## The Anti-Foundation Axiom In Constructive Set Theories (2003)

Venue: | Stanford University Press |

Citations: | 6 - 5 self |

### BibTeX

@INPROCEEDINGS{Rathjen03theanti-foundation,

author = {M. Rathjen},

title = {The Anti-Foundation Axiom In Constructive Set Theories},

booktitle = {Stanford University Press},

year = {2003},

pages = {87--108},

publisher = {CSLI Publ}

}

### OpenURL

### Abstract

. The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-wellfounded sets, or hypersets (cf. [17], [5]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [7]). Instead of the Foundation Axiom these set theories adopt the so-called Anti-Foundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and pr...

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Citation Context ...terpreted in \Sigma 1 1 -AC. Proof. This follows from the fact that the fixed point axioms of c ID 1 can be emulated in \Sigma 1 1 -AC by interpreting the fixed points as \Sigma 1 1 sets (cf. [1] and =-=[8]-=-). 3.4. Lower bounds. Lower bounds for the theories CZFA+\Sigma 1 -IND ! and CZFA + IND! can be established by interpreting suitable intuitionistic theories RA i ff of the ramified hierarchy up to lev... |

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Citation Context ...paper investigates the strength of AFA on the basis of various systems of constructive set theories, including ones with large set axioms. Constructive set theory grew out of Myhill's endeavours (cf. =-=[21]-=-) to discover a simple formalism that relates to Bishop's constructive mathematics as ZFC relates to classical Cantorian mathematics. Later on Supported by the Volkswagen-Stiftung (RiP program Oberwol... |

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Citation Context ...S 5 formula provably in CZF \Gamma . The latter principle is sometimes called the \Sigma Reflection Principle and can be proved like in Kripke-Platek set theory (one easily verifies that the proof of =-=[6]-=-, I.4.3 also works in CZF \Gamma ). \Sigma-IND ! enables one to define functions by \Sigma recursion on ! (cf. [6], I.6) and hence one can prove the existence of every primitive recursive function on ... |

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Citation Context ... one adds a strong system type S to ML 1 the proof theoretic strength does not increase. This can be seen by emulating ML 1 +S in the theory c ID 1 of positive arithmetic fixed points similarly as in =-=[14]-=-. Definition 3.13. Let L + be the language of Peano arithmetic augmented by a unary predicate symbol Q. For each formula OE(Q + ; u) of L + in which only the variable u occurs free and Q occurs only p... |

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Citation Context ...tion rule for it. The strength of a type theory ML 1 with the type constructors \Pi; \Sigma; +; I; N; N 0 ; N 1 and one universe U closed under these same constructors has been determined by Aczel in =-=[1]. ML -=-1 has the same strength as the theory \Sigma 1 1 -AC, a subsystem of second order arithmetic with the \Sigma 1 1 axiom of choice. Its proof theoretic ordinal is "' 0 0, where ' denotes the Veblen... |

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Citation Context ...the ffth level of the von Neumann hierarchy. Proposition 4.5. (ZFC) A set I is inaccessible if and only if I = Vsfor some strongly inaccessible cardinal . Proof. This is a consequence of the proof of =-=[24]-=-, Corollary 2.7. Proposition 4.6. Let EM denote the principle of excluded middle. The theories CZF \Gamma +INAC + EM and ZFC + 8ff 9 (ff !sis a strongly inaccessible cardinal) have the same proof theo... |

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Citation Context ...he type V then yields a realizability interpretation of CZF+INAC `a la Aczel, using the techniques of [24]. On the other hand, MLS can be interpreted in the classical set theory KPi by the methods of =-=[23]-=-, section 5. As CZF + REA has the same strength as KPi by [23], CZF +REA and CZF +INAC also have the same strength. Theorem 4.8. The proof theoretic ordinal of CZF \Gamma + INAC is the Feferman-Schutt... |

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Citation Context ...finition 3.14. Occasionally the theory c ID 1 is too `coarse' to obtain exact proof theoretic results. In those situations theories of natural numbers and ordinals which have been introduced by Jager =-=[18]-=- provide a versatile tool. To give an example, the notion of being (a code for) a small type is simulated in c ID 1 via a formula which is no longer positive in the fixed point predicates (Aczel's `tr... |

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Citation Context ...ing a code for an iterative set become \Sigma\Omega . The notion of being an element of a code for a small type is both \Sigma\Omega and \Pi\Omega . Details of the above constructions can be found in =-=[11]-=-. One then has to scrutinize all the type theoretic constructions of section and [19] to determine the kind of structural induction and recursion on N that is required for them. Closer inspection reve... |

4 |
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Citation Context ...atisfied: 1. I is a regular set, 2. !2I, 3. (8a2I) S a2I, 4. I is V -closed, 5. (8a; b 2 I) \Theta fx21 : a = bg 2 Isfx21 : a2bg 2 I . 3 6. (8a; b 2 I)(9c2I) \Theta c is full in mv( a b) : Proof. See =-=[26]-=-, Proposition 3.4. Viewed classically inaccessible sets are closely related to inaccessible cardinals. Let V ff denote the ffth level of the von Neumann hierarchy. Proposition 4.5. (ZFC) A set I is in... |

3 | Inaccessible set axioms may have little consistency strength
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Citation Context .... Due to the lack of 2-induction this system is proof-theoretically weak. On the other hand it is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics. =-=[12]-=- investigated the strength of CZF \Gamma plus the statement that every set is contained in an inaccessible set, INAC. [12] showed that CZF \Gamma + INAC has a realizability interpretation in type theo... |

2 |
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Citation Context ...e, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-wellfounded sets, or hypersets (cf. [17], =-=[5]-=-). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [7]). Instead of the Foundation Axiom the... |

2 |
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Citation Context ...lae which is closed under :; ; , number quantification, and bounded ordinal quantification) can be interpreted (asymmetrically) in the formal system RA!! ! by a method very similar to the one used in =-=[22], Theorem -=-5.2. It is also known that the proof-theoretic ordinal of RA i !"0 is "' 0 0 (cf. [15], chap. I., Theorem 3.2.13 and Theorem 3.1.11). As the latter ordinal is the proof-theoretic ordinal of ... |

1 |
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Citation Context ...t is known that the proof-theoretic ordinal of RA i !! ! is '!0; this follows from Theorem 3.2.13 and Theorem 3.1.11 in [15], chap. I. '!0 is the proof-theoretic ordinal of \Sigma 1 1 -DC 0 , too, by =-=[9]-=-. As it can be shown that '!0 is also the proof-theoretic ordinal of PA r\Omega + \Sigma\Omega -IND, (i) follows from Corollary 3.15 and Proposition 3.18. The proof that '!0 is the proof-theoretic ord... |

1 | Realizability interpretations for subsystems of CZF and proof theoretic strength - Crosilla - 1998 |

1 |
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Citation Context ...a 1 -IND ! and CZFA + IND! can be established by interpreting suitable intuitionistic theories RA i ff of the ramified hierarchy up to level ff in them. For the definition of the theories RA i ff see =-=[15], cha-=-pter II, 1.2. We assume that an ordinal representation system for " 0 has been formalized in CZF \Gamma + \Sigma 1 -IND ! as a decidable subset of ! (where A ` ! is said to be decidable if (8n2!)... |

1 |
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Citation Context ...n Martin-Lof type theory. The constructiveness of CZF was shown by Aczel by giving it an interpretation in Martin-Lof's intuitionistic theory of types (cf. [2, 3, 4]). I. Lindstrom [19] and L. Halnas =-=[16]-=- have shown that CZFA can be interpreted in Martin-Lof type theory as well. In this subsection we shall recall the interpretation of CZFA in Martin-Lof type theory, MLTT, as presented in [19]. In the ... |

1 | Realizing Mahlo set theory in type theory - Rathjen - 1998 |

1 | Kripke-Platek set theory and the anti-foundation axiom
- Rathjen
(Show Context)
Citation Context ...ction. This contrasts with Kripke-Platek set theory, KP. The theory KPA, which adopts AFA in place of the Foundation Axiom scheme, is proof-theoretically considerably stronger than KP as was shown in =-=[28]-=-. This paper also contains several other new results. 2. The anti-foundation axiom Definition 2.1. A graph will consist of a set of nodes and a set of edges, each edge being an ordered pair (x; y) of ... |