## Characterizing the interpretation of set theory in Martin-Löf type theory

### BibTeX

@MISC{Rathjen_characterizingthe,

author = {Michael Rathjen and Sergei Tupailo},

title = {Characterizing the interpretation of set theory in Martin-Löf type theory},

year = {}

}

### OpenURL

### Abstract

### Citations

463 | Foundations of Constructive Analysis
- Bishop
- 1967
(Show Context)
Citation Context ...ry with the axiom of choice relates to classical Cantorian mathematics. CST provides a standard set theoretical framework for the development of constructive mathematics in the style of Errett Bishop =-=[8]-=-. One of the hallmarks of constructive set theory is that it possesses (due to Aczel [1, 2, 3]) a canonical interpretation in Martin-Löf’s intuitionistic type theory (see [13, 14]) which is considered... |

290 |
Foundations of Constructive Mathematics
- Beeson
(Show Context)
Citation Context ...two results can be proved in the theory APP and thus hold true in any applicative structure. Thence applicative structure above satisfies the Abstraction Lemma and Recursion Theorem (see e.g. [10] or =-=[7]-=-). Lemma 4.6 (Abstraction lemma, cf. [7, VI.2.2]) For every application term t[x] there exists an application term λx.t[x] with FV(λx.t[x]) := {x1, . . . , xn} ⊆ FV(t[x])\{x} such that the following h... |

209 | Admissible Sets and Structures - Barwise - 1975 |

181 |
An intuitionistic theory of types: Predicative part. Logic Colloquium ‘73
- Martin-Löf
- 1975
(Show Context)
Citation Context ...he style of Errett Bishop [8]. One of the hallmarks of constructive set theory is that it possesses (due to Aczel [1, 2, 3]) a canonical interpretation in Martin-Löf’s intuitionistic type theory (see =-=[13, 14]-=-) which is considered to be the most acceptable foundational framework of ideas that make precise the constructive approach to mathematics. The interpretation employs the Curry-Howard ‘propositions-as... |

158 |
Intuitionistic Type Theory. Bibliopolis
- Martin-Löf
- 1984
(Show Context)
Citation Context ...he style of Errett Bishop [8]. One of the hallmarks of constructive set theory is that it possesses (due to Aczel [1, 2, 3]) a canonical interpretation in Martin-Löf’s intuitionistic type theory (see =-=[13, 14]-=-) which is considered to be the most acceptable foundational framework of ideas that make precise the constructive approach to mathematics. The interpretation employs the Curry-Howard ‘propositions-as... |

125 | The Type Theoretic Interpretation of Constructive Set Theory: Choice Principles
- Aczel
- 1982
(Show Context)
Citation Context ...tandard set theoretical framework for the development of constructive mathematics in the style of Errett Bishop [8]. One of the hallmarks of constructive set theory is that it possesses (due to Aczel =-=[1, 2, 3]-=-) a canonical interpretation in Martin-Löf’s intuitionistic type theory (see [13, 14]) which is considered to be the most acceptable foundational framework of ideas that make precise the constructive ... |

57 |
Constructive set theory
- Myhill
- 1975
(Show Context)
Citation Context ...atized by suitable forms of the axiom of choice. Keywords: Constructive Set Theory, Mart. 1 Introduction The general topic of Constructive Set Theory (CST ) originated in John Myhill’s endeavour (see =-=[17]-=-) to discover a simple formalism that relates to Bishop’s constructive mathematics as classical Zermelo-Fraenkel Set Theory with the axiom of choice relates to classical Cantorian mathematics. CST pro... |

52 | Notes on constructive set theory
- Aczel, Rathjen
- 2001
(Show Context)
Citation Context ...nd J a = {x | 〈a, x〉 ∈ J}, it holds I(Φ) = � J a , and for each a, a J a = ΓΦ( � J x ). J is uniquely determined by the above, and its stages J a will be denoted by Γ a Φ . Proof. [2], section 4.2 or =-=[4]-=-, Theorem 5.1. ✷ Lemma 2.8 (CZF) There exists a smallest ΠΣ-closed class, i.e. a smallest class Y such that the following holds: (i) n ∈ Y for all n ∈ N; (ii) ω ∈ Y; (iii) � x∈A Bx ∈ Y and � x∈A Bx ∈ ... |

27 |
Set-theoretic foundations for constructive analysis
- Friedman
(Show Context)
Citation Context ...oying clause (5), one can define the set of equivalence classes of Cauchy sequences, i.e., the set of reals. 3. Definition 5.4 clause (5) is related to the abstraction axiom of Friedman’s system B in =-=[11]-=-. Lemma 5.6 1. (CZF) Every mathematical set term is a set. 2. (CZF + REA) Every generalized mathematical set term is a set. Proof : We proceed by induction on the clauses for the definition of mathema... |

26 |
Axiom of choice and complementation
- Diaconescu
- 1975
(Show Context)
Citation Context ...r hand, it has been observed that the full axiom of choice cannot be added to systems of extensional constructive set theory without yielding constructively unacceptable cases of excluded middle (see =-=[9]-=-). In extensional intuitionistic set theories, a proof of a statement ∀x ∈ A ∃y ∈ B φ(x, y), in general, provides only a function F , which when fed a proof p witnessing x ∈ A, yields F (p) ∈ B and φ(... |

15 | The strength of some Martin–Löf type theories, Archive for Mathematical Logic - Rathjen - 1994 |

11 |
Feferman: Constructive Theories of Functions and classes
- unknown authors
- 1979
(Show Context)
Citation Context ...he next two results can be proved in the theory APP and thus hold true in any applicative structure. Thence applicative structure above satisfies the Abstraction Lemma and Recursion Theorem (see e.g. =-=[10]-=- or [7]). Lemma 4.6 (Abstraction lemma, cf. [7, VI.2.2]) For every application term t[x] there exists an application term λx.t[x] with FV(λx.t[x]) := {x1, . . . , xn} ⊆ FV(t[x])\{x} such that the foll... |

9 |
The lack of definable witnesses and provably recursive functions in intuitionistic set theories
- Friedman, Scedrov
- 1985
(Show Context)
Citation Context ...son) IZF has the DP and the NEP. (iii) (Friedman) IZF does not have the EP. (iv) (Rathjen) CZF and CZF + REA have the DP and the NEP. Proof : (i) is proved in [17]. For (ii) see [6] and for (iii) see =-=[12]-=-. (iv) is shown in [22]. ⊓⊔ The question of whether CZF satisfies the existence property is currently unanswered. Friedman’s proof of the failure of EP for IZF seems to single out Collection as the cu... |

6 |
Continuity in intuitionistic set theories
- Beeson
- 1979
(Show Context)
Citation Context ... and the EP. (ii) (Beeson) IZF has the DP and the NEP. (iii) (Friedman) IZF does not have the EP. (iv) (Rathjen) CZF and CZF + REA have the DP and the NEP. Proof : (i) is proved in [17]. For (ii) see =-=[6]-=- and for (iii) see [12]. (iv) is shown in [22]. ⊓⊔ The question of whether CZF satisfies the existence property is currently unanswered. Friedman’s proof of the failure of EP for IZF seems to single o... |

4 |
set Recursion, Annals of Pure and Applied Logic 71
- Moss, Power
- 1995
(Show Context)
Citation Context ...ssical standpoint, E℘-computability is related to power recursion, where the power set operation is regarded to be an initial function. The latter notion has been studied by Moschovakis [15] and Moss =-=[16]-=-. There is a lot of leeway in setting up E℘-recursion. The particular schemes we use are especially germane to our situation. Our construction will provide a specific set-theoretic model for the eleme... |

4 |
Set recursion, in: Generalized Recursion Theory
- Normann
- 1978
(Show Context)
Citation Context ...lication of sets to sets, i.e. we would like to assign a meaning to the symbol {a}(x) where a and x are sets. In generalized recursion theory this is known as E-recursion or set recursion (see, e.g., =-=[18]-=- or [23, Ch.X]). However, we shall introduce an extended notion of E-computability, christened E℘-computability, rendering the function exp(a, b) = a b is computable as well, (where a b denotes the se... |

4 | Choice principles in constructive and classical set theories
- Rathjen
- 2006
(Show Context)
Citation Context ...m a type-theoretic point of view has turned out to be valuable heuristic tool for finding new constructive choice principles. For more information on choice principles in the constructive context see =-=[20]-=-. 2.1 Some constructive choice principles In many a text on constructive mathematics, axioms of countable choice and dependent choices are accepted as constructive principles. This is, for instance, t... |

4 | Realization of analysis into explicit mathematics
- Tupailo
(Show Context)
Citation Context ....33. The proof builds on the proof of Theorem 4.13. ⊓⊔ 5 The formulae-as-classes interpretation and validity in H(Y ∗ ) The following considerations are reminiscent of Definition 3.8 and Theorem 3 of =-=[25]-=-. Definition 5.1 A formula is said to be CC if no unbounded quantifier in it occurs in the antecedent of an implication. Note that bounded as well as prenex (i.e. bounded preceded by a string of quant... |

4 | Realization of constructive set theory into explicit mathematics: a lower bound for impredicative Mahlo operation - Tupailo |

3 | Moschovakis: Recursion in the universe of sets, mimeographed note - N - 1976 |

2 |
The formulae-as-classes interpretation of constructive set theory. To appear in
- Rathjen
(Show Context)
Citation Context ...n itself via a formulae-as-classes interpretation. This is done for bounded formulae in section 3 and for arbitrary formulae in section 4 via a notion of extended set recursive functions (building on =-=[21]-=-). Section 5 deals with the question of how the formulae-as-classes interpretation can be characterized via an inner model construction on the basis of ΠΣ−AC and ΠΣW−AC, respectively. Section 6 featur... |

2 | Sacks: Higher Recursion Theory - E - 1990 |

1 |
The Disjunction and numerical existence property for constructive Zermelo-Fraenkel set theory. To appear
- Rathjen
(Show Context)
Citation Context ... the NEP. (iii) (Friedman) IZF does not have the EP. (iv) (Rathjen) CZF and CZF + REA have the DP and the NEP. Proof : (i) is proved in [17]. For (ii) see [6] and for (iii) see [12]. (iv) is shown in =-=[22]-=-. ⊓⊔ The question of whether CZF satisfies the existence property is currently unanswered. Friedman’s proof of the failure of EP for IZF seems to single out Collection as the culprit. However, that pr... |