## Categorical models of intuitionistic theories of sets and classes (2004)

Citations: | 3 - 2 self |

### BibTeX

@TECHREPORT{Forssell04categoricalmodels,

author = {H. Forssell},

title = {Categorical models of intuitionistic theories of sets and classes},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

The thesis consists of three sections, developing models of intuitionistic set theory in suitable categories. First, the categorical framework in which models are constructed is reviewed, and the theory of all such models, called Basic Intuitionistic Set Theory (BIST), is stated; second, we give a notion of an ideal over a category, with which one can build a model of BIST in which a given topos occurs as the sets; and third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending BIST.

### Citations

44 | Notes on constructive set theory
- Aczel, Rathjen
- 2001
(Show Context)
Citation Context ...rmulas is provable 16S S S in BIST, and there are some nice closure properties for formulas that are simple in BIST. We state these results below, but postpone the proofs until section 3.2 (see also =-=[1]-=-). Proposition 1.5.4 (!-Sep) BIST ⊢ (S(x) ∧ ∀y ∈ x.!φ) → Lemma 1.5.5 The following hold in BIST: 1. !⊥ 2. !φ∧!ψ →!(φ ∧ ψ) 3. !φ∧!ψ →!(φ ∨ ψ) 4. (S(x) ∧ ∀y ∈ x.!φ) → !(∃y ∈ x. φ) 5. (S(x) ∧ ∀y ∈ x.!φ) ... |

38 |
Algebraic Set Theory
- Joyal, Moerdijk
- 1995
(Show Context)
Citation Context .... . . . . . 57 iii� Introduction We begin with a brief sketch (elaborated in section 1 below) of the leading ideas of algebraic set theory, as it was recently presented in [2], and first proposed in =-=[8]-=- (see also [3, 11, 10, 6]). The basic tool of algebraic set theory is the notion of a category with class structure, or class category for short, which provides an axiomatic framework in which models ... |

34 |
Lane and Ieke Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Universitext
- Mac
- 1992
(Show Context)
Citation Context ...en. For each i ∈ I, denote by ci the colimit cocone monomorphism ci : yCi � lim −→ I(yCi), and by yfi : yCi � yA the composite φ ◦ ci. Since φ is an epimorphism in Sh(E), φ is locally surjective (see =-=[9]-=-), so we may chose a finite epimorphic family (ek)k≤n with target A in E such that for all ek : Dk � A, ek is in the image of φDk : −→ limI(yCi(Dk)) � HomE(Dk, A). We may choose, therefore, an ik ∈ I ... |

17 | Bernays-Gödel type theory - Butz |

16 | Presheaf models for constructive set theories, From Sets and Types to Topology and Analysis
- Gambino
- 2005
(Show Context)
Citation Context ... iii� Introduction We begin with a brief sketch (elaborated in section 1 below) of the leading ideas of algebraic set theory, as it was recently presented in [2], and first proposed in [8] (see also =-=[3, 11, 10, 6]-=-). The basic tool of algebraic set theory is the notion of a category with class structure, or class category for short, which provides an axiomatic framework in which models of set theory are constru... |

9 |
Relating topos theory and set theory via categories of classes
- Awodey, Butz, et al.
- 2003
(Show Context)
Citation Context ... . . . . . . . . . . . . . . . . . . . 57 iii� Introduction We begin with a brief sketch (elaborated in section 1 below) of the leading ideas of algebraic set theory, as it was recently presented in =-=[2]-=-, and first proposed in [8] (see also [3, 11, 10, 6]). The basic tool of algebraic set theory is the notion of a category with class structure, or class category for short, which provides an axiomatic... |

7 |
Foundations of Set Theory, volume 67
- Fraenkel, Bar-Hillel
- 1958
(Show Context)
Citation Context ... the consistency of ZF. (Another difference is that NBG is finitely axiomatizable, while MK is not, but we shall not be concerned with that issue. More on theories of sets and classes can be found in =-=[5]-=-.) We saw in section 1.5.1 that any class category with a universe models the set theory BIST. In section 1.4.4 it was stated that the (small) syntactic category of any set theory including BIST is a ... |

7 | Elementary Axioms for Categories of Classes (Extended Abstract - Simpson - 1999 |

3 |
Class Categories and Polymorphic Π1 Types
- Rummelhoff
- 2006
(Show Context)
Citation Context ... iii� Introduction We begin with a brief sketch (elaborated in section 1 below) of the leading ideas of algebraic set theory, as it was recently presented in [2], and first proposed in [8] (see also =-=[3, 11, 10, 6]-=-). The basic tool of algebraic set theory is the notion of a category with class structure, or class category for short, which provides an axiomatic framework in which models of set theory are constru... |

2 |
Fraenkel and Yehoshua Bar-Hillel. Foundations of Set Theory
- Abraham
- 1958
(Show Context)
Citation Context ...wo “membership” predicates, ∈ and ε, which takes sets, respectively classes, on the right and sets on the left. We give the following informally presented axioms for NBG, based on the presentation in =-=[4]-=- but omitting choice axioms: ZF axioms All axioms of ZF except Separation and Replacement. Class Extensionality Classes which have the same elements are equal. I.e. ∀X, Y .(∀z. zεX ↔ zεY ) → X = Y Cla... |

2 |
Sketches of an Elephant, volume 43 and 44 of Oxford Logic Guides
- Johnstone
- 2002
(Show Context)
Citation Context ...a class category as defined in [2] is required to have a universal object (see 1.5.1) 1.2 Class Logic Any topos is a class category where all maps are small. We describe a variant of topos logic (see =-=[7]-=-) adapted to suit class categories in general (see also [10]): A class signature Σ is defined by specifying (i) a set of type constants, ΣC; (ii) a set of typed function symbols, ΣF; and (iii) a set o... |