The basis for continuous hierarchy

The basis for continuous hierarchy is that things change in time. Anything in physical reality is different every time you measure it--every concrete set is different. The question is, do things we think of (abstract sets) change in time when we measure them. Our understanding of a situation may change. We may get more insight into ourselves or a situation. We have a set in our minds that is the set of all squares. There are an infinite number of squares in the square set--assuming our mental universe is unbounded in size. There is no such thing as a concrete set of squares...they are all imperfect examples of something in the abstract set of squares.

Continuous hierarchy is not about abstract sets, it's about concrete sets. I cannot argue about stuff that is infinite in nature. It would take me too long.

Measurement and Continuous Hierarchy

Since measurement is an essential component of continuous hierarchy, let us consider two measurements made at exactly the same time at a set. By the uncertainty principle, observation affects the measurements. So only one observer is different than two observers. The question is, will the two observers achieve the same measurement? We have set A, observer o1, and observer o2. They observe at time t. Thus:

A(o1, t) - A(o2, t) = {}(o1 - o2, t) Thus the effect of time is negligible (but it might affect the empty set), but the effects of the differences between the observers implies that we don't get the empty set. If o1 = o2, then we get the empty set.

Deltas between Sets Measured in Time

In continuous hierarchy, set subtraction computes a delta between sets. We don't yet know if this result is a set or not. For example, if you have a set A, and the subtraction operator -, the A - A is not the empty set. Each time you measure A, you get a different result, so the two references to A are references to different sets in time. So you might look at the equations A(t1) - A(t2) = {}(t2 - t1) As t1 approaches t2, A - A becomes the empty set.

Differential Recursion, Real Recursive Functions lead to Continuous Hierarchy

So, if you consider taking the time step in a recursive function to the smallest possible time, say dt, we get a smooth recursive function. This is "real recursive functions" or "differential recursion." I claim that if you apply this same technique to sets, you will get continuous hierarchy. What is the delta between sets? Reducing the delta between sets to something close to 0 will lead to continuous hierarchy.

Self-Reference versus Self-Awareness

It seems like I've been chasing after self-reference for quite a while, and now I need to chase after self-awareness.

Continuous Hierarchy: What we are trying to achieve

With continuous hierarchy, we are trying to get rid of recursion. The universe is not recursive, it is continuous. What we need is a continuous recursion, without levels. This works out to continuous self-reference, instead of leveled self-reference. Or no self-reference at all. No self-reference at all seems rather impossible at this point--people refer to themselves all the time, when they say "I", so let's attempt continuous self-reference. Continuous self-reference seems like narcissism, or schizophrenia, but without the mirror.

metaghastcar

So if you had a car that ran on meta, how far would it go?