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## Mechanising Set Theory: Cardinal Arithmetic and the Axiom of Choice (1995)

### Citations

596 |
A Computational Logic.
- Boyer, Moore
- 1979
(Show Context)
Citation Context ...f results have been mechanised using the Mizar Proof Checker and published in the Mizar journal [6]. However, the problem of mechanising mathematics is far from solved. The Boyer/Moore Theorem Prover =-=[2, 3]-=- has yielded the most impressive results [22, 23]. It has been successful because of its exceptionally strong support for recursive deÆnitions and inductive reasoning. But its lack of quantiÆers force... |

497 |
Set theory: an introduction to independence proofs. Studies in Logic and Foundations of Mathematics
- Kunen
- 1980
(Show Context)
Citation Context ...nising mathematics include IMPS [5] and Coq [4]. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanised most of the Ærst chapter of Kunen =-=[11]-=- and a paper by Abrial and LafÆtte [1]. GraÀbczewski has mechanised the Ærst two chapters of Rubin and Rubin's famous monograph [21], proving equivalent eight forms of the Wellordering Theorem and twe... |

470 | Isabelle: A Generic Theorem Prover
- Paulson
- 1994
(Show Context)
Citation Context ...ious contortions when they are formalised. Most automated reasoning systems are Ærst-order at best, while mathematics makes heavy use of higher-order notations. We have conducted our work in Isabelle =-=[18]-=-, which provides for higher-order syntax. Other recent systems that have been used for mechanising mathematics include IMPS [5] and Coq [4]. We describe below machine proofs concerning cardinal arithm... |

425 |
A Computational Logic Handbook.
- Boyer, Moore
- 1988
(Show Context)
Citation Context ...f results have been mechanised using the Mizar Proof Checker and published in the Mizar journal [6]. However, the problem of mechanising mathematics is far from solved. The Boyer/Moore Theorem Prover =-=[2, 3]-=- has yielded the most impressive results [22, 23]. It has been successful because of its exceptionally strong support for recursive deÆnitions and inductive reasoning. But its lack of quantiÆers force... |

199 |
Naive set theory.
- Halmos
- 1974
(Show Context)
Citation Context ... theorem jIj 8i2I i < + ( S i2I i) < + (1) You need not understand the details of how this is used in order to follow the paper. 1 Not many set theory texts cover such material well. Elementary texts =-=[8, 24]-=- never get far enough, while advanced texts such as Kunen [11] race through it. But Kunen's rapid treatment is nonetheless clear, and mentions all the essential elements. The desired result (1) follow... |

101 |
Axiomatic Set Theory
- Suppes
- 1972
(Show Context)
Citation Context ... theorem jIj 8i2I i < + ( S i2I i) < + (1) You need not understand the details of how this is used in order to follow the paper. 1 Not many set theory texts cover such material well. Elementary texts =-=[8, 24]-=- never get far enough, while advanced texts such as Kunen [11] race through it. But Kunen's rapid treatment is nonetheless clear, and mentions all the essential elements. The desired result (1) follow... |

87 | imps: An Interactive Mathematical Proof System.
- Farmer, Guttman, et al.
- 1993
(Show Context)
Citation Context ...y use of higher-order notations. We have conducted our work in Isabelle [18], which provides for higher-order syntax. Other recent systems that have been used for mechanising mathematics include IMPS =-=[5]-=- and Coq [4]. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanised most of the Ærst chapter of Kunen [11] and a paper by Abrial and LafÆ... |

54 |
Benthem Jutting. Checking Landau's "Grundlagen
- van
- 1979
(Show Context)
Citation Context ...to note passages that seem unusually hard to mechanise, and discuss some of them below.In conducting these proofs, particularly from Rubin and Rubin, we have tried to follow the footsteps of Jutting =-=[10]-=-. During the 1970s, Jutting mechanised a mathematics textbook using the AUTOMATH system [12]. He paid close attention to the text – which described the construction of the real and complex numbers sta... |

38 |
Automated Deduction in von Neumann-Bernays-Godel Set Theory
- Quaife
- 1992
(Show Context)
Citation Context ...of AC. We have conducted these proofs using an implementation of Zermelo-FrÒnkel (ZF) set theory in Isabelle. Compared with other Isabelle/ZF proofs [13, 15, 16] and other automated set theory proofs =-=[20]-=-, these are deep, abstract and highly technical results. We have tried to reproduce the mathematics faithfully. This does not mean slavishly adhering to every detail of the text, but attempting to pre... |

28 |
Residual theory in © -calculus: a formal development
- Huet
- 1994
(Show Context)
Citation Context ...995 1 Introduction A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [23], number theory [22], group theory [25], -calculus =-=[9]-=-, etc. An especially wide variety of results have been mechanised using the Mizar Proof Checker and published in the Mizar journal [6]. However, the problem of mechanising mathematics is far from solv... |

22 | Constructing Recursion Operators in Intuitionistic Type Theory
- Paulson
- 1986
(Show Context)
Citation Context ... between the orders hA� <Ai and hB� <Bi; it follows that their order types are equal. Sum, product and inverse image are useful building blocks for well-orderings; this follows Paulson's earlier work =-=[14]-=- within Constructive Type Theory.Cardinal = OrderType + Fixedpt + Nat + Sum + consts Least :: "(i=>o) => i" (binder "LEAST " 10) eqpoll, lepoll, lesspoll :: "[i,i] => o" (infixl 50) cardinal :: "i=>i... |

16 | The Unexpected Hanging and Other Mathematical Diversions - Gardner - 1991 |

14 |
Towards the mechanization of the proofs of some classical theorems of set theory
- Abrial, Laffitte
- 1993
(Show Context)
Citation Context ... Coq [4]. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanised most of the Ærst chapter of Kunen [11] and a paper by Abrial and LafÆtte =-=[1]-=-. GraÀbczewski has mechanised the Ærst two chapters of Rubin and Rubin's famous monograph [21], proving equivalent eight forms of the Wellordering Theorem and twenty forms of AC. We have conducted the... |

14 |
Experimenting with Isabelle in ZF set theory
- Noël
- 1993
(Show Context)
Citation Context ...forms of the Wellordering Theorem and twenty forms of AC. We have conducted these proofs using an implementation of Zermelo-FrÒnkel (ZF) set theory in Isabelle. Compared with other Isabelle/ZF proofs =-=[13, 15, 16]-=- and other automated set theory proofs [20], these are deep, abstract and highly technical results. We have tried to reproduce the mathematics faithfully. This does not mean slavishly adhering to ever... |

10 |
Computer proofs in group theory.
- Yu
- 1990
(Show Context)
Citation Context ...n.pl 10 August 1995 1 Introduction A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [23], number theory [22], group theory =-=[25]-=-, -calculus [9], etc. An especially wide variety of results have been mechanised using the Mizar Proof Checker and published in the Mizar journal [6]. However, the problem of mechanising mathematics i... |

9 |
A mechanical proof of quadratic reciprocity
- Russinoff
- 1992
(Show Context)
Citation Context ...rabcze@mat.uni.torun.pl 10 August 1995 1 Introduction A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [23], number theory =-=[22]-=-, group theory [25], -calculus [9], etc. An especially wide variety of results have been mechanised using the Mizar Proof Checker and published in the Mizar journal [6]. However, the problem of mechan... |

8 |
et al. The Coq proof assistant user’s guide
- Dowek
- 1993
(Show Context)
Citation Context ...her-order notations. We have conducted our work in Isabelle [18], which provides for higher-order syntax. Other recent systems that have been used for mechanising mathematics include IMPS [5] and Coq =-=[4]-=-. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanised most of the Ærst chapter of Kunen [11] and a paper by Abrial and LafÆtte [1]. Gra... |

1 |
Published by Fondation Philippe le Hodey
- Mathematics
(Show Context)
Citation Context ...s in logic [23], number theory [22], group theory [25], -calculus [9], etc. An especially wide variety of results have been mechanised using the Mizar Proof Checker and published in the Mizar journal =-=[6]-=-. However, the problem of mechanising mathematics is far from solved. The Boyer/Moore Theorem Prover [2, 3] has yielded the most impressive results [22, 23]. It has been successful because of its exce... |

1 |
Set theory for veriÆcation: I. From foundations to functions
- Paulson
- 1993
(Show Context)
Citation Context ...forms of the Wellordering Theorem and twenty forms of AC. We have conducted these proofs using an implementation of Zermelo-FrÒnkel (ZF) set theory in Isabelle. Compared with other Isabelle/ZF proofs =-=[13, 15, 16]-=- and other automated set theory proofs [20], these are deep, abstract and highly technical results. We have tried to reproduce the mathematics faithfully. This does not mean slavishly adhering to ever... |

1 |
Set theory for veriÆcation: II. Induction and recursion
- Paulson
- 1993
(Show Context)
Citation Context ...forms of the Wellordering Theorem and twenty forms of AC. We have conducted these proofs using an implementation of Zermelo-FrÒnkel (ZF) set theory in Isabelle. Compared with other Isabelle/ZF proofs =-=[13, 15, 16]-=- and other automated set theory proofs [20], these are deep, abstract and highly technical results. We have tried to reproduce the mathematics faithfully. This does not mean slavishly adhering to ever... |

1 |
A Æxedpoint approach to implementing (co)inductive deÆnitions
- Paulson
- 1994
(Show Context)
Citation Context ....5]. For i 2 I let i be the least such that i 2 V [A] . From (1) we can prove jIj I V [A] + I ! V [A] + V [A] + This result allows V [A] + to serve as the bounding set for a least Æxedpoint deÆnition =-=[17]-=-.multiplication of inÆnite cardinals: = : This is Theorem 10.12 of Kunen. (In this paper, we refer only to his Chapter I.) The proof presents a challenging example of formalisation, as we shall see. ... |