Von Neumann hierarchy

The Von Neumann hierarchy or cumulative hierarchy is a term in set theory that names a construction by John von Neumann from 1928, namely a step-by-step structure of the entire universe of sets with the help of ordinal numbers and the iteration of power sets.

definition

The levels of ordinal numbers or Limes ordinal numbers are defined by transfinite recursion using the following recursion conditions: ${\displaystyle V_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \lambda }$

${\displaystyle {\begin{aligned}V_{0}\,&:=\,\emptyset \\V_{\alpha +1}\,&:=\,{\mathfrak {P}}\left(V_{\alpha }\right)\\V_{\lambda }\,&:=\,\bigcup _{\alpha <\lambda }V_{\alpha }\end{aligned}}}$

So is

${\displaystyle {\begin{aligned}V_{1}&=\{\emptyset \}\\V_{2}&=\{\emptyset ,\{\emptyset \}\}\\V_{3}&=\{\,\emptyset ,\{\emptyset \},\{\{\emptyset \}\},\{\emptyset ,\{\emptyset \}\}\,\}\\\vdots \\V_{\omega }&=\{x\mid x{\text{ ist erblich endlich}}\}\\\vdots \end{aligned}}}$

etc.

All sets in the are thus constructed from the empty set. The levels are transitive sets , and it applies to all ordinals , which explains the name of cumulative hierarchy. ${\displaystyle V_{\alpha }}$ ${\displaystyle V_{\alpha }\subset V_{\beta }}$ ${\displaystyle \alpha <\beta }$

The hierarchy

Within the Zermelo-Fraenkel set theory (ZF for short) it can be shown that every set lies in one level of the hierarchy: Describes the class of all sets, so it applies ${\displaystyle V:=\{x\mid x=x\}}$

${\displaystyle V=\bigcup _{\alpha \in \operatorname {Ord} }V_{\alpha }}$

Here the foundation axiom is essentially used in the context of epsilon induction . Conversely, the axiom of foundation also follows from the above statement, so both statements are equivalent (over the remaining axioms of ZF).

Furthermore, it can be shown that the class , understood as a subset of an assumed model from ZF without a foundation axiom, is a model for ZF. The same is therefore relatively consistent with the other axioms. ${\displaystyle \textstyle \bigcup _{\alpha \in \operatorname {Ord} }V_{\alpha }}$

Rank function

Since every set is in a suitable level , there is always a smallest ordinal number with and thus . This is called the rank,, of quantity . ${\displaystyle x}$ ${\displaystyle V_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle x\subseteq V_{\alpha }}$ ${\displaystyle x\in V_{\alpha +1}}$ ${\displaystyle \alpha }$ ${\displaystyle Rg(x)}$ ${\displaystyle x}$

Using transfinite induction over one can ${\displaystyle \alpha }$

${\displaystyle Rg(\alpha )\,=\,\alpha }$ for all ordinal numbers ${\displaystyle \alpha }$

demonstrate. For each set applies . The rank of a set is therefore always strictly greater than the rank of all of its elements. ${\displaystyle x}$ ${\displaystyle Rg(x)=\sup\{Rg(y)+1\mid y\in x\}}$ ${\displaystyle x}$

Applications

• ${\displaystyle V_{\omega }}$consists exactly of the hereditary finite sets . In apply, except for the infinity axiom all ZFC axioms. This shows that the axiom of infinity cannot be derived from the other ZFC axioms. ${\displaystyle V_{\omega }}$
• If a cardinal number is strongly unreachable , then is a model for ZFC. In this way, a model is obtained for the smallest strongly unreachable cardinal number in which there are no strongly unreachable cardinal numbers. The existence of strongly unreachable cardinal numbers cannot be derived in ZFC. ${\displaystyle \kappa }$ ${\displaystyle V_{\kappa }}$
• The levels play a role in the reflection principle , which is an important axiom in Scott's system of axioms . ${\displaystyle V_{\alpha }}$

Individual evidence

1. ↑ John von Neumann: About a consistency question in axiomatic set theory , 1928, in: Journal for pure and applied mathematics 160 (1929) 227–241. There, p. 236f the cumulative hierarchy, but nameless.
2. ^ Heinz-Dieter Ebbinghaus : Introduction to set theory , Spektrum Verlag 2003, ISBN 3-8274-1411-3 .
3. ^ Thomas Jech : Set Theory , Springer-Verlag (2003), ISBN 3-540-44085-2 , Theorem 12.12.