Von Neumann universe

The cumulative hierarchy is a collection of sets V_α indexed by the class of ordinal numbers; in particular, V_α is the set of all sets having ranks less than α. Thus there is one set V_α for each ordinal number α. V_α may be defined by transfinite recursion as follows:

• Let V₀ be the empty set:

  ${\displaystyle V_{0}:=\varnothing .}$

• For any ordinal number β, let V_β+1 be the power set of V_β:

  ${\displaystyle V_{\beta +1}:={\mathcal {P}}(V_{\beta }).}$

• For any limit ordinal λ, let V_λ be the union of all the V-stages so far:

  ${\displaystyle V_{\lambda }:=\bigcup _{\beta <\lambda }V_{\beta }.}$

A crucial fact about this definition is that there is a single formula φ(α,x) in the language of ZFC that states "the set x is in V_α".

The sets V_α are called stages or ranks.

An initial segment of the von Neumann universe. Ordinal multiplication is reversed from our usual convention; see Ordinal arithmetic.

The class V is defined to be the union of all the V-stages:

${\displaystyle V:=\bigcup _{\alpha }V_{\alpha }.}$

An equivalent definition sets

${\displaystyle V_{\alpha }:=\bigcup _{\beta <\alpha }{\mathcal {P}}(V_{\beta })}$

for each ordinal α, where ${\displaystyle {\mathcal {P}}(X)\!}$ is the powerset of ${\displaystyle X}$.

The rank of a set S is the smallest α such that ${\displaystyle S\subseteq V_{\alpha }\,.}$ Another way to calculate rank is:

${\displaystyle \operatorname {rank} (S)=\bigcup \{\operatorname {rank} (z)+1\mid z\in S\}}$.

Finite and low cardinality stages of the hierarchy

The first five von Neumann stages V₀ to V₄ 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.)

First 5 von Neumann stages

This sequence exhibits tetrational growth. The set V₅ contains 2¹⁶ = 65536 elements; the set V₆ contains 2⁶⁵⁵³⁶ elements, which very substantially exceeds the number of atoms in the known universe; and for any natural n, the set V_n+1 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_ω has the same cardinality as ω. The set V_ω+1 has the same cardinality as the set of real numbers.

The cumulative type hierarchy, also known as the von Neumann universe, is claimed by Gregory H. Moore (1982) to be inaccurately attributed to von Neumann.[15] The first publication of the von Neumann universe was by Ernst Zermelo in 1930.[7]

Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory[16] and von Neumann's own set theory (which later developed into NBG set theory).[17] In neither of these papers did he apply his transfinite recursive method to construct the universe of all sets. The presentations of the von Neumann universe by Bernays[10] and Mendelson[11] both give credit to von Neumann for the transfinite induction construction method, although not for its application to the construction of the universe of ordinary sets.

The notation V is not a tribute to the name of von Neumann. It was used for the universe of sets in 1889 by Peano, the letter V signifying "Verum", which he used both as a logical symbol and to denote the class of all individuals.[18] Peano's notation V was adopted also by Whitehead and Russell for the class of all sets in 1910.[19] The V notation (for the class of all sets) was not used by von Neumann in his 1920s papers about ordinal numbers and transfinite induction. Paul Cohen[20] explicitly attributes his use of the letter V (for the class of all sets) to a 1940 paper by Gödel,[21] although Gödel most likely obtained the notation from earlier sources such as Whitehead and Russell.[19]

There are two approaches to understanding the relationship of the von Neumann universe V to ZFC (along with many variations of each approach, and shadings between them). Roughly, formalists will tend to view V as something that flows from the ZFC axioms (for example, ZFC proves that every set is in V). On the other hand, realists are more likely to see the von Neumann hierarchy as something directly accessible to the intuition, and the axioms of ZFC as propositions for whose truth in V we can give direct intuitive arguments in natural language. A possible middle position is that the mental picture of the von Neumann hierarchy provides the ZFC axioms with a motivation (so that they are not arbitrary), but does not necessarily describe objects with real existence.

• Bernays, Paul (1991) [1958]. Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.
• Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York: Dover Publications. ISBN 978-0-486-46921-8.
• Gödel, Kurt (1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I". Monatshefte für Mathematik und Physik. 38: 173–198. doi:10.1007/BF01700692.
• Gödel, Kurt (1940). The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory. Annals of Mathematics Studies. Vol. 3. Princeton, N. J.: Princeton University Press.
• Howard, Paul; Rubin, Jean E. (1998). Consequences of the axiom of choice. Providence, Rhode Island: American Mathematical Society. pp. 175–221. ISBN 9780821809778.
• Jech, Thomas (2003). Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kunen, Kenneth (1980). Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
• Manin, Yuri I. (2010) [1977]. A Course in Mathematical Logic for Mathematicians. Graduate Texts in Mathematics. Vol. 53. Translated by Koblitz, N. (2nd ed.). New York: Springer-Verlag. pp. 89–96. doi:10.1007/978-1-4419-0615-1. ISBN 978-144-190-6144.
• Mendelson, Elliott (1964). Introduction to Mathematical Logic. Van Nostrand Reinhold.
• Mirimanoff, Dmitry (1917). "Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles". L'Enseignement Mathématique. 19: 37–52.
• Moore, Gregory H (2013) [1982]. Zermelo's axiom of choice: Its origins, development & influence. Dover Publications. ISBN 978-0-486-48841-7.
• Peano, Giuseppe (1889). Arithmetices principia: nova methodo exposita. Fratres Bocca.
• Roitman, Judith (2011) [1990]. Introduction to Modern Set Theory. Virginia Commonwealth University. ISBN 978-0-9824062-4-3.
• Rubin, Jean E. (1967). Set Theory for the Mathematician. San Francisco: Holden-Day. ASIN B0006BQH7S.
• Smullyan, Raymond M.; Fitting, Melvin (2010) [revised and corrected republication of the work originally published in 1996 by Oxford University Press, New York]. Set Theory and the Continuum Problem. Dover. ISBN 978-0-486-47484-7.
• von Neumann, John (1923). "Zur Einführung der transfiniten Zahlen". Acta Litt. Acad. Sc. Szeged X. 1: 199–208.. English translation: van Heijenoort, Jean (1967), "On the introduction of transfinite numbers", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 346–354
• von Neumann, John (1928a). "Die Axiomatisierung der Mengenlehre". Mathematische Zeitschrift. 27: 669–752. doi:10.1007/bf01171122.
• von Neumann, John (1928b). "Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre". Mathematische Annalen. 99: 373–391. doi:10.1007/bf01459102.
• Whitehead, Alfred North; Russell, Bertrand (2009) [1910]. Principia Mathematica. Vol. One. Merchant Books. ISBN 978-1-60386-182-3.
• Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre". Fundamenta Mathematicae. 16: 29–47. doi:10.4064/fm-16-1-29-47.