Von Neumann universe

The von Neumann hierarchy or cumulative hierarchy is a concept of set theory, which identifies a design by John von Neumann in 1928, namely a gradual build-up of the entire amount of the universe with the help of ordinals.

Definition

The steps to ordinals are defined by transfinite recursion on the following Rekursionsbedingungen:

  • , The empty set;
  • , The power set of;
  • For limit ordinals.

It is always advisable

Etc.

All amounts in the are so constructed from the empty crowd. The steps are transitive subsets, and it applies to all ordinals, this explains the name cumulative hierarchy.

The hierarchy

Within the Zermelo -Fraenkel set theory ( ZF short ) it can be shown that any amount is in a stage of the hierarchy: Indicates the class of all sets so true

Herein, the foundation axiom is used in the epsilon- induction essential. Conversely, it follows from the above statement and the foundation axiom, 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 of ZF without the foundation axiom, a model of ZF. The same is thus relatively consistent with the other axioms.

Rank function

Since each quantity is at an appropriate stage, there is always a least ordinal and thus. This is referred to as the rank, the amount.

Using transfinite induction on can

Show. For each quantity. The rank of a set is therefore always strictly greater than the rank of all its elements.

Applications

  • Consists exactly of the sets hereditarily finite. In apply, with the exception of the infinity axiom all ZFC axioms. Thus we have shown that the axiom of infinity can not be derived from the other axioms of ZFC.
  • Is an unattainable cardinal number, is a model of ZFC. In particular, we obtain in this way a model in which there is no unattainable cardinal numbers. The existence of an inaccessible cardinal numbers so can not be derived in ZFC.
  • The levels play a role in the reflection principle, which is an important axiom in Scott 's axiom system.
Ordinal number Axiom of regularity Epsilon-induction Heinz-Dieter Ebbinghaus
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