1,389 research outputs found
Classical Predicative Logic-Enriched Type Theories
A logic-enriched type theory (LTT) is a type theory extended with a primitive
mechanism for forming and proving propositions. We construct two LTTs, named
LTTO and LTTO*, which we claim correspond closely to the classical predicative
systems of second order arithmetic ACAO and ACA. We justify this claim by
translating each second-order system into the corresponding LTT, and proving
that these translations are conservative. This is part of an ongoing research
project to investigate how LTTs may be used to formalise different approaches
to the foundations of mathematics.
The two LTTs we construct are subsystems of the logic-enriched type theory
LTTW, which is intended to formalise the classical predicative foundation
presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has
also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA
as LTTs, we are able to compare them with LTTW. It is a consequence of the work
in this paper that LTTW is strictly stronger than ACAO.
The conservativity proof makes use of a novel technique for proving one LTT
conservative over another, involving defining an interpretation of the stronger
system out of the expressions of the weaker. This technique should be
applicable in a wide variety of different cases outside the present work.Comment: 49 pages. Accepted for publication in special edition of Annals of
Pure and Applied Logic on Computation in Classical Logic. v2: Minor mistakes
correcte
Semi-simplicial Types in Logic-enriched Homotopy Type Theory
The problem of defining Semi-Simplicial Types (SSTs) in Homotopy Type Theory
(HoTT) has been recognized as important during the Year of Univalent
Foundations at the Institute of Advanced Study. According to the interpretation
of HoTT in Quillen model categories, SSTs are type-theoretic versions of Reedy
fibrant semi-simplicial objects in a model category and simplicial and
semi-simplicial objects play a crucial role in many constructions in homotopy
theory and higher category theory. Attempts to define SSTs in HoTT lead to some
difficulties such as the need of infinitary assumptions which are beyond HoTT
with only non-strict equality types.
Voevodsky proposed a definition of SSTs in Homotopy Type System (HTS), an
extension of HoTT with non-fibrant types, including an extensional strict
equality type. However, HTS does not have the desirable computational
properties such as decidability of type checking and strong normalization. In
this paper, we study a logic-enriched homotopy type theory, an alternative
extension of HoTT with equational logic based on the idea of logic-enriched
type theories. In contrast to Voevodskys HTS, all types in our system are
fibrant and it can be implemented in existing proof assistants. We show how
SSTs can be defined in our system and outline an implementation in the proof
assistant Plastic
A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice
We present a Kleene realizability semantics for the intensional level of the
Minimalist Foundation, for short mtt, extended with inductively generated
formal topologies, Church's thesis and axiom of choice. This semantics is an
extension of the one used to show consistency of the intensional level of the
Minimalist Foundation with the axiom of choice and formal Church's thesis in
previous work. A main novelty here is that such a semantics is formalized in a
constructive theory represented by Aczel's constructive set theory CZF extended
with the regular extension axiom
Sets in homotopy type theory
Homotopy Type Theory may be seen as an internal language for the
-category of weak -groupoids which in particular models the
univalence axiom. Voevodsky proposes this language for weak -groupoids
as a new foundation for mathematics called the Univalent Foundations of
Mathematics. It includes the sets as weak -groupoids with contractible
connected components, and thereby it includes (much of) the traditional set
theoretical foundations as a special case. We thus wonder whether those
`discrete' groupoids do in fact form a (predicative) topos. More generally,
homotopy type theory is conjectured to be the internal language of `elementary'
-toposes. We prove that sets in homotopy type theory form a -pretopos. This is similar to the fact that the -truncation of an
-topos is a topos. We show that both a subobject classifier and a
-object classifier are available for the type theoretical universe of sets.
However, both of these are large and moreover, the -object classifier for
sets is a function between -types (i.e. groupoids) rather than between sets.
Assuming an impredicative propositional resizing rule we may render the
subobject classifier small and then we actually obtain a topos of sets
Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice
Here we consider an extension MFcind of the Minimalist Foundation MF for
predicative constructive mathematics with the addition of inductive and
coinductive definitions sufficient to generate Sambin's Positive topologies,
i.e. Martin-L\"of-Sambin formal topologies equipped with a Positivity relation
(used to describe pointfree formal closed subsets). In particular the
intensional level of MFcind, called mTTcind, is defined by extending with
coinductive definitions another theory mTTind extending the intensional level
mTT of MF with the sole addition of inductive definitions. In previous work we
have shown that mTTind is consistent with Formal Church's Thesis CT and the
Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show
the expectation that the addition of coinductive definitions to mTTind does not
increase its consistency strength by reducing the consistency of mTTcind+CT+AC
to the consistency of CZF+REA through various interpretations. We actually
reach our goal in two ways. One way consists in first interpreting
mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension
Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent
Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of
Martin-L\"of's type theory with Palmgren's superuniverse S. A last step
consists in interpreting MLS* back into CZF+REA. The alternative way consists
in first interpreting mTTcind+AC+CT directly in a version of Martin-L\"of's
type theory with Palmgren's superuniverse extended with CT, which is then
interpreted back to CZF+REA. A key benefit of the first way is that the theory
CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the
extensional level of MFcind. Finally, all the theories considered, except
mTTcind+AC+CT, are shown to be of the same proof-theoretic strength.Comment: arXiv admin note: text overlap with arXiv:1905.1196
The generalised type-theoretic interpretation of constructive set theory
We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation
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