• Corpus ID: 208643529

Zermelo Fraenkel Set Theory in Higher-Order Logic

@article{Paulson2019ZermeloFS,
  title={Zermelo Fraenkel Set Theory in Higher-Order Logic},
  author={Lawrence Charles Paulson},
  journal={Arch. Formal Proofs},
  year={2019},
  volume={2019}
}
This entry is a new formalisation of ZFC set theory in Isabelle/HOL. It is logically equivalent to Obua’s HOLZF [2]; the point is to have the closest possible integration with the rest of Isabelle/HOL, minimising the amount of new notations and exploiting type classes. There is a type V of sets and a function elts :: V ⇒ V set mapping a set to its elements. Classes simply have type V set, and the predicate small identifies those classes that correspond to actual sets. Type classes connected… 
Isabelle/HOL/GST: A Formal Proof Environment for Generalized Set Theories
TLDR
This paper presents Isabelle/HOL support for GSTs, which are treated as type classes that combine features that specify kinds of mathematical objects, e.g., sets, ordinal numbers, functions, etc.
Formalizing Ordinal Partition Relations Using Isabelle/HOL
TLDR
An overview of a formalization project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory by Erdős–Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all.
Wetzel: Formalisation of an Undecidable Problem Linked to the Continuum Hypothesis
. In 1964, Paul Erdős published a paper [5] settling a question about function spaces that he had seen in a problem book. Erdős proved that the answer was yes if and only if the continuum hypothesis
A Formalised Theorem in the Partition Calculus
A paper on ordinal partitions by Erdős and Milner [1] has been formalised using the proof assistant Isabelle/HOL, augmented with a library for Zermelo– Fraenkel set theory. The work is part of a
Automated Reasoning: 10th International Joint Conference, IJCAR 2020, Paris, France, July 1–4, 2020, Proceedings, Part II
TLDR
This paper discusses the design of a hierarchy of structures which combine linear algebra with concepts related to limits, like topology and norms, in dependent type theory, and presents and discusses a solution, coined forgetful inheritance, based on packed classes and unification hints.
Formalization of Forcing in Isabelle/ZF
TLDR
The theory of forcing in the set theory framework of Isabelle/ZF is formalized and Paulson’s ZF-Constructibility library is remodularized.

References

SHOWING 1-2 OF 2 REFERENCES
Partizan Games in Isabelle/HOLZF
TLDR
This work formalizes PGs in Higher Order Logic extended with ZF axioms (HOLZF) using Isabelle, a mechanical proof assistant, and formalizes the induction principle that Conway uses throughout his proofs about games, and proves its correctness.
Addition and multiplication of sets
TLDR
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.