An Extended Predicative Definition of the Mahlo Universe

Setzer, Anton

doi:10.1515/9783110324907.315

Cited by 4 publications

(6 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…With [16] we have not reached the limit of this methodology. We have developed draft versions which reach at least the strength of Kripke Platek set theory extended by Π 3 -reflection, and it is likely that we can go far beyond with that strength.…”

Section: Feferman's System Of Explicit Mathematics and The Extended Pmentioning

confidence: 91%

“…Therefore in MLTT we do not have access to the collection of all terms, which in general are only partial objects. We regard this version ( [16]) as being predicative (in an extended sense) and believe that this theory can be validated. Feferman's theory has been developed in second order logic 16 , and optimised towards a short and concise theory.…”

Section: Feferman's System Of Explicit Mathematics and The Extended Pmentioning

confidence: 93%

“…We regard this version ( [16]) as being predicative (in an extended sense) and believe that this theory can be validated. Feferman's theory has been developed in second order logic 16 , and optimised towards a short and concise theory. While this makes metamathematical investigations easy and makes it easily accessible to non-specialists, it causes problems when validating the provable statements 17 .…”

Section: Feferman's System Of Explicit Mathematics and The Extended Pmentioning

confidence: 93%

“…In [16] Kahle and the author have published an extended predicative version of the Mahlo universe. This version is developed in Feferman's system of explicit mathematics [5,6].…”

Section: Feferman's System Of Explicit Mathematics and The Extended Pmentioning

confidence: 99%

“…Not much of second order logic is actually used, its use is mainly for convenience rather than need 17. We note that this is the opinion of the second author of[16] only, who is the author of the current article.…”

mentioning

confidence: 94%

See 4 more Smart Citations

The Use of Trustworthy Principles in a Revised Hilbert’s Program

Setzer

2015

Gentzen's Centenary

Self Cite

View full text Add to dashboard Cite

After the failure of Hilbert's original program due to Gödel's second incompleteness theorem, relativized Hilbert's programs have been suggested. While most metamathematical investigations are focused on carrying out mathematical reductions, we claim that in order to give a full substitute for Hilbert's program, one should not stop with purely mathematical investigations, but give an answer to the question why one should believe that all theorems proved in certain mathematical theories are valid.We suggest that, while it is not possible to obtain absolute certainty, it is possible to develop trustworthy core principles using which one can prove the correctness of mathematical theories. Trust can be established by both providing a direct validation of such principles, which is necessarily non-mathematical and philosophical in nature, and at the same time testing those principles using metamathematical investigations. We investigate three approaches for trustworthy principles, namely ordinal notation systems built from below, Martin-Löf type theory, and Feferman's system of explicit mathematics. We will review what is known about the strength up to which direct validation can be provided.

show abstract

Section: Feferman's System Of Explicit Mathematics and The Extended Pmentioning

confidence: 91%

Section: Feferman's System Of Explicit Mathematics and The Extended Pmentioning

confidence: 93%

Section: Feferman's System Of Explicit Mathematics and The Extended Pmentioning

confidence: 93%

“…In [16] Kahle and the author have published an extended predicative version of the Mahlo universe. This version is developed in Feferman's system of explicit mathematics [5,6].…”

Section: Feferman's System Of Explicit Mathematics and The Extended Pmentioning

confidence: 99%

mentioning

confidence: 94%

See 3 more Smart Citations

The Use of Trustworthy Principles in a Revised Hilbert’s Program

Setzer

2015

Gentzen's Centenary

Self Cite

View full text Add to dashboard Cite

show abstract

Containers, monads and induction recursion

Ghani

Hancock

2014

Math. Struct. Comp. Sci.

View full text Add to dashboard Cite

Induction recursion offers the possibility of a clean, simple and yet powerful meta-language for the type system of a dependently typed programming language. At its crux, induction recursion allows us to define a universe, that is a set U of codes and a decoding function T : U → D which assigns to every code u : U, a value T, u of some type D, e.g. the large type Set of small types or sets. The name induction recursion refers to the build-up of codes in U using inductive clauses, simultaneously with the definition of the function T, by structural recursion on codes. Our contribution is to (i) bring out explicitly algebraic structure which is less visible in the original type-theoretic presentation – in particular showing how containers and monads play a pivotal role within induction recursion; and (ii) use these structures to present a clean and high level definition of induction recursion suitable for use in functional programming

show abstract

The extended predicative Mahlo universe in Martin-Löf type theory

Dybjer

Setzer

2023

Journal of Logic and Computation

View full text Add to dashboard Cite

This paper addresses the long-standing question of the predicativity of the Mahlo universe. A solution, called the extended predicative Mahlo universe, has been proposed by Kahle and Setzer in the context of explicit mathematics. It makes use of the collection of untyped terms (denoting partial functions) which are directly available in explicit mathematics but not in Martin-Löf type theory. In this paper, we overcome the obstacle of not having direct access to untyped terms in Martin-Löf type theory by formalizing explicit mathematics with an extended predicative Mahlo universe in Martin-Löf type theory with certain indexed inductive-recursive definitions. In this way, we can relate the predicativity question to the fundamental semantics of Martin-Löf type theory in terms of computation to canonical form. As a result, we get the first extended predicative definition of a Mahlo universe in Martin-Löf type theory. To this end, we first define an external variant of Kahle and Setzer’s internal extended predicative universe in explicit mathematics. This is then formalized in Martin-Löf type theory, where it becomes an internal extended predicative Mahlo universe. Although we make use of indexed inductive-recursive definitions that go beyond the type theory $\mathbf {IIRD}$ of indexed inductive-recursive definitions defined in previous work by the authors, we argue that they are constructive and predicative in Martin-Löf’s sense. The model construction has been type-checked in the proof assistant Agda.

show abstract

An Extended Predicative Definition of the Mahlo Universe

Cited by 4 publications

References 54 publications

The Use of Trustworthy Principles in a Revised Hilbert’s Program

The Use of Trustworthy Principles in a Revised Hilbert’s Program

Containers, monads and induction recursion

The extended predicative Mahlo universe in Martin-Löf type theory

Contact Info

Product

Resources

About