Wikipedia is a madhouse. I get lost in articles everytime I go there, and today was no exception. While doing a bit of brush up for writing this, I got sidetracked into reading up on Non-standard Analysis and Hyperreal Numbers. I’d like to note that I think there’s a set-theoretical definition of an infinite number.

Assume some definition of sets. I’m not going to get into deciding on which particular system for defining ‘set’ is best, I think this applies to all. The first thing I need is a concept of the Natural Numbers, N. To provide this, I don’t think I’ll use the standard model of attempting to define a set which defines a natural number. That seems to me to be wholly circular. A natural number seems intuitively to describe a relation between sets, and therefore would best be described as a function. The set of natural numbers would be the set of values denoted by a size function spanning the domain of all sets. It seems apparent that a value is not a set, and therefore the set of natural numbers cannot be a set of sets. That, for me, calls into question the conventional definition.

How, then, can we define a natural number in order to describe the set of all natural numbers? I would suggest we do so by counting, which is the main purpose of natural numbers anyway. One simple way to progress would be this: we may construct a new set, N, whose members denote different sizes of sets. To do so, we start with the member 0, which denotes the size of the empty set.

We know that for any non-empty set, s, there exists at least one subset whose only subsets are the empty set and itself. From any set we examine, remove one such subset. Add to our description of N a new member: 1. If the set left after the removal of the subset is empty, the function Size(s) maps to 1. Otherwise, remove another such set and add a new member to N: 11. Size(s) maps to 11. Repeat this procedure, adding a new member to N with a new 1 attached or until all that remains is the empty set. The final addition to N is the size of set s.

This is equivalent to me creating a set N whose members are symbolic constructions which are assembled according to a simple rule. The generated set is not, I’d argue, the set of natural numbers. It is merely a set whose construction is readily understandable to us and about whose members we can easily deduce a great deal. Having generated this set of symbols, we may then use the above algorithm to generate a function, Size(), which maps from (a subset of {sets}->N). I’d say that the set of Natural Numbers, **N**, is the set of all such mappings.

I say Size() maps from (a subset of {sets} -> N) in order to avoid including infinite sets. Specifically, Size(N) is not an element of N. This is demonstrable by noting that we generate a new element of N for every element we remove. Also of note is that, for the generated subset of N, n, from Size(s), Size(n) ~= Size(s). That’s because of the inclusion of the 0 symbol, to provide a size for the empty set.

To return to the problem of Size(N), which is equivalent to the question of Size(**N**), we run into cardinality issues. What’s interesting is there seems no way to generate a new set, say R, such that Size(N) where Size maps from (subset of {Sets} -> R) maps to a member of R and Size(R) also maps to a member of R. That’s pretty much the essence of Cantor’s Theorem.

And finally, I wanted to touch on the Second Derivative, which has come up a lot in economic discussions about the current economic downturn. The equity markets are a pretty common proxy for the health of the economy, so let’s examine the second derivative in terms of equities. First, what’s being discussed is really standard calculus/physics. The dimension being analyzed is position as a function of time: p = f(t), where p is position and t is time. From there, the first derivative is velocity, change in position per change in time: v = f'(t). The second derivative is acceleration: the change in velocity per change in time: a = f”(t).

A common recession is often described as “U” shaped, which is ideally described by a square: p = f(t) = (m)t^2 + n(t) + c where m, n, and c are constants that morph the shape of the U. This is known as a quadratic equation and if you took Algerba 2, you probably spent a lot of time working on it. If you took physics, you likely worked on things like constant acceleration from gravity and may recall that this produces a quadratic equation.

Well, the second derivative of a quadratic is a constant. Since the economists are hellbent on finding a change in the second derivative, from positive to negative, I have to assume they are not, in fact, talking about a quadratic function describing change in value of equity indices.

By discussing the second derivative, I get the nasty feeling that they’re bandying about math terms that don’t make as much sense for their data as we might wish. Perhaps they are discussing us having had an L-Shaped recovery, in which case acceleration is something like (1/t^3). Think of that like a skateboarder on a half-pipe. The component of their motion which is affected by acceleration declines as time passes, which can be seen by tendency of p towards 0 as t -> infinity.

I feel like the examination of technical information fails to capture properly the manner of change in an economy. Specifically, what if the value of the equity market was driven by multiple variables? In that case, examining the value of the second derivative, while it will provide insight into changes in velocity, may not give us the full picture.

In addition, velocity may be changed by a variety of factors. An externality which induces a constant velocity against drag, say throwing a ball up into the air, will demonstrate an initial reduction in velocity to 0 followed by an increase in the absolute value of velocity. Acceleration remains constant. In the L-shaped case, acceleration declines towards 0, but remains positive. Velocity is also positive, but tends towards 0.

So I’m left to wonder about the relevance of this”second derivative” beyond a fancy way of pointing at the much more mainstream, and therefore vague, acceleration.