Wednesday, July 01, 2009

The Dark Theorem of Economics

Tyler Cowen:

Christopher J. Ayres sends me a paper.with the following abstract:

I begin by proving the "Dark Theorem of Economics," from which it follows that the foundations of economic theory rely on the Axiom of Choice (AC). All current solution concepts in game theory
also require the theorems implied by AC. In particular, lexicographic utility, lexicographic probability, the real line being well-ordered, and the existence of a universal space are all equivalent to AC; therefore any argument to disprove their existence must be false. Any proofs using properties that fail under AC must be redone. The concept of Nash Equilibrium becomes either a tautology (in the absence of AC) or violates rationality (in the presence of AC); we provide an example demonstrating this. Knowledge, Common Knowledge, Epistemics, Game Theory, and Macroeconomics (through the failure of Rational Expectations) must be rebuilt. Any economics …field or concept relying on these must also be rebuilt. I begin this process with the de…finition of "Fundamental Game."

I joked with Chris that the other people who pursued this line of inquiry met with unfavorable ends. But that doesn't mean he is wrong. Fortunately, I am a pragmatist when it comes to the foundations of economic theory or lack thereof. It's hard enough to define what a number is, so if you push on the foundations of micro theory, don't expect a completely comfortable journey.

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