Why does DD matter?

DD (discrete derivative) matters because the tree built to produce it row after row goes on to infinity.

It would be impossible to talk about the whole set of binary sequences, both finite and infinite, if we tried to do it individually. No, we need to see the whole set at once. How else besides looking at its tree structure? Isn’t that what Cantor “proved”?

That structure is what produces the set. Understand the tree structure and you understand the set. (By the way, when I use the word “set” it is most certainly not what Cantor and the rest had in mind….this set goes off to infinity…while its structure is fixed, the set itself is never complete..it is “bounded” only by its structure, not by any other sort of non-moving horizon…like axiom, the “member relation”, etc….there are “holes” all the way down)

The reason the DD tree is so important is that the ordering of binary strings it produces has the distinguishing feature of being both “linear” in the sense that there is a one-after-the other aspect (endowed by its relation to the Dyadic tree where a linear order is endowed by its parent/child relation to the natural numbers in binary) while guaranteeing that neighbors differ by exactly one bit change. It is that “oneness” that reveals its genetic connection to N and yields countability.

The role of the anti-DD tree is to tie it all together…we have an operator (the DD tree) and its inverse (the anti-DD tree)…we get a group: identity (the Dyadic tree). We get a model. Thank you, Professor Skolem.