Die Widerspruchsfreiheit der allgemeinen Mengenlehre

  title={Die Widerspruchsfreiheit der allgemeinen Mengenlehre},
  author={Wilhelm Ackermann},
  journal={Mathematische Annalen},
  • W. Ackermann
  • Published 1 December 1937
  • Mathematics
  • Mathematische Annalen

Bounded finite set theory

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

Neostability in countable homogeneous metric spaces

Hereditarily Finite Sets in Constructive Type Theory

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.

Addition and multiplication of sets

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.

Counting extensional acyclic digraphs

Cumulative hierarchies and computability over universes of sets

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

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.

A fine structure for the hereditarily finite sets




Untersuchungen über die Grundlagen der Mengenlehre. I

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