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Realistic Assumptions and Godel

A good post from Peter Dorman about the dangers of unrealistic assumptions.   This, however, I disagree with:

 I take it as axiomatic that the economic world is far too complex and variegated to be comprehended or forecasted by any single model.  Sometimes one set of factors is paramount, and particular model captures its dynamic, and then another set takes over, and if you continue to follow the first model you’re toast.  There are complicated times when you need a bunch of models all at once to make sense of what’s going on, even when they disagree with each another in certain respects. [emphasis mine]

That right there is the problem.  You can’t take it as an assumption that no finite set of assumptions can describe the world.  You need to do better than that.   I’ve been dying for someone to try and make this argument since this chameleon paper came out.

Setup

Let’s think about this carefully.  Let’s suppose, for simplicity, that the world can be described by the unit interval, [0,1].   For the moment don’t worry too much about what I mean by this, it’ll become clear later.

I want to describe this world, what happens in it, given some initial condition x in [0,1], and so I develop a model with some set of assumptions, which together imply that the domain for validity of that model is some subset, S, of [0,1] (for simplicity, also assume that S is connected).  That is, since I have made assumptions with my model, not every set of initial conditions can be used to map to a prediction in a well-defined way.

Cantor’s Description of [0,1]

OK.  So here’s my question: given a finite vocabulary to describe the set of initial conditions, can I exactly specify the boundary of S?  The answer is no.

To understand why, consider describing S with the following vocabulary: start with the entire interval [0,1]; if a point is in the left half of the interval assign it as a “0”, otherwise assign it a “1”; repeat this process for each half interval and assign the {0,1} to the left of the string you’ve already produced.   Basically, I’m describing [0,1] as a 2-adic number, so the point 0 on [o,1] is written as ….000000.   Which you might read as the instruction “to find point 0, always take the left interval”.

Could you describe the boundaries of S using this method?

Assumptions as p-adic numbers

You might be tempted to look at this set-up and ask yourself why I’m describing [0,1] in such an inefficient way.   The reason is that my description of [0,1] using 2-adic numbers is isomorphic to the use of axioms.

Think of it this way.  An axiom is a string of letters/symbols.   There are a finite number of letters/symbols and I string a finite number of them together to make arbitrarily long statements.  That means that I can assign every possible axiom a natural number.   Let’s do that.

So, consider that if I’m using only axiom 2 and no others.   I can write that as the set of all 2-adic numbers with a “1” in the second position; i.e. {…010,…011,…110,…111}.  This is a cantor set, and it is also the domain of validity of the assumption 2 (as I have set things up).   There’s no way that a finite set of these things could describe the boundary of S.

What’s the Take Away?

I started off with a model with a well-behaved domain of validity S.   It was nice and well-behaved.   Then, I explored trying to describe that domain of validity using a finite language and finding that the two don’t match.

What does that mean?  I don’t have to assume that no finite set of axioms can describe our economic system, and its not just that the economic system is complex (although it undoubtedly is).  Every model is an approximation by its very nature, because our ability to describe that model is an approximation.

This argument isn’t a proof, I’d need to clean things up a bit, but the intuition here is basically the same as the Godel incompleteness theorem (as an aside, I use the same basic reasoning as the basis for my moral theory, but that’ll have to wait until I write a post about my philosophy, which I plan to do someday…).  The only real insight is that a scientific model is a kind of Turing Machine and so every model has an approximation problem.

 

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