Talk:Von Neumann universe

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Does that mean that every element of every set is always a set itself, too? Are sets like 'the set of current or former U.S. Presidents', whose elements aren't sets, disallowed? If so, what the f*** does it mean "The band Metallica has four members"? The most obvious answer is that the sets {James Hetfield; Lars Ulrich; Kirk Hammett; Robert Trujillo} and 4 = {0; 1; 2; 3} = {{}; {{}}; {{}; {{}}}; {{}; {{}}; {{}; {{}}}}} are equipotent. But here I learn that the former is not a set, as its elements aren't themselves sets, so what does it mean?

In the Von Neumann universe, if a set has any elements at all, then the elements must themselves be sets. "Sets" containing elements which are not sets are not allowed in this universe by definition. What you are probably looking for are universes with urelements, which allow sets containing members that are not themselves sets. - Gauge 05:46, 24 Jan 2005 (UTC)