likely no interesting phenomena to occur when the universe has reached its
final state,
that is if all matter has collapsed into neutron stars and black holes.
According to the maximum estimate that final collapse is reached 10^10^76 years from now, if the universe is still there...
In the end the total number, as the universe keeps expanding in 11 dimensions,
is maximally
1E438*(10^(10^76-51))^11 = (1E1E76)^11 = 1E1E836
Planck units big.
(If your universe was part of a
multiverse
with less than
1E1E835 universes with similar laws,
the total of Planck-like units would still be about
1E1E836)
In
holographic
theories of the universe every choice is not binary, but creates a whole new universe itself. Then the final number of the holographic universe would be in the range
4^^5 < 1E1E1E836 < 5^^5.
This proves that our universe is not very big compared to the countable numbers we can create with the first three operators of the bigO function O(a,b,c).
Systems rule above numbers, a system number horizon is typically an operator level bigger than the number horizon for a free variable, as a system contains these numbers as choices. Else the holes inbetween the variable level numbers will get too big or remain too small, and the system level cannot relate the variable level numbers anymore, as was hinted at in last weblog.
Our universe or physical multiverse, because it appears physically as a bunch of free variables bound by laws, must be a sub-system of a bigger system
ruled by a God if you will, a
Brahma heaven.
Then that Brahma heaven system, to still have some grip on the lower level physical universe,
can be represented by a number that is only an operator higher than ours.
About O(4,4,4) = 4^^^4 will do,
but not exactly...
God's number,
as it represents His super-system, will have such an amount of apparent randomness that no mathematical formula or number system in this universe can express it exactly.
I call my estimate
O(4,4,4)
the Alpha-4-Mann number, because it is related to the
Ackermann numbers.
The best estimate of the Number of God is hereby given.
posted on Thursday, October 25, 2007 6:37 PM
Feedback
# re: On the number horizon 7 - bigO and God's Number 10/26/2007 3:01 PM marco
Leuk, maar ik ben er nog niet aan toegekomen om deel 1 tot 6 te internaliseren.
zie verder http://blogger.xs4all.nl/mxlml/archive/2007/10/16/305674.aspx onderaan
# re: On the number horizon 7 - bigO and God's Number 10/26/2007 3:31 PM marco
A bit strange to equate units of length with three dimensional (cubic or spherical) space. You should have taken the radius of the universe instead while that is the length of time, plus or minus the initial acceleration of the universal core before the so called big bang.
# re: On the number horizon 7 - bigO and God's Number 10/26/2007 4:46 PM Fran6 van NovaLoka
Hi Marco, biertje al achter de kiezen?
The initial cosmic inflation that happened 10^-35 seconds after the Big Bang is left out of the equation. Indeed I first calculated the present state of the cosmos (without the errors you mention).
Cosmic inflation expanded the universe by a factor 10^26 (still not much). But I don't know if inflation continued beyond the boundaries of our universe, or if Planck units are appropriate in that 'region'.
http://en.wikipedia.org/wiki/Big_Bang#Overview
http://en.wikipedia.org/wiki/Cosmic_inflation
My work on numbers here suggests cosmic inflation might be the result of 'setting up' the universe as a number system:
- First the numbers are small, exact and clear. The size of formulable numbers increases very fast though: inflation rules.
- Then a number horizon is reached where most numbers are uncertain and probabilistic: quantum mechanics rule.
- In the last phase the holes that fall inbetween probabilities ripps the universe apart: phantom energy rules.
Highly speculative, I agree...
Cheers!
# re: On the number horizon 7 - bigO and God's Number 1/4/2008 3:15 PM Richard Gill
This is related to Seth Lloyds work on the ultimate laptop. What is the greatest computing power which you can get out of - say - 1 Kilo of matter? The answer - it will take about 100 years by Moore's law, till we get there.
http://www.ar-tiste.com/qcomp_onion/jan2002/UltimateLaptop.htm
It is fascinating how many quite different approaches to computing the computing power of the universe come up with very similar answers.
Since the universe is finite and discrete we just need a finite number of binary random choices to simulate the whole universe. I call this sequence of bits, the G number (Gill, or God perhaps?). Given this number, the course of the whole universe is totally deterministic.
Well thanks for telling me how long the G number is!
# re: On the number horizon 7 - bigO and God's Number 1/4/2008 3:40 PM freeZotic van NovaLoka
Might be..
I hear 't Hooft doubts whether the number of directions in a Planck-size volume is 3 degrees of freedom.
't Hooft thinks it is just 1 degree of freedom, so then the Gill number would be 2 exponents less, but then Gill came to talk with him for half an hour and now he is confused ;-|
The idea is, the universe can be isomorphic to a number that is
"not so big".
And that systems that use numbers as input variables are Gödel-numerically not so much bigger than the input variable itself.
I just make a "tongue in cheek" guess as to the size of the God-universe-system.
Of course if the universe is an input number, so can the God-universe-system that uses this number be an input number of an even bigger system. And so on... POLYTHEISM!
All these systems are easily expressible by my bigO O(a,b,c) function as explained above.
But the first three variables of bigO are just the start!
(Notice all these are 'only' countable numbers :o)
# re: On the number horizon 7 - bigO and God's Number 1/4/2008 4:04 PM freeZotic van NovaLoka
Man exists, but not without the universe. I guess even a foton implies the whole history of the universe, to describe it.
I propose an upper limit to the size of the description of the universe: your G number, my universe-number.
Then a cascade of bigger upper limits represent "God systems" that use smaller universe-numbers as input variables.
In such worlds the Gods may live.
The numbers exist, so why not the universes they perfectly represent?