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Power-tower paradox

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The power-tower paradox is an apparent paradox (so-named by Robert Munafo) that arises in the calculation of very large numbers, usually as the result of a rounding error.[1] The paradox is visible in Hypercalc and other software using a level-index system to represent numbers.

A (rather small) example of the paradox would be an approximation of \(50^{10^{10^{10}}}\) using a "stack" of base-10 exponents:

\begin{eqnarray*} 50^{10^{10^{10}}} &=& 10^{\log_{10}50 \cdot 10^{10^{10}}} \\ &\approx& 10^{1.698970004 \cdot 10^{10^{10}}} = 10^{10^{\log_{10} 1.698970004 + \log_{10} 10^{10^{10}}}} \\ &=& 10^{10^{\log_{10} 1.698970004 + 10^{10}}} \\ &\approx& 10^{10^{0.230185711 + 10^{10}}} = 10^{10^{10000000000.230185711}} \\ &=& 10^{10^{1.0000000000230185711 \cdot 10^{10}}} = 10^{10^{10^{\log_{10} 1.0000000000230185711 + \log_{10} 10^{10}}}} \\ &=& 10^{10^{10^{\log_{10} 1.0000000000230185711 + 10}}} \approx 10^{10^{10^{10.0000000000099967971146562514}}} \end{eqnarray*}

To compare this against \(10^{10^{10^{10}}}\) would take high levels of precision and even higher levels for larger "stacks," so a computer with low precision may believe that \(50^{10^{10^{10}}} \approx 10^{10^{10^{10}}}\). Munafo's original example required about 100 digits of precision to compare accurately; far beyond the scope of most modern desktop computers.

This paradox is a result of the fact that alterations to the "top" of a power tower are amplified in the full calculation of the tower, and this amplification becomes larger when the tower becomes taller.

The phenomenon is very common in almost all googological functions, not just power-tower arithmetic. For example, Graham's number is \(3 \{\{1\}\} 65\) (expansion) with the middle \(3\) replaced with \(4\). It is tempting to say that \(g_{64} \approx 3 \{\{1\}\} 65\), but in fact \(g_{64}\) is arithmetically much larger. We can only say that \(g_{64}\) is "in the neighborhood of", fundamentally or "googologically" close to, or "comparable" to \(3 \{\{1\}\} 65\); googologists will usually write \(g_{64} \approx 3 \{\{1\}\} 65\) anyway.

Sources Edit

  1. Power-tower paradox

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