Power-tower paradox

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.Robert Munafo. Power-tower paradox Retrieved 2023-09-03. 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^{10.0000000000099968384}}}
\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 precision limits of standard floating-point representations used in modern 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.
Sbiis Saibian calls this paradoxical property "pandemonium with power towers" and derived several lemmas.Sbiis Saibian. 3.3.1 Pandemonium With Power Towers Large numbers. Retrieved 2023-09-03. Fish used Taylor expansion for the first-order approximation of the power tower.Fish. Approximation based on power-tower paradox 2023-09-03.
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 ==
Category:Paradoxes
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