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Cake day: February 6th, 2024

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  • If the growth is superexponential, we make it so that each successive doubling takes 10% less time.

    (From AI 2027, as quoted by titotal.)

    This is an incredibly silly sentence and is certainly enough to determine the output of the entire model on its own. It necessarily implies that the predicted value becomes infinite in a finite amount of time, disregarding almost all other features of how it is calculated.

    To elaborate, suppose we take as our ā€œbase modelā€ any function f which has the property that lim_{t → āˆž} f(t) = āˆž. Now I define the concept of ā€œsuper-fā€ function by saying that each subsequent block of ā€œvirtual timeā€ as seen by f, takes 10% less ā€œreal timeā€ than the last. This will give us a function like g(t) = f(-log(1 - t)), obtained by inverting the exponential rate of convergence of a geometric series. Then g has a vertical asymptote to infinity regardless of what the function f is, simply because we have compressed an infinite amount of ā€œvirtual timeā€ into a finite amount of ā€œreal timeā€.