Inset theory and related branches ofmathematics, thevon Neumann universe, orvon Neumann hierarchy of sets, denoted byV, is theclass ofhereditarywell-founded sets. This collection, which is formalized byZermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named afterJohn von Neumann, although it was first published byErnst Zermelo in 1930.

Therank of a well-founded set is defined inductively as the smallestordinal number greater than the ranks of all members of the set.^[1] In particular, the rank of theempty set is zero, and every ordinal has a rank equal to itself. The sets inV are divided into thetransfinite hierarchyV_α , calledthe cumulative hierarchy, based on their rank.

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

• LetV₀ be theempty set:$${\displaystyle V_0:=\varnothing .}$$
• For anyordinal number β, letV_β+1 be thepower set ofV_β:$${\displaystyle V_{\beta +1}:=\mathcal{P}(V_{\beta }).}$$
• For anylimit ordinal λ, letV_λ be theunion of all theV-stages so far:

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

The setsV_α are calledstages orranks.

The classV is defined to be the union of all theV-stages:

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

Therank of a setS is the smallest α such that${\displaystyle S\subseteq V_{\alpha }.}$ In other words,${\displaystyle \mathcal{P}(V_{\alpha })}$ is the set of sets with rank ≤α. The stageV_α can also be characterized as the set of sets with rank strictly less than α, regardless of whether α is 0, a successor ordinal, or a limit ordinal:

This gives an equivalent definition ofV_α by transfinite recursion.

Substituting the above definition ofV_α back into the definition of the rank of a set gives a self-contained recursive definition:

In other words,

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

The first five von Neumann stagesV₀ toV₄ 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.)

This sequence exhibitstetrational growth. The setV₅ contains 2¹⁶ = 65536 elements; the setV₆ contains 2⁶⁵⁵³⁶ elements, which very substantially exceeds thenumber of atoms in the observable universe; and for any natural${\displaystyle n}$, the setV_n+1 contains${\displaystyle 2\uparrow \uparrow n}$ elements usingKnuth's up-arrow notation. So the finite stages of the cumulative hierarchy cannot physically be written down explicitly after stage 5. The setV_ω has the same cardinality as ω. The setV_ω+1 has the same cardinality as the set of real numbers.

If ω is the set ofnatural numbers, thenV_ω is the set ofhereditarily finite sets, which is amodel of set theory without theaxiom of infinity.^[2][3]

V_ω+ω is theuniverse of "ordinary mathematics", and is a model ofZermelo set theory (but not a model ofZF).^[4] A simple argument in favour of the adequacy ofV_ω+ω is the observation thatV_ω+1 is adequate for the integers, whileV_ω+2 is adequate for the real numbers, and most other normal mathematics can be built as relations of various kinds from these sets without needing theaxiom of replacement to go outsideV_ω+ω.

If κ is aninaccessible cardinal, thenV_κ is a model ofZermelo–Fraenkel set theory (ZFC) itself, andV_κ+1 is a model ofMorse–Kelley set theory.^[5][6] (Note that every ZFC model is also a ZF model, and every ZF model is also a Z model.)

V is not "theset of all (naive) sets" for two reasons. First, it is not a set; although each individual stageV_α is a set, their unionV is aproper class. Second, the sets inV are only the well-founded sets. Theaxiom of foundation (or regularity) demands that every set be well founded and hence inV, and thus in ZFC every set is inV. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (an example isAczel's anti-foundation axiom). These non-well-founded set theories are not commonly employed, but are still possible to study.

A third objection to the "set of all sets" interpretation is that not all sets are necessarily "pure sets", which are constructed from the empty set using power sets and unions. Zermelo proposed in 1908 the inclusion ofurelements, from which he constructed a transfinite recursive hierarchy in 1930.^[7] Such urelements are used extensively inmodel theory, particularly in Fraenkel-Mostowski models.^[8]

The von Neumann universe satisfies the following two properties:

• ${\displaystyle \mathcal{P}(x)\in V}$ for everyset${\displaystyle x\in V}$.
• ${\displaystyle \bigcup x\in V}$ for everysubset${\displaystyle x\subseteq V}$.

Indeed, if${\displaystyle x\in V}$, then${\displaystyle x\in V_{\alpha }}$ for some ordinal${\displaystyle \alpha }$. Any stage is atransitive set, hence every${\displaystyle y\in x}$ is already${\displaystyle y\in V_{\alpha }}$, and so every subset of${\displaystyle x}$ is a subset of${\displaystyle V_{\alpha }}$. Therefore,${\displaystyle \mathcal{P}(x)\subseteq V_{\alpha +1}}$ and${\displaystyle \mathcal{P}(x)\in V_{\alpha +2}\subseteq V}$. For unions of subsets, if${\displaystyle x\subseteq V}$, then for every${\displaystyle y\in x}$, let${\displaystyle {\beta }_y}$ be the smallest ordinal for which${\displaystyle y\in V_{{\beta }_y}}$. Because by assumption${\displaystyle x}$ is a set, we can form the limit${\displaystyle \alpha =sup\{{\beta }_y:y\in x\}}$. The stages are cumulative, and therefore again every${\displaystyle y\in x}$ is${\displaystyle y\in V_{\alpha }}$. Then every${\displaystyle z\in y}$ is also${\displaystyle z\in V_{\alpha }}$, and so${\displaystyle \cup x\subseteq V_{\alpha }}$ and${\displaystyle \cup x\in V_{\alpha +1}}$.

Hilbert's paradox implies that no set with the above properties exists .^[9] For suppose${\displaystyle V}$ was a set. Then${\displaystyle V}$ would be a subset of itself, and${\displaystyle U=\cup V}$ would belong to${\displaystyle V}$, and so would${\displaystyle \mathcal{P}(U)}$. But more generally, if${\displaystyle A\in B}$, then${\displaystyle A\subseteq \cup B}$. Hence,${\displaystyle \mathcal{P}(U)\subseteq \cup V=U}$, which is impossible in models of ZFC such as${\displaystyle V}$ itself.

Interestingly,${\displaystyle x}$ is a subset of${\displaystyle V}$ if, and only if,${\displaystyle x}$ is a member of${\displaystyle V}$. Therefore, we can consider what happens if the union condition is replaced with${\displaystyle x\in V⟹\cup x\in V}$. In this case, there are no known contradictions, and anyGrothendieck universe satisfies the new pair of properties. However, whether Grothendieck universes exist is a question beyond ZFC.

The formulaV = ⋃_αV_α is often considered to be a theorem, not a definition.^[10][11] Roitman states (without references) that the realization that theaxiom of regularity is equivalent to the equality of the universe of ZF sets to the cumulative hierarchy is due to von Neumann.^[12]

Since the classV may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed byGödel's incompleteness theorems, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.^[13]

The integrity of the von Neumann universe depends fundamentally on the integrity of theordinal numbers, which act as the rank parameter in the construction, and the integrity oftransfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers.^[14] The integrity of the construction ofV by transfinite induction may be said to have then been established in Zermelo's 1930 paper.^[7]

The cumulative type hierarchy, also known as the von Neumann universe, is claimed by Gregory H. Moore (1982) to be inaccurately attributed tovon Neumann.^[15] The first publication of the von Neumann universe was byErnst 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 intoNBG 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 notationV is not a tribute to the name of von Neumann. It was used for the universe of sets in 1889 by Peano, the letterV signifying "Verum", which he used both as a logical symbol and to denote the class of all individuals.^[18] Peano's notationV was adopted also by Whitehead and Russell for the class of all sets in 1910.^[19] TheV 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 letterV (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 universeV to ZFC (along with many variations of each approach, and shadings between them). Roughly, formalists will tend to viewV as something that flows from the ZFC axioms (for example, ZFC proves that every set is inV). 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 inV 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.^{[citation needed]}

• 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 byKoblitz, 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.

• John von Neumann
• Set-theoretic universes

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