## A Framework for Formalizing Set Theories Based on the Use of Static Set Terms

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@MISC{Avron_aframework,

author = {Arnon Avron},

title = {A Framework for Formalizing Set Theories Based on the Use of Static Set Terms},

year = {}

}

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### Abstract

To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus ” for set theory, it is essentially based on just two set-theoretical principles: extensionality and comprehension (to which we add ∈-induction and optionally the axiom of choice). Comprehension is formulated as: x ∈{x | ϕ} ↔ϕ, where {x | ϕ} is a legal set term of the theory. In order for {x | ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL). Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas. 1

### Citations

1526 |
Foundations of Databases
- Abiteboul, Hull, et al.
- 1995
(Show Context)
Citation Context ... our systems is the safety relation it employs. Now the idea of using such relations is due to the similarity (noted first in [4]) between issues of safety and domain independence in database theory (=-=[2,24]-=-), and issues of set-existence and absoluteness in set theory. This similarity allows us to apply in the context of set theories the purely syntactic approach to safety of formulas that has been devel... |

1064 |
Principles of Database and Knowledge-Base Systems. Volume II: The New Technologies
- Ullman
- 1989
(Show Context)
Citation Context ... our systems is the safety relation it employs. Now the idea of using such relations is due to the similarity (noted first in [4]) between issues of safety and domain independence in database theory (=-=[2,24]-=-), and issues of set-existence and absoluteness in set theory. This similarity allows us to apply in the context of set theories the purely syntactic approach to safety of formulas that has been devel... |

274 |
Set Theory - An Introduction to Independence Proofs
- Kunen
- 1980
(Show Context)
Citation Context ... unified treatment of two important subjects of set theory: axiomatization and absoluteness (the latter is a crucial issue in independence proofs and in the study of models of set theories – see e.g. =-=[17]-=-). In the usual approaches these subjects are completely separated. Absoluteness is investigated mainly from a syntactic point of view, axiomatizations – from a semantic one. Here both are given the s... |

216 |
Mathematical Logic
- Shoenfield
- 1967
(Show Context)
Citation Context ... However, whenever they are intended to denote sets (rather than classes) they are introduced (at least partially) in a dynamic way, based for example on the “extension by definitions” procedure (see =-=[20]-=-, Sect. 4.6): In order to be able to introduce some set term for a set (as well as a new operation on sets) it is necessary first to justify this introduction by proving a corresponding existence theo... |

112 |
The fine structure of the constructible hierarchy
- Jensen
- 1972
(Show Context)
Citation Context ...ystem obtained from Gandy’s “Basic Set Theory” BST ([12]) by the addition of the ∈ −induction schema. 6 The class of rudimentary set functions was introduced independently by Gandy ([12]) and Jensen (=-=[15]-=-). See also [8], Sect. IV.1.s3.2 Generalized Absoluteness For simplicity of presentation, we assume the cumulative universe V of ZF , and formulate our definitions accordingly. It is easy to see that ... |

72 |
Basic Set Theory
- Levy
- 1979
(Show Context)
Citation Context ... on sets) it is necessary first to justify this introduction by proving a corresponding existence theorem. (The same is basically true in case set terms are officially used to denote “classes”, as in =-=[18]-=-, Sect. I.4.) The very useful complete separation we have in first-order logic between the (easy) check whether a given expression is a well-formed term or formula, and the (difficult) check whether i... |

52 | Languages which capture complexity classes
- Immerman
- 1987
(Show Context)
Citation Context ...ch assume infinity, and set theories which are valid also in the universe of hereditarily finite sets, can again be reduced to differences in the underlying syntactic safety relations. Definition 9. (=-=[14, 22]-=-) Let L be a (first-order) language. The language LT C is obtained from L by adding the following clause to the definition of a formula: If ϕ is a formula, x, y are distinct variables, and t, s are te... |

47 |
Foundations without foundationalism. A case for Second-Order Logic
- Shapiro
- 1991
(Show Context)
Citation Context ...ch assume infinity, and set theories which are valid also in the universe of hereditarily finite sets, can again be reduced to differences in the underlying syntactic safety relations. Definition 9. (=-=[14, 22]-=-) Let L be a (first-order) language. The language LT C is obtained from L by adding the following clause to the definition of a formula: If ϕ is a formula, x, y are distinct variables, and t, s are te... |

45 |
Foundations of Set Theory
- Fraenkel, Bar-Hillel, et al.
- 1973
(Show Context)
Citation Context ...al set notations and constructs as found in textbooks on naive or axiomatic set theory (and only such notations). Our starting point is what is known as the “ideal calculus” for naive set theory (see =-=[10]-=-, Sect. III.1). This very simple calculus is based on just two set-theoretical principles: extensionality and full comprehension. It thus exactly reflects our initial, immediate intuitions concerning ... |

31 |
The Incompleteness Theorems
- Smullyan
(Show Context)
Citation Context ...nductively be defined by using the clauses of Definition 3 and the assumption that x < t ≻b x if x �∈ F v(t). The set {ϕ | ϕ ≻b ∅} is a straightforward extension of Smullyan’s set of Σ0 formulas (see =-=[23]-=-, P. 41), which can serve as a basis for the usual arithmetical hierarchy. It is interesting to note that a succinct inductive definition of ≻b can be given which is almost identical to that of the ba... |

25 |
The Recursive Unsolvability of the Decision Problem for the Class of Definite Formulas
- Paola
- 1969
(Show Context)
Citation Context ...hich are mentioned in the query. Practical database query languages are designed so that only d.i. queries can be formulated in them. Unfortunately, it easily follows from Trakhtenbrot’s Theorem (see =-=[9]-=-) that it is undecidable which formulas are d.i. (or “safe” in any other reasonable notion of safety of queries, like “finite and computable”). Therefore all commercial query languages (like SQL) allo... |

24 | Finitary inductively presented logics
- FEFERMAN
- 1989
(Show Context)
Citation Context ...n in [3] concerning TC are: 1. If L contains a constant 0 and a (symbol for) a pairing function, then all types of finitary inductive definitions of relations and functions (as defined by Feferman in =-=[11]-=-) are available in LTC. 2. Let V0 be the smallest set including 0 and closed under the operation of pairing. Let U be the smallest set of first-order terms in a language with a constant for 0 and a fu... |

23 |
Cantorian Set Theory and Limitation of Size
- Hallett
- 1984
(Show Context)
Citation Context ... are believed to be consistent impose constraints on the use of this principle. In all textbooks the choice of these constraints is guided by semantic intuitions (like the limitation of size doctrine =-=[10, 16]-=-), especially the question: what operations on sets are “safe”. Since it is one of our main purposes to remain as close to the “ideal calculus” as possible, on one hand, and we aim at computerized sys... |

22 | Axioms of set theory - Shoenfield - 1977 |

18 |
Set Theory for Computing
- Cantone, Omodeo, et al.
- 2001
(Show Context)
Citation Context ...initions, and introduction of new symbols is reduced to using abbreviations. 1 The closest attempt I am aware of to develop a language for sets that employs static set terms can be found Sect. 5.1 of =-=[7]-=-. However, the construction there is rather complicated, and far remoted from actual mathematical practice. (The terms have the form: {tn+1 : x0C0t0, x1C1t1, . . . , xnCntn | ϕ}, where each Ci is eith... |

15 |
Neue Fassung des Widerspruchsfreiheit für die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften, NewSeries 4:19–44
- Gentzen
- 1938
(Show Context)
Citation Context ...nd T C. Then a subset S of V0 is recursively enumerable iff there exists a formula ϕ(x) of PT C + such that S = {x ∈ V0 | ϕ(x)}.s3. By generalizing a particular case which has been used by Gentzen in =-=[13]-=-, mathematical induction can be presented as a logical rule of languages with T C. Indeed, Using a Gentzen-type format, a general form of this principle can be formulated as follows: Γ, ψ, ϕ ⇒ ∆, ψ[x ... |

11 |
em Set-theoretic functions for elementary syntax
- Gandy
- 1974
(Show Context)
Citation Context ...P . Conversely, if P is a rudimentary predicate then there is a formula ϕ such that ϕ ≻RST ∅ and ϕ defines P . Theorem 3. RST is equivalent to the system obtained from Gandy’s “Basic Set Theory” BST (=-=[12]-=-) by the addition of the ∈ −induction schema. 6 The class of rudimentary set functions was introduced independently by Gandy ([12]) and Jensen ([15]). See also [8], Sect. IV.1.s3.2 Generalized Absolut... |

11 |
Transitive closure and the mechanization of mathematics
- Avron
- 2003
(Show Context)
Citation Context ...sible the uniformity of our framework, it is most natural to use as the underlying logic a logic which is stronger than FOL, but still reasonably manageable from a computational point of view. Now in =-=[3]-=- it was argued that languages and logics with transitive closure operation TC provide the best framework for the formalization of mathematics. Following this suggestion seems particularly suitable in ... |

9 |
Ackermann’s set theory equals ZF
- Reinhardt
- 1970
(Show Context)
Citation Context ... connected with absoluteness properties of ϕ occurs also (though with a very different formalization) in Ackermann’s set theory [1], which turned out to be equivalent (once one adds regularity) to ZF =-=[19]-=-. The connections (if any) between Ackermann’s approach and the present one are yet to be determined, and will be investigated in the future. (I am grateful to an anonymous referee for bringing Ackerm... |

7 |
Finitary Inductively Presented Logics, in: Logic Colloquium
- Feferman
- 1988
(Show Context)
Citation Context ... in [3] concerning T C are: 1. If L contains a constant 0 and a (symbol for) a pairing function, then all types of finitary inductive definitions of relations and functions (as defined by Feferman in =-=[11]-=-) are available in LT C. 2. Let V0 be the smallest set including 0 and closed under the operation of pairing. Let U be the smallest set of first-order terms in a language with a constant for 0 and a f... |

7 |
Zur Axiomatik der Mengenlehre
- Ackermann
- 1956
(Show Context)
Citation Context ...d perhaps be noted that the idea that existence of sets {x | ϕ} might be connected with absoluteness properties of ϕ occurs also (though with a very different formalization) in Ackermann’s set theory =-=[1]-=-, which turned out to be equivalent (once one adds regularity) to ZF [19]. The connections (if any) between Ackermann’s approach and the present one are yet to be determined, and will be investigated ... |

4 | Safety Signatures for First-order Languages and Their
- Avron
(Show Context)
Citation Context ...ds only on the safety relation used by each. Hence also the differences in strength between the systems will mainly be due to the differences between their safety relations. 2. It is easy to see (see =-=[4]-=-) that our assumptions concerning the underlying logic and the comprehension schema together imply that the above formulation of the extensionality axiom is equivalent to the more usual one: ∀z(z ∈ x ... |

4 | Constructibility and decidability versus domain independence and absoluteness
- Avron
- 2008
(Show Context)
Citation Context ...s that if b is an element of S1, and S2 |= a ∈ b, then a belongs to S1, and S1 |= a ∈ b. In other words: the formula x ∈ y should be d.i. with respect to {x} (but not with respect to {y}). In [4] and =-=[6]-=- this observation was used for developing a general framework forsdomain independence and absoluteness, and it was shown that this framework has deep applications in computability theory. The similari... |

3 |
Formalizing Set Theory as It Is Actually Used
- Avron
- 2004
(Show Context)
Citation Context ... facts that y ∈ z ≻RST y, and that P (z) = {x | ∀y(y ∈ x → y ∈ z)}. The proof of the second part is similar to that of Theorem 6. ⊓⊔ Another method (which may look more natural and is the one used in =-=[5]-=-) to add the power of the powerset axiom to the systems described above, is to extend the language by taking ⊆ as an extra primitive binary relation symbol. A definition of a system which is equivalen... |

2 |
Zur Axiomatik der Mengenlehre, Mathematische Annalen 131
- unknown authors
- 1956
(Show Context)
Citation Context ...d perhaps be noted that the idea that existence of sets {x | ϕ} might be connected with absoluteness properties of ϕ occurs also (though with a very different formalization) in Ackermann’s set theory =-=[1]-=-, which turned out to be equivalent (once one adds regularity) to ZF [19]. The connections (if any) between Ackermann’s approach and the present one are yet to be determined, and will be investigated ... |

2 |
Transitive closure and the mechanization of mathematics. In: Thirty Five Years of Automating Mathematics
- Avron
- 2003
(Show Context)
Citation Context ...sible the uniformity of our framework, it is most natural to use as the underlying logic a logic which is stronger than FOL, but still reasonably manageable from a computational point of view. Now in =-=[3]-=- it was argued that languages and logics with transitive closure operation T C provide the best framework for the formalization of mathematics. Following this suggestion seems particularly suitable in... |