## Mechanizing set theory: Cardinal arithmetic and the axiom of choice (1996)

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Venue: | Journal of Automated Reasoning |

Citations: | 16 - 9 self |

### BibTeX

@ARTICLE{Paulson96mechanizingset,

author = {Lawrence C. Paulson and Krzysztof Grabczewski},

title = {Mechanizing set theory: Cardinal arithmetic and the axiom of choice},

journal = {Journal of Automated Reasoning},

year = {1996},

pages = {291--323}

}

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### Abstract

Abstract. Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I. Furthermore, we have proved the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice, and involves highly technical material. The definitions used in the proofs are

### Citations

530 |
A computational logic
- Boyer, Moore
- 1979
(Show Context)
Citation Context ... been mechanized using the Mizar Proof Checker, including the theorem κ ⊗ κ = κ discussed below [2]. However, the problem of mechanizing mathematics is far from solved. The Boyer/Moore Theorem Pr=-=over [3, 4]-=- has yielded the most impressive results [25, 26]. It has been successful because of its exceptionally strong support for recursive definitions and inductive reasoning. But its lack of quantifiers for... |

500 |
T.: Introduction to HOL: A Theorem Proving Environment for Higher Order Logic: Cambridge
- Melham
- 1993
(Show Context)
Citation Context ...higher-order notations. We have conducted our work in Isabelle [20], which provides for higher-order syntax. Other recent systems that have been used for mechanizing mathematics include IMPS [6], HOL =-=[8]-=- and Coq [5]. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanized most of the first chapter of Kunen [12] and a paper by Abrial and Laf... |

420 | Isabelle: A generic theorem prover
- Paulson
- 1994
(Show Context)
Citation Context ...ous contortions when they are formalized. Most automated reasoning systems are first-order at best, while mathematics makes heavy use of higher-order notations. We have conducted our work in Isabelle =-=[20]-=-, which provides for higher-order syntax. Other recent systems that have been used for mechanizing mathematics include IMPS [6], HOL [8] and Coq [5]. We describe below machine proofs concerning cardin... |

395 |
A Computational Logic Handbook
- Boyer, Moore
- 1988
(Show Context)
Citation Context ... been mechanized using the Mizar Proof Checker, including the theorem κ ⊗ κ = κ discussed below [2]. However, the problem of mechanizing mathematics is far from solved. The Boyer/Moore Theorem Pr=-=over [3, 4]-=- has yielded the most impressive results [25, 26]. It has been successful because of its exceptionally strong support for recursive definitions and inductive reasoning. But its lack of quantifiers for... |

274 |
Set Theory - An Introduction to Independence Proofs
- Kunen
- 1980
(Show Context)
Citation Context ...hematics include IMPS [6], HOL [8] and Coq [5]. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanized most of the first chapter of Kunen =-=[12] an-=-d a paper by Abrial and Laffitte [1]. Gra¸bczewski has mechanized the first two chapters of Rubin and Rubin’s famous monograph [24], proving equivalent eight forms of the Well-ordering Theorem and ... |

182 |
Isabelle: The next 700 theorem provers
- Paulson
- 1990
(Show Context)
Citation Context ...logic, etc. Our work is based upon Isabelle’s implementation of Zermelo-Frænkel (ZF) set theory, itself based upon an implementation of first-order logic. Isabelle/ZF arose from early work by Pauls=-=on [17] and-=- Noël [15]; it is described in detail elsewhere [18, 21]. There are two key ideas behind Isabelle: − Expressions are typed λ-terms. Thus the syntax is higher-order, giving a uniform treatment of q... |

126 |
Naive Set Theory
- Halmos
- 1960
(Show Context)
Citation Context ...quantifier Figure 1. ASCII notation for ZF 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 =-=[9, 27] n-=-ever get far enough, while advanced texts such as Kunen [12] race through it. But Kunen’s rapid treatment is nonetheless clear, and mentions all the essential elements. The desired result (1) follow... |

124 | Unification under a mixed prefix
- Miller
- 1992
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Citation Context ... 5 The statement of Global Choice can be obtained by Skolemizing the trivial theorem ∀c.c �= 0 → (∃x .x∈ c). This is a standard example showing that Skolemization can be unsound in higher-or=-=der logic [13]-=-. 6 Such figures can be regarded only as a rough guide. Many of the theorems properly belong in the main libraries. Small changes to searching commands can have a drastic effect on the run time. For c... |

81 |
Axiomatic Set Theory
- Suppes
- 1960
(Show Context)
Citation Context ...quantifier Figure 1. ASCII notation for ZF 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 =-=[9, 27] n-=-ever get far enough, while advanced texts such as Kunen [12] race through it. But Kunen’s rapid treatment is nonetheless clear, and mentions all the essential elements. The desired result (1) follow... |

78 | IMPS: An interactive mathematical proof system
- Farmer, Guttman, et al.
- 1990
(Show Context)
Citation Context ...y use of higher-order notations. We have conducted our work in Isabelle [20], which provides for higher-order syntax. Other recent systems that have been used for mechanizing mathematics include IMPS =-=[6]-=-, HOL [8] and Coq [5]. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanized most of the first chapter of Kunen [12] and a paper by Abria... |

47 |
Benthem Jutting. Checking Landau’s “Grundlagen” in the Automath system
- van
- 1977
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Citation Context ...to note passages that seem unusually hard to mechanize, and discuss some of them below. In conducting these proofs, particularly from Rubin and Rubin, we have tried to follow the footsteps of Jutting =-=[11]. -=-During the 1970s, Jutting mechanized a mathematics textbook using the AUTOMATH system [14]. He paid close attention to the text — which described the construction of the real and complex numbers sta... |

44 | Set theory for verification: I. From foundations to functions
- Paulson
- 1993
(Show Context)
Citation Context ...forms of the Well-ordering Theorem and twenty forms of AC. We have conducted these proofs using an implementation of ZermeloFrænkel (ZF) set theory in Isabelle. Compared with other Isabelle/ZF proofs=-= [15, 18, 21]-=- and other automated set theory proofs [23], these are deep, abstract and highly technical results. We have tried to reproduce the mathematics faithfully. This does not mean slavishly adhering to ever... |

44 |
A fixedpoint approach to implementing (co)inductive definitions
- Paulson
(Show Context)
Citation Context ...to contain an error; we used an alternative justification of their Property 6.4. The inductive definition involves fixedpoints and some non-trivial proofs, but Isabelle’s inductive definition packag=-=e [19]-=- automates this process. Abrial and Laffitte envisaged the definition and related proofs to depend implicitly on its successor parameter. In Isabelle this parameter must be explicit in all definitions... |

41 | Set theory for verification: II. Induction and recursion
- Paulson
- 1995
(Show Context)
Citation Context ...forms of the Well-ordering Theorem and twenty forms of AC. We have conducted these proofs using an implementation of ZermeloFrænkel (ZF) set theory in Isabelle. Compared with other Isabelle/ZF proofs=-= [15, 18, 21]-=- and other automated set theory proofs [23], these are deep, abstract and highly technical results. We have tried to reproduce the mathematics faithfully. This does not mean slavishly adhering to ever... |

35 |
Automated deduction in von Neumann–Bernays–Gödel set theory
- Quaife
- 1992
(Show Context)
Citation Context ... of AC. We have conducted these proofs using an implementation of ZermeloFrænkel (ZF) set theory in Isabelle. Compared with other Isabelle/ZF proofs [15, 18, 21] and other automated set theory proofs=-= [23]-=-, 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... |

34 | Countable sets and Hessenberg’s theorem
- Bancerek
- 1991
(Show Context)
Citation Context ... number theory [25], group theory [28], λ-calculus [10], etc. An especially wide variety of results have been mechanized using the Mizar Proof Checker, including the theorem κ ⊗ κ = κ discussed =-=below [2]-=-. However, the problem of mechanizing mathematics is far from solved. The Boyer/Moore Theorem Prover [3, 4] has yielded the most impressive results [25, 26]. It has been successful because of its exce... |

27 |
Residual theory in -calculus: A formal development
- Huet
- 1994
(Show Context)
Citation Context ... 1 1. Introduction A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [26], number theory [25], group theory [28], -calculus =-=[10]-=-, etc. An especially wide variety of results have been mechanized using the Mizar Proof Checker, including the theorem \Omegas=sdiscussed below [2]. However, the problem of mechanizing mathematics is ... |

22 | Constructing recursion operators in intuitionistic type theory
- Paulson
- 1986
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Citation Context ...rphism between the orders 〈A, <A〉 and 〈B,<B〉; it follows that their order types are equal. Sum, product and inverse image are useful for expressing well-orderings; this follows Paulson’s ear=-=lier work [16] wi-=-thin Constructive Type Theory. 4.4. CARDINAL NUMBERS Figure 2 presents the Isabelle/ZF definitions of cardinal numbers, following Kunen’s §10. The Isabelle theory file extends some Isabelle theorie... |

20 | Residual theory in λ-calculus: A formal development
- Huet
- 2013
(Show Context)
Citation Context ...1 1. Introduction A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [26], number theory [25], group theory [28], λ-calculus=-= [10], etc-=-. An especially wide variety of results have been mechanized using the Mizar Proof Checker, including the theorem κ ⊗ κ = κ discussed below [2]. However, the problem of mechanizing mathematics is... |

18 |
Benthem Jutting. Checking Landau's "Grundlagen
- van
- 1977
(Show Context)
Citation Context ...to note passages that seem unusually hard to mechanize, and discuss some of them below. In conducting these proofs, particularly from Rubin and Rubin, we have tried to follow the footsteps of Jutting =-=[11]-=-. During the 1970s, Jutting mechanized a mathematics textbook using the AUTOMATH system [14]. He paid close attention to the text --- which described the construction of the real and complex numbers s... |

15 |
The Unexpected Hanging and Other Mathematical Diversions, Simon and
- Gardner
- 1969
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Citation Context ...osite corners from a chessboard, can we cover the remaining 62 squares with 31 dominos? The usual proof that the answer is “no” seems impossible to formalize without disproportionate efforts. Gard=-=ner [7] desc-=-ribes a number of similar puzzles. Mechanizing the reverse induction mentioned above, and the construction from x of some y such that x ∪ (y × y) ⊆ y, is routine. All the difficulties lie in prov... |

14 |
Experimenting with Isabelle in ZF set theory
- No»el
- 1993
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Citation Context ...forms of the Well-ordering Theorem and twenty forms of AC. We have conducted these proofs using an implementation of ZermeloFrænkel (ZF) set theory in Isabelle. Compared with other Isabelle/ZF proofs=-= [15, 18, 21]-=- and other automated set theory proofs [23], these are deep, abstract and highly technical results. We have tried to reproduce the mathematics faithfully. This does not mean slavishly adhering to ever... |

12 |
Towards the mechanization of the proofs of some classical theorems of set theory
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- 1993
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Citation Context ...oq [5]. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanized most of the first chapter of Kunen [12] and a paper by Abrial and Laffitte =-=[1]. G-=-ra¸bczewski has mechanized the first two chapters of Rubin and Rubin’s famous monograph [24], proving equivalent eight forms of the Well-ordering Theorem and twenty forms of AC. We have conducted t... |

9 |
et al. The Coq proof assistant user's guide
- Dowek
- 1993
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Citation Context ... notations. We have conducted our work in Isabelle [20], which provides for higher-order syntax. Other recent systems that have been used for mechanizing mathematics include IMPS [6], HOL [8] and Coq =-=[5]-=-. We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC). Paulson has mechanized most of the first chapter of Kunen [12] and a paper by Abrial and Laffitte [1]. G... |

9 |
A mechanical proof of quadratic reciprocity
- Russinoff
- 1992
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Citation Context ... GrabczewskisMechanizing Set Theory 1 1. Introduction A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [26], number theory =-=[25], grou-=-p theory [28], λ-calculus [10], etc. An especially wide variety of results have been mechanized using the Mizar Proof Checker, including the theorem κ ⊗ κ = κ discussed below [2]. However, the p... |

9 |
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- Yu
- 1990
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Citation Context ...izing Set Theory 1 1. Introduction A growing corpus of mathematics has been checked by machine. Researchers have constructed computer proofs of results in logic [26], number theory [25], group theory =-=[28], λ-c-=-alculus [10], etc. An especially wide variety of results have been mechanized using the Mizar Proof Checker, including the theorem κ ⊗ κ = κ discussed below [2]. However, the problem of mechanizi... |