I'm not a mathematician, so go easy. I dropped out of high school. But I'm a programmer / logician / writer of algorithms. And I'm currently engaged in one of my most complex projects to date, which involves building up something like Markov chains within one another to lead people to find unexpected, complex results out of their own psychological preferences.
I'm raising this question not because I need help with my project, but because I stumbled on an unexpected feature of my data, and then had a corresponding dream about it.
The feature I stumbled on was that it's quite easy to record all the events and conditions leading up to a happening, but it's quite another thing to record all the ones that didn't lead to it, let alone all the ones that didn't lead to something not happening. And let's not forget all the ones that didn't lead to something not happening which caused a whole chain of other things not to happen.
The dream In the dream, K[0] is a set of things that had never been imagined happening. K[1] is a set of things that you could imagine, because they diverged from the chain of events at some point where you could still conceive of their results. But they didn't actually happen. K[2] is the much smaller set of things that actually did happen. From K[2] you can begin to infer K[1]. As soon as you imagine something from K[0], it becomes part of K[1]. For all intents and purposes, K[0] is infinite, even after you imagine parts out of it. K[1] is only limited by your imagination, and K[2] is the reality upon which your imagination is based.
So in this dream, K[1] acts as a kind of thicker or thinner membrane around reality, describing the contour of K[0] which most closely match K[2]. Depending on how much of K[0] you can imagine and reference to something happening in K[2].
What I'm interested in is, is what is the relationship between these sets?
For example, how can you prove that K[1] doesn't exist until you imagine it, and that K[0] is still unknown even though you can pull an element out randomly, by imagining it, and insert it into K[1]?
Apologies if this makes no sense, just a human brain trying to make order out of chaos.