The Vsauce numbers were coined (not explicitly) by Michael "Vsauce" Stevens in a YouTube video discussing the different types of "infinity".[1] They were used to emphasize the fact that \(\aleph_0\) can never be reached by any finite operation of finite numbers. In the order they appeared:

\(\text{Vsauce}_1=\left((\text{googolplex}!)^{\text{googolplex}^\text{googolplex}}\right)^2\times G\)

\(\text{Vsauce}_2=\mathcal P(\text{millinillion})^{\text{googolplex}^{\text{googolplex}^\text{googolplex}}}\)

Since these numbers were only spoken aloud, there is some ambiguity as to their exact values. While they are thus subject to some variation, the intonations while saying each number were interpreted to produce the above expressions. The numbers and constructions he referenced (for example millinillion and factorial), in order to convey a feeling of enormity, consequently rendered the Vsauce numbers as salad numbers. Even if they are interpreted differently in a way that maximizes their values, \(\text{Vsauce}_1\) will barely exceed Graham's number, easily dominated by \(g_{65}\), and \(\text{Vsauce}_2\) is vastly exceeded by, say, googolquinplex.

Approximations in other notations

Notation Vsauce1 Vsauce2
Chained arrow notation \(3\rightarrow3\rightarrow64\rightarrow2\) \(10\rightarrow7\rightarrow2\)
Graham's function \(g_{64}\) \(g_1\)
BEAF \(\{3,3,64,2\}\) \(\{10,7,2\}\)
Hyperfactorial array notation \((64![1])!\) \(10!1\)
Fast-growing hierarchy \(f_2(f_{\omega+1}(64))\) \(f_3(7)\)
Slow-growing hierarchy \(g_{\Gamma_0\varepsilon_0}(65)\) \(g_{\varepsilon_0}(6)\)

Sources

  1. Stevens, Michael. How To Count Past Infinity (timestamps are 3:25 and 19:50, respectively). Retrieved 2019-06-19.
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