## A Finite First-Order Theory of Classes

Citations: | 6 - 0 self |

### BibTeX

@MISC{Kirchner_afinite,

author = {Florent Kirchner},

title = {A Finite First-Order Theory of Classes},

year = {}

}

### OpenURL

### Abstract

Abstract. We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. This allows us to give finite first-order axiomatizations of arithmetic and real analysis, and a presentation of arithmetic in deduction modulo that has a finite number of rewrite rules. Overall, this formalization relies on a weak calculus of explicit substitutions to provide a simple and finite framework. 1

### Citations

48 | Set theory for verification: I. From foundations to functions - Paulson - 1993 |

37 | Automated deduction in von Neumann-Bernays-Gödel set theory - Quaife - 1992 |

24 | Set theory in first-order logic: Clauses for Gödel's axioms - Boyer, Lusk, et al. - 1986 |

20 | Functional back-ends within the lambda-sigma calculus
- Hardin, Maranget, et al.
- 1996
(Show Context)
Citation Context ...here is one such axiom for each function (resp. predicate) symbol of arity n (resp. m) in the language L ws , which is finite. – This axiom system implements a weak calculus of explicit substitutions =-=[5]-=-: the substitutions are propagated over the elements of the language via the symbols ∈ and ·[·], and no lift is introduced by the P or C binders (axioms (T ws 10) and (T ws 11)). 3.2 Expressiveness We... |

20 | Arithmetic as a theory modulo
- Dowek, Werner
- 2005
(Show Context)
Citation Context ... the theory, HA, we consider here a slightly more elaborate presentation of the theory where the universe of discourse is not restricted to natural numbers. This theory, called HAN , was presented in =-=[7]-=- by Dowek and Werner. 7sDefinition 5 (HAN). The theory HAN of arithmetic is defined in first-order logic using the symbols 0, Succ, +, ×, Pred, =, Null and N. It counts the axioms: N(0) ∀x,(N(x) ⇒ N(S... |

17 |
Éléments de Mathématique I: Théorie des Ensembles, Fascicule de Résultats”, Hermann & Cie
- Bourbaki
- 1939
(Show Context)
Citation Context ...e work we present in the previous sections is related to von Neumann, Bernays and Gödel’s formalism for set theory (NBG) [1] that rehabilitated the notion of class used by 19th century mathematicians =-=[10]-=-. However it improves on a couple of points: – Classes and the NBG approach have always been associated with set theory [10]. We generalize it to any theory that has axiom schemes. – By clarifying the... |

14 | A mechanically verified, sound and complete theorem prover for first order logic. In: TPHOLs
- Ridge, Margetson
- 2005
(Show Context)
Citation Context ...s on weaker frameworks, such as first-order logic. The maturity of these frameworks make them very secure centerpieces of formal tools in general, and proof checkers in particular; some designs (e.g. =-=[3, 4]-=-) have already taken advantage of them. It is essential in this project to be able to implement strong theories, such as arithmetic, real analysis or set theory, with a finite number of axioms. The ma... |

13 | Computer proofs in Gödel’s class theory with equational definitions for composite and cross - Belinfante - 1999 |

5 |
Introduction to Mathematical Logic, 4th ed
- Mendelson
- 1997
(Show Context)
Citation Context ...heme of comprehension to this new structure. In the end, however, the theory still has an axiom scheme that generates an infinite number of axioms. This is not a zero-sum game, though. Previous works =-=[1, 2]-=- have shown that, in the case of set theory, it is possible to reduce the comprehension axiom scheme to a finite number of axioms. However we believe that the notion of class is independent of set the... |

3 | Cut elimination for Zermelo’s set theory
- Dowek, Miquel
- 2007
(Show Context)
Citation Context ...eory. The same holds for the binary replacement axiom scheme of ZermeloFraenkel’s theory, or the three schemes that result from Dowek and Miquel’s encoding of set theory in a theory of pointed graphs =-=[6]-=-. We detail two examples: arithmetic and real analysis. 4.1 A Finite Theory of Arithmetic In the following, we will explore Heyting’s arithmetic. While our formalism applies to the original formulatio... |

2 |
Fellowship: who needs a manual anyway
- Kirchner
- 2005
(Show Context)
Citation Context ...s on weaker frameworks, such as first-order logic. The maturity of these frameworks make them very secure centerpieces of formal tools in general, and proof checkers in particular; some designs (e.g. =-=[3, 4]-=-) have already taken advantage of them. It is essential in this project to be able to implement strong theories, such as arithmetic, real analysis or set theory, with a finite number of axioms. The ma... |

2 |
C.: Theorem proving modulo, revised version. Rapport de Recherche 4861, Institut National de Recherche en Informatique et en Automatique
- Dowek, Hardin, et al.
- 2003
(Show Context)
Citation Context ...med-down theory. HA + is equivalent to HA ws N , and counts 26 axioms instead of 29. 4.2 Arithmetic as a Theory Modulo A theory modulo is a theory in which formulas are identified modulo a congruence =-=[8]-=-. In particular, the thoery arithmetic has been expressed in such a framework [7], but this formalization had an infinite number of rewrite rules. The goal of this section is to show how our system al... |

1 |
A finite first-order presentation of set theory
- Vaillant
(Show Context)
Citation Context ...heme of comprehension to this new structure. In the end, however, the theory still has an axiom scheme that generates an infinite number of axioms. This is not a zero-sum game, though. Previous works =-=[1, 2]-=- have shown that, in the case of set theory, it is possible to reduce the comprehension axiom scheme to a finite number of axioms. However we believe that the notion of class is independent of set the... |

1 |
J.M.: Cours de Mathématiques. Tome 2 : Analyse. Dunod
- Lelong-Ferrand, Arnaudies
- 1972
(Show Context)
Citation Context ...ter science are too numerous to cite here. Real numbers are used, e.g., in exact arithmetic, programming languages, formal calculus and formal systems.The following formal development is quite common =-=[9]-=-, and is used e.g. in the proof assistant Coq to implement the theory of real numbers. Definition 9 (IR cs ). The language of the theory of real numbers IR cs is formed by the symbols 0, 1, +, ×, the ... |