The space of possible Everythings.
Consider the set of all possible <node-length> strings which are indexed by <title-length> strings, and their connections, established by the normal Everything linking syntax.
Let's pluck some figures out of the air and say node-length is 64k (or less) and title-length is (up to) 256. We also need to define the character set for our strings - let's go not go unicode just yet, and allow only 256 characters.
There are then 25665536 possible node bodies, but only 256256 node titles. Because we index on the title, we are only allowed to use 256256 nodes in any possible Everything.
We could try and solve this by giving each node-text
a unique id number - but then we have the problem that the vast majority of these numbers would be longer than the texts that must contain them in order to link. If we had the numbers in decimal, these would be much longer than the texts themselves (in almost all cases) since the texts are equivalent to 64k digit numbers in base 256, which are shorter than their decimal equivalents.
So we consider instead the space of "possible Everythings", or E-space.
With the arbitrary figures inserted, we get (please see note 1 for the calculation!)
(25665536 + 1)256256
possible Everythings (ignoring 'tiny and niggling considerations' like case-sensitivity and the forbidden html tags.)
This is a large number: just the part in brackets has 157827 digits. This needs to be raised to the power of 256256, or
3231700607 1311007300 7148766886 6995196044 4102669715 4840321303 4542752465 5138867890 8931972014 1152291346 3688717960 9218980194 9411955915 0490921095 0881523864 4828312063 0877367300 9960917501 9775038965 2106796057 6383840675 6827679221 8642619756 1618380943 3847617047 0581645852 0363050428 8757589154 1065808607 5523991239
3038552191 4333389668 3424206849 7478656456 9494856176 0353263220 5807780565 9331026192 7084603141 5025859286 4177116725 9436037184 6185735759 8351152301 6459044036 9761323328 7231227125 6847108202 0972515710 1726931323 4696785425 8065669793 5045997268 3529986382 1552516638 9437335543 6021354332 2960464531 8478604952 1481935558
My calculator blew up.
So I got a better calculator2 ... which tells me
the number is about
105.1 x 10621. That is, it has about 5 x 10621 digits.
This is, interestingly enough, considerably larger than the number of possible histories of the universe which are distinct at the particle level, which has only 2.1 x 10343 digits.(see note 3)
We can think of the current state of Everything, or any of the other 105 x 10621 possible Everythings, as nodes in E-space.
E2 node-edits are edges connecting these E-space nodes: every time a noder nodes or an editor edits, E2 moves from node to node in E-space.
Inside E-space are Everything-instances containing the equivalents of the marvels hinted at in Borges' Library of Babel:
Everything: the minutely detailed history of the future, the best users' autobiographies, the faithful metanode of [its own] Everything, thousands and thousands of false metanodes, the demonstration of the fallacy of those metanodes [...] nodes of apology and prophecy which vindicated for all time the acts of every noder in Everything and retained prodigious arcana for his XP.
(But not, we may add, a compass-and-ruler method of squaring the circle, the full and accurate decimal expansion of pi, the proof of the Continuum Hypothesis in ZFC, the disproof of the Continuum Hypothesis in ZFC ... )
Perhaps, somewhere at the other end of E-space, a lucky noder will find the One True Everything: fully populated, fully connected (?), maximally cool, optimally redundant, every node a perfect node. Would she be "analogous to a god?"
1. (Many thanks to Chris for considerable assistance with the combinatorics, here, which were more complicated and very wrong before I enlisted his aid.)
Call the number of possible node-bodies N and the number of titles T; then there are NT unique fully populated Everythings - each FPE is like a T-length digit in base N.
For an Everything with t titles, there are Nt
choices of node bodies, and T!/((T-t)! t!) ways of picking the combinations of titles and node bodies.
Call an Everything with only one node 1-populated. Then in order to count the total number of Everythings (under this crude schema) we need to sum the number of all t-populated Everythings where t is less than or equal to T, giving
/ Nt -----
Thankfully, this simplifies nicely to (N + 1)T
What we're doing in allowing less than fully populated Everythings is to add one extra node body (call it the null node) and pretend all the unused titles 'point to' it, so each Everything effectively becomes a T-digit number in base N + 1.
2. See Hypercalc.
3. Based on taking 102.1x10343 as "the number of universe-timeline wave-functions that exist simultaneously from the viewpoint of an observer outside our universe." This number is the factorial of the single perturbation count. Information from http://home.earthlink.net/~mrob/pub/numbers-7.html. The number of digits in the number of possible Everythings is about the square of the number of digits in this number.
See large number for expert exposition of tricky exponent questions!
If have you a better name for the 'digits squared' relation, or indeed, if you can add to the model, please /msg me, or add below!