09-14-2026, 04:02 PM
Summary
Joel David Hamkins’s March 2025 Notices of the AMS article, “The Nearly Perfect Prediction Theorem,” presents a striking result of Christopher Hardin and Alan Taylor: there exists a prediction strategy that can predict the present—and even the immediate future—value of every function almost always correctly. The surprising part is that the strategy requires no assumptions such as continuity, smoothness, or randomness about the function. Instead, one fixes a well-ordering of all possible functions and, whenever the past history is known, chooses the earliest function in that ordering that remains compatible with the observations. The resulting prediction is wrong only on a countable, nowhere-dense set of exceptional points.
The theorem becomes even more counterintuitive because the exceptional failures are extremely sparse: after every real time ss, there is an interval on which the predictions are correct, and there can be arbitrarily long intervals of perfect prediction. The underlying argument is based on the fact that whenever a prediction fails, the chosen “best” compatible function must move upward in the fixed well-ordering; such changes cannot happen too often because a well-order does not admit an infinitely descending chain. The result is therefore less a practical forecasting method than a beautiful example of how set theory, the axiom of choice, and an abstract ordering of mathematical possibilities can produce almost-perfect prediction without any computational access to the future.
Main takeaway: the theorem does not mean that we can realistically predict the stock market, weather, or arbitrary physical systems. Its power is mathematical: it demonstrates that, under a highly nonconstructive choice principle, “prediction” can be extraordinarily accurate in a formal sense even when the function being predicted is completely arbitrary.
ARTICLE
Joel David Hamkins’s March 2025 Notices of the AMS article, “The Nearly Perfect Prediction Theorem,” presents a striking result of Christopher Hardin and Alan Taylor: there exists a prediction strategy that can predict the present—and even the immediate future—value of every function almost always correctly. The surprising part is that the strategy requires no assumptions such as continuity, smoothness, or randomness about the function. Instead, one fixes a well-ordering of all possible functions and, whenever the past history is known, chooses the earliest function in that ordering that remains compatible with the observations. The resulting prediction is wrong only on a countable, nowhere-dense set of exceptional points.
The theorem becomes even more counterintuitive because the exceptional failures are extremely sparse: after every real time ss, there is an interval on which the predictions are correct, and there can be arbitrarily long intervals of perfect prediction. The underlying argument is based on the fact that whenever a prediction fails, the chosen “best” compatible function must move upward in the fixed well-ordering; such changes cannot happen too often because a well-order does not admit an infinitely descending chain. The result is therefore less a practical forecasting method than a beautiful example of how set theory, the axiom of choice, and an abstract ordering of mathematical possibilities can produce almost-perfect prediction without any computational access to the future.
Main takeaway: the theorem does not mean that we can realistically predict the stock market, weather, or arbitrary physical systems. Its power is mathematical: it demonstrates that, under a highly nonconstructive choice principle, “prediction” can be extraordinarily accurate in a formal sense even when the function being predicted is completely arbitrary.
ARTICLE
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│ KONSTANTINOS MICHAILIDIS │
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│ KONSTANTINOS MICHAILIDIS │
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