## Universes in Explicit Mathematics (1999)

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Venue: | Annals of Pure and Applied Logic |

Citations: | 9 - 5 self |

### BibTeX

@ARTICLE{Jäger99universesin,

author = {Gerhard Jäger and Reinhard Kahle and Thomas Studer},

title = {Universes in Explicit Mathematics},

journal = {Annals of Pure and Applied Logic},

year = {1999},

volume = {14},

pages = {41--55}

}

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### Abstract

This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman's T 0 . 1 Introduction In some form or another, universes play an important role in many systems of set theory and higher order arithmetic, in various formalizations of constructive mathematics and in logics for computation. One aspect of universes is that they expand the set or type formation principles in a natural and perspicuous way and provide greater expressive power and proof-theoretic strength. The general idea behind universes is quite simple: suppose that we are given a formal system Th comprising certain set (or type) existence principles which are justified on specific philosophical grounds. Then it may be a...

### Citations

355 |
Intuitionistic Type Theory
- Martin-Lof
- 1984
(Show Context)
Citation Context ...verses in Martin-Löf type theory are generated by specific introduction and (sometimes) elimination rules and can be regarded as the constructive versions of certain regular cardinals. See Martin-Lö=-=f [21]-=-, Palmgren [25], Rathjen [26] and Setzer [27] for more information about this approach. In the framework of explicit mathematics, universes have first been considered by Feferman [7] in connection wit... |

214 |
Subsystems of second order Arithmetic
- Simpson
- 1999
(Show Context)
Citation Context ...th Hancock’s conjecture and by 1sMarzetta [22] for designing an explicit analogue of Friedman’s theory ATR0 of arithmetic transfinite recursion (cf. e.g. Friedman, McAloon and Simpson [9] and Simp=-=son [28]) and-=- Jäger’s theory KPl 0 of iterated admissible sets without foundation (cf. e.g. Jäger [10, 11]). More about universes in explicit mathematics can be found, for example, in Jäger and Strahm [15] an... |

92 |
A language and axioms for explicit mathematics
- Feferman
- 1975
(Show Context)
Citation Context ...hasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Feferman’s T0. 2 Explicit mathematics Explicit mathematics has been introduced in Feferma=-=n [5]-=- as a framework for Bishop style constructive mathematics. The relationship between explicit mathematics, other formalizations of constructive mathematics and subsystems of analysis and an interesting... |

72 |
Foundations of constructive mathematics. Metamathematical studies., volume 6 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3
- Beeson
- 1985
(Show Context)
Citation Context .... . . , sn we write: ℜ(�s, � U) := ℜ(s1, U1) ∧ · · · ∧ ℜ(sn, Un), ℜ(�s) := ℜ(s1) ∧ · · · ∧ ℜ(sn). The logic of systems of explicit mathematics is Beeson’s classical l=-=ogic of partial terms (cf. Beeson [1]-=- or Troelstra and van Dalen [30]) for individuals and classical logic for types. Now we introduce the theory EETJ which provides a framework for explicit elementary types with join. The nonlogical axi... |

68 | Iterated Inductive Definitions and Subsystems of Analysis - Buchholz, Feferman, et al. - 1981 |

48 | Systems of explicit mathematics with non-constructive µ-operator
- Feferman, Jäger
- 1993
(Show Context)
Citation Context ...inal formulation of explicit mathematics, elementary comprehension is not dealt with by a finite axiomatization, but directly as an infinite axiom schema. According to a theorem in Feferman and Jäger=-= [8]-=-, reformulated in Lemma 1 below, this schema of uniform elementary comprehension is provable from our finite axiomatization. Join is not needed for this argument. In the following we assume that z1, z... |

39 |
Set Theory: an Introduction to Large Cardinals
- Drake
- 1974
(Show Context)
Citation Context ...and stronger extensions of Th. In classical set theory this process is related to what is inherent in the usual reflection principles yielding the existence of certain large cardinals (cf. e.g. Drake =-=[4]).-=- In theories for iterated admissible sets, admissibles act as universes and provide for recursive analogues of large cardinals (cf. e.g. Jäger [11]). Universes in Martin-Löf type theory are generate... |

35 |
Theories for Admissible Sets: A Unifying Approach to Proof Theory
- Jäger
- 1986
(Show Context)
Citation Context ...xistence of certain large cardinals (cf. e.g. Drake [4]). In theories for iterated admissible sets, admissibles act as universes and provide for recursive analogues of large cardinals (cf. e.g. Jäger=-= [11])-=-. Universes in Martin-Löf type theory are generated by specific introduction and (sometimes) elimination rules and can be regarded as the constructive versions of certain regular cardinals. See Marti... |

34 |
A finite combinatorial principle which is equivalent to the l-consistency of predicative analysis
- FRIEDMAN, MCALOON, et al.
- 1982
(Show Context)
Citation Context ...in connection with Hancock’s conjecture and by 1sMarzetta [22] for designing an explicit analogue of Friedman’s theory ATR0 of arithmetic transfinite recursion (cf. e.g. Friedman, McAloon and Simp=-=son [9] and -=-Simpson [28]) and Jäger’s theory KPl 0 of iterated admissible sets without foundation (cf. e.g. Jäger [10, 11]). More about universes in explicit mathematics can be found, for example, in Jäger a... |

29 |
Iterated inductive fixed-point theories: application to Hancock’s conjecture, The Patras Symposion
- Feferman
- 1982
(Show Context)
Citation Context .... See Martin-Löf [21], Palmgren [25], Rathjen [26] and Setzer [27] for more information about this approach. In the framework of explicit mathematics, universes have first been considered by Feferman=-= [7] in -=-connection with Hancock’s conjecture and by 1sMarzetta [22] for designing an explicit analogue of Friedman’s theory ATR0 of arithmetic transfinite recursion (cf. e.g. Friedman, McAloon and Simpson... |

24 |
The proof-theoretic analysis of transfinitely iterated fixed point theories, The Journal of Symbolic Logic 64
- Jäger, Kahle, et al.
- 1999
(Show Context)
Citation Context ...[29] and Kahle [19], respectively. The fixed point theory � ID<ω is studied in Feferman [7]. Transfinitely iterated fixed point theories are introduced and analyzed in Jäger, Kahle, Setzer and Str=-=ahm [14]. Th-=-eorem 12 1. The theory T0 + (Lim) + (L-UG) + (Uℓ-Lin) + (Uℓ-Con) is consistent and of the same proof-theoretic strength as T0. 2. This proof-theoretic equivalence remains true if, on both sides, i... |

21 | Upper bounds for metapredicative Mahlo in explicit mathematics and admissible set theory
- Jäger, Strahm
(Show Context)
Citation Context ...on [28]) and Jäger’s theory KPl 0 of iterated admissible sets without foundation (cf. e.g. Jäger [10, 11]). More about universes in explicit mathematics can be found, for example, in Jäger and St=-=rahm [15]-=- and Strahm [29], always in connection with theories of predicative or metapredicative strength. Universes are also crucial for dealing with Mahloness in explicit mathematics, as shown in the forthcom... |

21 | On applicative theories
- Jäger, Kahle, et al.
- 1999
(Show Context)
Citation Context ... . , sn) belongs to W . This notion is analogue to the strictness of definedness, implemented in the logic of partial terms, and the so-called N-strictness for the natural numbers, discussed in Kahle =-=[20]-=-. By reflecting name strictness on universes, one obtains name strict universes. In the next section, we introduce a form of name induction saying that all names have to be constructed by the use of g... |

21 | Well-ordering proofs for Martin-Löf type theory with W-type and one universe., Annals of Pure and applied Logic 92
- Setzer
- 1998
(Show Context)
Citation Context ...d by specific introduction and (sometimes) elimination rules and can be regarded as the constructive versions of certain regular cardinals. See Martin-Löf [21], Palmgren [25], Rathjen [26] and Setzer=-= [27] f-=-or more information about this approach. In the framework of explicit mathematics, universes have first been considered by Feferman [7] in connection with Hancock’s conjecture and by 1sMarzetta [22]... |

17 |
Recursion theory and set theory, a marriage of convenience
- Feferman
- 1978
(Show Context)
Citation Context ...ns of constructive mathematics and subsystems of analysis and an interesting interplay between set-theoretic and recursion-theoretic models of explicit mathematics have first been studied in Feferman =-=[5, 6]. I-=-n the following, we do not work with Feferman’s original formalization of systems of explicit mathematics. Instead, we treat them as theories of types and names as developed in Jäger [12]. Our theo... |

15 |
Induction in the elementary theory of types and names
- Jäger
- 1988
(Show Context)
Citation Context ...eferman [5, 6]. In the following, we do not work with Feferman’s original formalization of systems of explicit mathematics. Instead, we treat them as theories of types and names as developed in Jäg=-=er [12]-=-. Our theories of types and names are formulated in the second order language L for individuals and types. It comprises individual variables a, b, c, f, u, v, w, x, y, z, . . . as well as type variabl... |

13 | Extending the system T0 of explicit mathematics: the limit and Mahlo axioms
- Jäger, Studer
(Show Context)
Citation Context ... connection with theories of predicative or metapredicative strength. Universes are also crucial for dealing with Mahloness in explicit mathematics, as shown in the forthcoming paper Jäger and Studer=-= [16]-=-. In Kahle [18], universes are studied for Frege structures, i.e. truth theories corresponding to explicit mathematics. The purpose of this article is to clarify several principle aspects of universes... |

13 |
Predicative Theories of Types and Names
- Marzetta
- 1994
(Show Context)
Citation Context ...[27] for more information about this approach. In the framework of explicit mathematics, universes have first been considered by Feferman [7] in connection with Hancock’s conjecture and by 1sMarzett=-=a [22] for -=-designing an explicit analogue of Friedman’s theory ATR0 of arithmetic transfinite recursion (cf. e.g. Friedman, McAloon and Simpson [9] and Simpson [28]) and Jäger’s theory KPl 0 of iterated adm... |

9 |
Constructivism in mathematics. Vol
- Troelstra, Dalen
- 1988
(Show Context)
Citation Context ... := ℜ(s1, U1) ∧ · · · ∧ ℜ(sn, Un), ℜ(�s) := ℜ(s1) ∧ · · · ∧ ℜ(sn). The logic of systems of explicit mathematics is Beeson’s classical logic of partial terms (cf. Beeson [1]=-= or Troelstra and van Dalen [30]-=-) for individuals and classical logic for types. Now we introduce the theory EETJ which provides a framework for explicit elementary types with join. The nonlogical axioms of EETJ can be divided into ... |

8 |
Applikative Theorien und Frege-Strukturen. PhD thesis, Institut für Informatik und angewandte Mathematik
- Kahle
- 1997
(Show Context)
Citation Context ...h theories of predicative or metapredicative strength. Universes are also crucial for dealing with Mahloness in explicit mathematics, as shown in the forthcoming paper Jäger and Studer [16]. In Kahle=-= [18]-=-, universes are studied for Frege structures, i.e. truth theories corresponding to explicit mathematics. The purpose of this article is to clarify several principle aspects of universes in explicit ma... |

7 | The strength of Martin-Löf type theory with a superuniverse. Part I
- Rathjen
- 2000
(Show Context)
Citation Context ...ory are generated by specific introduction and (sometimes) elimination rules and can be regarded as the constructive versions of certain regular cardinals. See Martin-Löf [21], Palmgren [25], Rathjen=-= [26] a-=-nd Setzer [27] for more information about this approach. In the framework of explicit mathematics, universes have first been considered by Feferman [7] in connection with Hancock’s conjecture and by... |

6 |
The strength of admissibility without foundation, The
- Jäger
- 1984
(Show Context)
Citation Context ...s theory ATR0 of arithmetic transfinite recursion (cf. e.g. Friedman, McAloon and Simpson [9] and Simpson [28]) and Jäger’s theory KPl 0 of iterated admissible sets without foundation (cf. e.g. Jä=-=ger [10, 11])-=-. More about universes in explicit mathematics can be found, for example, in Jäger and Strahm [15] and Strahm [29], always in connection with theories of predicative or metapredicative strength. Univ... |

6 |
Applikative Theorien und explizite Mathematik. Lecture Notes iam-97-001, Institut für Informatik und angewandte Mathematik
- Jäger, Wirz
- 1997
(Show Context)
Citation Context ..., one can prove that U(S) ∧ ℜ(a, S) → a /∈ S. Note that in explicit mathematics, the names of a type do not form a type. This is proved in various places, for example in Cantini and Minari [3]=-=, Jäger [13]-=- and Jansen [17]; join is not needed for this argument. In connection with universes, a stronger result is possible: each type has so many names that not all of them can be contained in a single unive... |

5 |
Uniform limit in explicit mathematics with universes
- Kahle
- 1997
(Show Context)
Citation Context ... natural models for (Lim). The proof-theoretic strengths of (Lim) in the context of elementary comprehension and join plus type or formula induction on the natural numbers have been analyzed in Kahle =-=[19]-=- and Strahm [29]. Although, in many situations, (Lim) is proof-theoretically equivalent to its obvious nonuniform version as studied in Marzetta [22] and Marzetta and Strahm [23], 9ssometimes there ar... |

5 | The µ quantification operator in explicit mathematics with universes and iterated fixed point theories with ordinals. Archive for Mathematical Logic
- Marzetta, Strahm
- 1998
(Show Context)
Citation Context ...een analyzed in Kahle [19] and Strahm [29]. Although, in many situations, (Lim) is proof-theoretically equivalent to its obvious nonuniform version as studied in Marzetta [22] and Marzetta and Strahm =-=[23]-=-, 9ssometimes there are subtle differences between (Lim) and its nonuniform version, which will be discussed elsewhere. There are, of course, many universes which contain a given name a. The universe ... |

5 | First steps into metapredicativity in explicit mathematics
- Strahm
- 1999
(Show Context)
Citation Context ...er’s theory KPl 0 of iterated admissible sets without foundation (cf. e.g. Jäger [10, 11]). More about universes in explicit mathematics can be found, for example, in Jäger and Strahm [15] and Str=-=ahm [29],-=- always in connection with theories of predicative or metapredicative strength. Universes are also crucial for dealing with Mahloness in explicit mathematics, as shown in the forthcoming paper Jäger ... |

2 |
Uniform inseparability in explicit mathematics
- Cantini, Minari
- 1999
(Show Context)
Citation Context ...a 5 In EETJ, one can prove that U(S) ∧ ℜ(a, S) → a /∈ S. Note that in explicit mathematics, the names of a type do not form a type. This is proved in various places, for example in Cantini and=-= Minari [3],-=- Jäger [13] and Jansen [17]; join is not needed for this argument. In connection with universes, a stronger result is possible: each type has so many names that not all of them can be contained in a ... |

2 |
Axioms for universes. Handwritten notes
- Minari
(Show Context)
Citation Context ...possible: each type has so many names that not all of them can be contained in a single universe, or, in other words, no universe is large enough to contain all names of a given type (see also Minari =-=[24]). 8sLemma -=-6 In EETJ, one can prove that U(S) → ∃x(ℜ(x, T ) ∧ x /∈ S). Proof Let S be a universe and choose a name a of S. Then j(a, λx.x) is a name of the type The next step is to prove the equivalen... |

1 | Ontologische Aspekte expliziter Mathematik. Diploma thesis, Institut für Informatik und angewandte Mathematik - Jansen - 1997 |

1 |
On universes in type theory. To appear in Twenty-five Years of Type Theory
- Palmgren
(Show Context)
Citation Context ...n-Löf type theory are generated by specific introduction and (sometimes) elimination rules and can be regarded as the constructive versions of certain regular cardinals. See Martin-Löf [21], Palmgre=-=n [25], -=-Rathjen [26] and Setzer [27] for more information about this approach. In the framework of explicit mathematics, universes have first been considered by Feferman [7] in connection with Hancock’s con... |