# Die Widerspruchsfreiheit der allgemeinen Mengenlehre

```@article{Ackermann1937DieWD,
title={Die Widerspruchsfreiheit der allgemeinen Mengenlehre},
author={Wilhelm Ackermann},
journal={Mathematische Annalen},
year={1937},
volume={114},
pages={305-315}
}```
• W. Ackermann
• Published 1 December 1937
• Mathematics
• Mathematische Annalen
104 Citations

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We define an axiom schema IΔ0S for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic, IΔ0 , and use some of its basic model theory to establish some

### Hereditarily Finite Sets in Constructive Type Theory

• Economics, Mathematics
ITP
• 2016
The axiomatization takes the empty set and adjunction as primitives and comes with a strong induction principle and the set operations of ZF are constructed and the basic theory of finite ordinals and cardinality is developed.

### Finitary Set Theory

• L. Kirby
• Mathematics
Notre Dame J. Formal Log.
• 2009
The use of the adjunction operator (adding a single new element to an existing set) as a basis for building a finitary set theory allows a simplified axiomatization for the first-order theory of hereditarily finite sets and a rigorous characterization of the primitive recursive set functions.

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The natural partial ordering associated with addition of sets is shown to be a tree, which allows us to prove that any set has a unique representation as a sum of additively irreducible sets and that the non-empty elements of any model of set theory can be partitioned into infinitely many submodels, each isomorphic to the original model.

### Cumulative hierarchies and computability over universes of sets

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• 2008
Various metamathematical investigations, beginning with Fraenkel’s historical proof of the independence of the axiom of choice, called for suitable deﬁnitions of hierarchical universes of sets. This

### Executable Set Theory and Arithmetic Encodings in Prolog

A digraph representation of Hereditarily Finite Sets with Urelements is implemented and the surprising possibility of internally sharing isomorphic objects, independently of their language level types and meanings is uncovered.

### Ackermann encoding, bisimulations and OBDDs

• Computer Science
Theory and Practice of Logic Programming
• 2004
An alternative way to represent graphs via OBDDs using the notion of Ackermann encoding of hereditarily finite sets into natural numbers is proposed and a method to compute at the same time the maximum bisimulation and the OBDD representation of a given graph is presented.