Editing Von Neumann universe (section)

===Finite and low cardinality stages of the hierarchy===

The first five von Neumann stages ''V''<sub>0</sub> to ''V''<sub>4</sub> may be visualized as follows. (An empty box represents the empty set. A box containing only an empty box represents the set containing only the empty set, and so forth.)

[[File:Von Neumann universe 4.png|upright=3<!--[[WP:IAR]]: 1.8 would be too small-->|frameless|center|First 5 von Neumann stages]]

This sequence exhibits [[tetrational]] growth. The set ''V''<sub>5</sub> contains 2<sup>16</sup> = 65536 elements; the set ''V''<sub>6</sub> contains 2<sup>65536</sup> elements, which very substantially exceeds the [[Observable universe#Matter content—number of atoms|number of atoms in the known universe]]; and for any natural ''n'', the set ''V''<sub>''n''+1</sub> contains 2 ⇈ ''n'' elements using [[Knuth's up-arrow notation]]. So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage 5. The set ''V''<sub>&omega;</sub> has the same cardinality as &omega;. The set ''V''<sub>&omega;+1</sub> has the same cardinality as the set of real numbers.