### BibTeX

@MISC{Avron_anew,

author = {Arnon Avron},

title = {A New Approach to Predicative Set Theory},

year = {}

}

### OpenURL

### Abstract

We suggest a new basic framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domain-independence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is type-free, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1

### Citations

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1066 |
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(Show Context)
Citation Context ...k has been introduced. This framework unifies different notions of “safety” of formulas, coming from different areas of mathematics and computer science, like: domain independence in database theory (=-=[1,48]-=-), decidability of arithmetical formulas in computability theory and metamathematics, and absoluteness in set theory. In the next definition we review this Framework. Notation. Let σ be a first-order ... |

283 |
Set Theory: An Introduction to Independence Proofs
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Citation Context ...nce it makes the l.u.b. principle valid for predicatively acceptable sets of reals. 3Note on the other hand that ϕ is predicative for ∅ iff it is absolute in the usual sense of set theory. (see e.g. =-=[33]-=-). The main problem in formulating a predicative, type-free, set theory is how to syntactically impose this predicativity property on formulas without introducing syntactic types or levels. The soluti... |

216 | Mathematical Logic - Shoenfield - 1967 |

191 |
Admissible Sets and Structures
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Citation Context ...ould in fact suffice for (most of) applicable mathematics. Now PZF0 is relatively a week system. Thus it can easily be interpreted in Kripke-Platek set theory KP together with the infinity axiom (see =-=[7,31,11]-=-) 9 . However, it should again be emphasized that PZF as a whole is open-ended, and transcends any given formal system. Note 4 In addition to having TC (which is the major difference between our under... |

153 | Logic and the challenge of computer science
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(Show Context)
Citation Context ...erty 7 from the list in Theorem 1. 6yond FOL (First-Order Logic) by introducing an operation TC for transitive closure 6 . The corresponding language and semantics are defined as follows (see, e.g., =-=[30,29,47,28,13]-=-): Definition 3 Let σ be a signature for a first-order language with equality. The language L1 TC (σ) is defined like the usual first-order language which is based on σ, but with the addition of the f... |

113 |
The fine structure of the constructible hierarchy
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Citation Context ...T, and connect it (and ≻RST) with the class of rudimentary set functions — a refined version of Gödel basic set functions (from [27]) which was independently introduced by Gandy in [25] and Jensen in =-=[32]-=- (See also [10]). Theorem 2 (1) If F is an n-ary rudimentary function, then there exists a formula ϕF with the following properties: (a) Fv(ϕF) ⊆ {y, x1, . . .,xn} (b) ϕF ≻RST {y} (c) F(x1, . . .,xn) ... |

99 |
Proof Theory
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Citation Context ...would be much more natural to use terms of our systems which provably denote in them von Neumann’s ordinals. We expect that every ordinal less than Γ0, the Feferman-Schütte ordinal for predicativity (=-=[15,17,44,45]-=-), should be obtainable in this way. Decoding: Although {ωn | n ∈ N} and λn ∈ N.ωn are not definable in PZF, {�ωn� | n ∈ N} and λn ∈ N.�ωn� are definable, where �ωn� is some natural Gödel code in HF f... |

90 | A survey of the project AUTOMATH - Bruijn - 1980 |

58 |
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Citation Context ...wever, the use of levels makes it impossible to develop mathematics in RTT, and so Russell had to add a special axiom of reducibility which practically destroyed the predicative nature of his system (=-=[37]-=-). The principle was then taken again by Weyl in [50], but instead of Russell’s ramified hierarchy, Weyl adopted the second principle, (NAT), which also goes back to Poincaré. Weyl’s predicativist pro... |

52 | Languages which capture complexity classes
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(Show Context)
Citation Context ...erty 7 from the list in Theorem 1. 6yond FOL (First-Order Logic) by introducing an operation TC for transitive closure 6 . The corresponding language and semantics are defined as follows (see, e.g., =-=[30,29,47,28,13]-=-): Definition 3 Let σ be a signature for a first-order language with equality. The language L1 TC (σ) is defined like the usual first-order language which is based on σ, but with the addition of the f... |

47 |
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Citation Context ...erty 7 from the list in Theorem 1. 6yond FOL (First-Order Logic) by introducing an operation TC for transitive closure 6 . The corresponding language and semantics are defined as follows (see, e.g., =-=[30,29,47,28,13]-=-): Definition 3 Let σ be a signature for a first-order language with equality. The language L1 TC (σ) is defined like the usual first-order language which is based on σ, but with the addition of the f... |

46 |
Systems of predicative analysis
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(Show Context)
Citation Context ...fied hierarchy, Weyl adopted the second principle, (NAT), which also goes back to Poincaré. Weyl’s predicativist program was later extensively pursued by Feferman, who in a series of papers (see e.g. =-=[15,17,19,20]-=-) developed proof systems for predicative mathematics. Feferman’s systems are less complex than RTT, and he has shown that a very large part of classical analysis can be developed within them. He furt... |

34 |
Theories for Admissible Sets: A Unifying Approach to Proof Theory
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(Show Context)
Citation Context ...ould in fact suffice for (most of) applicable mathematics. Now PZF0 is relatively a week system. Thus it can easily be interpreted in Kripke-Platek set theory KP together with the infinity axiom (see =-=[7,31,11]-=-) 9 . However, it should again be emphasized that PZF as a whole is open-ended, and transcends any given formal system. Note 4 In addition to having TC (which is the major difference between our under... |

26 |
The Consistency of the Continuum Hypothesis
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- 1940
(Show Context)
Citation Context ...ng theorem and its two corollaries determine the expressive power of LRST, and connect it (and ≻RST) with the class of rudimentary set functions — a refined version of Gödel basic set functions (from =-=[27]-=-) which was independently introduced by Gandy in [25] and Jensen in [32] (See also [10]). Theorem 2 (1) If F is an n-ary rudimentary function, then there exists a formula ϕF with the following propert... |

25 |
Computable set theory
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- 1989
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Citation Context ...straightforward way. 2 The use of such terms, albeit in a somewhat cumbersome form, more complicated than that actually used in mathematical texts, is also a major feature of the systems developed in =-=[8,9]-=-. 22 The Main Ideas 2.1 Interpreting and Implementing Principle (PRE) According to our approach, a predicative set theory need not exclude the possibility that “arbitrary (undefinable) sets of intege... |

21 |
Kritische Untersuchungen über die Grundlagen der Analysis (Velt
- Weyl, Kontinuum
(Show Context)
Citation Context ...of FOL with a transitive closure operation). 1 Introduction The predicativist program for the foundations of mathematics, initiated by Poincaré in [35,36] 1 , and first seriously developed by Weyl in =-=[50]-=-, seeks to establish certainty in mathematics without revolutionizing it (as the intuitionistic program does). The program as is usually conceived nowadays (following Weyl and Feferman) is based on th... |

16 |
Set Theory for Computing. From Decision Procedures to Declarative Programming with Sets
- Cantone, Omodeo, et al.
- 2001
(Show Context)
Citation Context ...straightforward way. 2 The use of such terms, albeit in a somewhat cumbersome form, more complicated than that actually used in mathematical texts, is also a major feature of the systems developed in =-=[8,9]-=-. 22 The Main Ideas 2.1 Interpreting and Implementing Principle (PRE) According to our approach, a predicative set theory need not exclude the possibility that “arbitrary (undefinable) sets of intege... |

16 |
Mixed Computations
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- 1984
(Show Context)
Citation Context ...ould in fact suffice for (most of) applicable mathematics. Now PZF0 is relatively a week system. Thus it can easily be interpreted in Kripke-Platek set theory KP together with the infinity axiom (see =-=[7,31,11]-=-) 9 . However, it should again be emphasized that PZF as a whole is open-ended, and transcends any given formal system. Note 4 In addition to having TC (which is the major difference between our under... |

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 ...lows a very natural treatment of induction as a logical rule, as well as a neat extension of the safety relation - see below. 7(3) By generalizing a particular case which has been used by Gentzen in =-=[26]-=-, mathematical induction can be presented as a logical rule of languages with TC. Indeed, Using a Gentzen-type format, a general form of this principle can be formulated as follows: Γ, ψ, ϕ ⇒ ∆, ψ(y/x... |

15 |
On Transitive Closure Logic
- Grädel
- 1992
(Show Context)
Citation Context |

11 |
Transitive closure and the mechanization of mathematics
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- 2003
(Show Context)
Citation Context ...ike forming the notion of an ancestor from the notion of a parent) should be taken as a major ingredient of our logical abilities (even prior to our understanding of the natural numbers). In fact, in =-=[2]-=- it was argued that this concept is the key for understanding finitary inductive definitions and reasoning, and evidence was provided for the thesis that systems which are based on it provide the righ... |

11 |
em Set-theoretic functions for elementary syntax
- Gandy
- 1974
(Show Context)
Citation Context ...reditarily finite sets, is a model of RST. Hence ω is not definable in LRST, and so TC is indeed necessary for its definition. 11 Note 5 RST can be shown to be equivalent to Gandy’s basic set theory (=-=[25]-=-), and to the system called BST0 in [43]. The following theorem and its two corollaries determine the expressive power of LRST, and connect it (and ≻RST) with the class of rudimentary set functions — ... |

11 | On bounded set theory
- Sazonov
- 1995
(Show Context)
Citation Context ...T. Hence ω is not definable in LRST, and so TC is indeed necessary for its definition. 11 Note 5 RST can be shown to be equivalent to Gandy’s basic set theory ([25]), and to the system called BST0 in =-=[43]-=-. The following theorem and its two corollaries determine the expressive power of LRST, and connect it (and ≻RST) with the class of rudimentary set functions — a refined version of Gödel basic set fun... |

10 | Predicative Foundations of Arithmetic
- Feferman, Hellman
- 1995
(Show Context)
Citation Context ...ched with TC is equivalent in its expressive power to the language of weak SOL. So taking “transitive closure” as primitive is equivalent to taking “finite set” as primitive (which is the approach of =-=[23]-=-, though the system presented there is essentially first-order). We prefer the former as primitive, because it allows a very natural treatment of induction as a logical rule, as well as a neat extensi... |

9 |
Weyl vindicated: Das Kontinuum seventy years later
- Feferman
- 1998
(Show Context)
Citation Context ...fied hierarchy, Weyl adopted the second principle, (NAT), which also goes back to Poincaré. Weyl’s predicativist program was later extensively pursued by Feferman, who in a series of papers (see e.g. =-=[15,17,19,20]-=-) developed proof systems for predicative mathematics. Feferman’s systems are less complex than RTT, and he has shown that a very large part of classical analysis can be developed within them. He furt... |

8 |
A more perspicuous formal system for predicativity
- Feferman
- 1979
(Show Context)
Citation Context ...fied hierarchy, Weyl adopted the second principle, (NAT), which also goes back to Poincaré. Weyl’s predicativist program was later extensively pursued by Feferman, who in a series of papers (see e.g. =-=[15,17,19,20]-=-) developed proof systems for predicative mathematics. Feferman’s systems are less complex than RTT, and he has shown that a very large part of classical analysis can be developed within them. He furt... |

8 | Computer aided reasoning - Trybulec, Blair - 1985 |

7 |
Finitary Inductively Presented Logics, in: Logic Colloquium
- Feferman
- 1988
(Show Context)
Citation Context ... in [2] concerning TC are: (1) If σ 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 =-=[21]-=-) are available in L1 TC (σ). This result, in turn, allows for presenting a simple version of Feferman’s framework FS0, demonstrating that TC-logics provide an excellent framework for mechanizing form... |

7 |
Set Theory — An Operational Approach
- Sanchis
- 1996
(Show Context)
Citation Context ...ily implies that PA, the first-order Peano’s Arithmetics, has a natural interpretation in PZF0 (see Proposition 4 for a partial proof). However, the availability of ω alone is 12 Thus both Sanchis in =-=[42]-=- and Weaver in [49] argue that classical logic is unsuitable for dealing with the whole of V , and intuitionistic logic should be used for it instead. 17not sufficient for getting the full power of m... |

4 | Safety Signatures for First-order Languages and Their
- Avron
(Show Context)
Citation Context ...oducing syntactic types or levels. The solution suggested here to this problem comes from the observation that this is an instance of a more general task, not peculiar only to set Theory. In fact, in =-=[3]-=- and [6] an appropriate purely logical framework that can be used for this task has been introduced. This framework unifies different notions of “safety” of formulas, coming from different areas of ma... |

4 | Constructibility and decidability versus domain independence and absoluteness
- Avron
- 2008
(Show Context)
Citation Context ...syntactic types or levels. The solution suggested here to this problem comes from the observation that this is an instance of a more general task, not peculiar only to set Theory. In fact, in [3] and =-=[6]-=- an appropriate purely logical framework that can be used for this task has been introduced. This framework unifies different notions of “safety” of formulas, coming from different areas of mathematic... |

3 |
Formalizing Set Theory as It Is Actually Used
- Avron
- 2004
(Show Context)
Citation Context ...d by adding to the definition of LPZF the following three conditions: (1) ϕ ≻TZF ∅ for every formula ϕ. (2) x ⊆ t ≻TZF {x} if x ̸∈ Fv(t). (3) ∃yϕ ∧ ∀y(ϕ → ψ) ≻TZF X if ψ ≻TZF X, and X ∩ Fv(ϕ) = ∅. In =-=[4,5]-=- it was shown that a first-order system which is equivalent to ZF (but more natural and easier to mechanize than the usual presentation of ZF) is obtained from TZF if the underlying logic is changed t... |

3 |
Systems of Predicative Analysis II
- Feferman
- 1968
(Show Context)
Citation Context |

3 |
A derivation on number theory from ancestral theory
- Myhill
- 1952
(Show Context)
Citation Context ...iables within the scope of an operator which binds them. Otherwise the rules/axioms concerning the quantifiers and terms 8 The resulting system is equivalent to Myhill’s system for ancestral logic in =-=[34]-=-. 9 KP itself includes the ∆0-collection schema, which is not predicatively justified. 11remain unchanged (for example: ∀xϕ → ϕ{t/x} is valid for every term t which is free for x in ϕ). We also assum... |

2 | From Kant to Hilbert - Ewald - 1996 |

2 |
Predicatively Reducible Systems of Set Theory, in Axiomatic Set Theory
- Feferman
- 1974
(Show Context)
Citation Context ...ms with previous works concerned with predicative set theory. This includes first of all Feferman’s various systems for predicative mathematics, especially his system PS1E for predicative set theory (=-=[16,18]-=-), and his system W from [20]. Also relevant are the proof-theoretic investigations of systems of Kripke-Platek set theory by Jäger, Pohlers, and Rathjen (a partial list), as well as the works on cons... |

2 |
Dernières pensées; Flammarion
- Poincaré
- 1913
(Show Context)
Citation Context ...erlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1 Introduction The predicativist program for the foundations of mathematics, initiated by Poincaré in =-=[35,36]-=- 1 , and first seriously developed by Weyl in [50], seeks to establish certainty in mathematics without revolutionizing it (as the intuitionistic program does). The program as is usually conceived now... |

2 | Isabell/HOL A proof Assistant for Higher Order Logic, LNCS 2283 - Nipkow, Paulson, et al. - 2002 |

1 | A Framework for Formalizing Set Theories Based on the Use of Static Set Terms
- Avron
- 2008
(Show Context)
Citation Context ...terparts of our various claims can be formulated and proved, but we leave this task to another opportunity. The straightforward proof of the following proposition was practically given in Note 1 (see =-=[5]-=- for a proof of a stronger claim): Proposition 2 V is a model of PZF. 4 The Expressive Power of PZF 4.1 Some Standard Notations for Sets In LPZF we can introduce as abbreviations most of the standard ... |

1 |
Predicative Provability in Set Theory
- Feferman
- 1966
(Show Context)
Citation Context ...ms with previous works concerned with predicative set theory. This includes first of all Feferman’s various systems for predicative mathematics, especially his system PS1E for predicative set theory (=-=[16,18]-=-), and his system W from [20]. Also relevant are the proof-theoretic investigations of systems of Kripke-Platek set theory by Jäger, Pohlers, and Rathjen (a partial list), as well as the works on cons... |

1 |
Mathématiques et la Logique, II, III, Revue de Métaphysique et Morale 14
- Poincaré, Les
- 1906
(Show Context)
Citation Context ...erlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1 Introduction The predicativist program for the foundations of mathematics, initiated by Poincaré in =-=[35,36]-=- 1 , and first seriously developed by Weyl in [50], seeks to establish certainty in mathematics without revolutionizing it (as the intuitionistic program does). The program as is usually conceived now... |

1 |
Letter à Monnsieur le rédacteur de la Revue général de Sciences
- Richard
- 1905
(Show Context)
Citation Context ...a totality it constitutes a set. Email address: aa@cs.tau.ac.il (Arnon Avron). URL: http://www.cs.tau.ac.il/ aa/ (Arnon Avron). 1 Though its kernel can be found in Richard’s discussion of his paradox =-=[38]-=-. Preprint submitted to Elsevier Science 28 May 2009The first of these principles, (PRE), was interpreted by Russell according to his philosophical views of logic, [39], [40], and incorporated as the... |

1 |
Mathematical Logic as based on a theory of logical types
- Russelll
- 1908
(Show Context)
Citation Context ...iscussion of his paradox [38]. Preprint submitted to Elsevier Science 28 May 2009The first of these principles, (PRE), was interpreted by Russell according to his philosophical views of logic, [39], =-=[40]-=-, and incorporated as the ramified type theory (RTT) in Principia Mathematica ([51]). In RTT objects are divided into types, and each higher-order type is further divided into levels. However, the use... |

1 | Essays in Analysis, edited by D - Russelll - 1973 |

1 |
Mathematical Conceptualism, unpublished manuscript (available from http://www.math.wustle.edu/nweaver/concept.pdf
- Weaver
(Show Context)
Citation Context ..., the first-order Peano’s Arithmetics, has a natural interpretation in PZF0 (see Proposition 4 for a partial proof). However, the availability of ω alone is 12 Thus both Sanchis in [42] and Weaver in =-=[49]-=- argue that classical logic is unsuitable for dealing with the whole of V , and intuitionistic logic should be used for it instead. 17not sufficient for getting the full power of mathematical inducti... |

1 |
Paradoxes de la logique, Revue de Métaphisique et de Morale
- Russell, Les
- 1906
(Show Context)
Citation Context ...s a basic well understood mathematical concept, and as a totality it constitutes a set. The first of these principles, (PRE), was interpreted by Russell according to his philosophical views of logic, =-=[38]-=-, [39], and incorporated as the ramified type theory (RTT) in Principia Mathematica ([50]). In RTT objects are divided into types, and each higher-order type is further divided into levels. However, t... |

1 |
Mathematical Logic as based on a theory of logical types
- Russell
- 1908
(Show Context)
Citation Context ...sic well understood mathematical concept, and as a totality it constitutes a set. The first of these principles, (PRE), was interpreted by Russell according to his philosophical views of logic, [38], =-=[39]-=-, and incorporated as the ramified type theory (RTT) in Principia Mathematica ([50]). In RTT objects are divided into types, and each higher-order type is further divided into levels. However, the use... |

1 | Essays in Analysis, edited by D - Russell - 1973 |

1 |
trans. by J. Bolduc as Mathematics and Science: Last Essays
- Poincaré, Pensées, et al.
- 1913
(Show Context)
Citation Context ...erlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1 Introduction The predicativist program for the foundations of mathematics, initiated by Poincaré in =-=[34,35]-=- 1 , and first seriously developed by Weyl in [49], seeks to establish certainty in mathematics without revolutionizing it (as the intuitionistic program does). The program as is usually conceived now... |