A representation is CombinatoriallyComplete if it doesn't rely on the order of its symbols to express its semantics. Ordinary integers are a good example. There is no information implicit in the permutations internal to the ZF representation of a number; 1+3 == 2+2 == 3+1 == 4 == {}{{}}{{}{{}}}{{}{{}}{{}{{}}}}. An example of a representation that is not CombinatoriallyComplete is the AdjacencyMatrix? of a graph.
Hey, the ZFC 4 looks pretty cool when you write it out longhand like that, don't it? For those who ain't hip to that, you just start with the empty trip on MemesAreDigital which gives you the empty set for a universe - {} which will do for a one - then pull the same trick inside and outside that {}{{}} for a two, then do it to all the empties again to get three {}{{}}{{}{{}}} ... and keep right on subdividing to make a four. You can pretend its rings or you can pretend its PowerSets, which amounts to the same thing for the (zero dimensional) scalars here.
Now we use a whole lot of information day to day, these here words included, that rely on permutations to express their semantics. The notion on MemesAreDigital would require us to find ways to quantize all these permutational birds into a CombinatoriallyComplete representation ... because there's plainly nothing digital about this texty crap you're reading now ...
You might wonder, hey, how could I do that if I had more than one dimension to my number? Shucks, that's easy, LawsOfForm shows you how, go look it up ...
See also CannibalsAndMissionaries, SurrealNumbers

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