128 research outputs found
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets
We give a sharpening of a recent result of Aschenbrenner and Pong about the maximal order type of the term ordering on the finite multisets over a wpo. Moreover we discuss an approach to compute maximal order types of well-partial orders which are related to tree embeddings
In Praise of Impredicativity: A Contribution to the Formalization of Meta-Programming
Processing programs as data is one of the successes of functional and logic programming. Higher-order functions, as program-processing programs are called in functional programming, and meta-programs, as they are called in logic programming, are widespread declarative programming techniques. In logic programming, there is a gap between the meta-programming practice and its theory: The formalizations of meta-programming do not explicitly address its impredicativity and are not fully adequate. This article aims at overcoming this unsatisfactory situation by discussing the relevance of impredicativity to meta-programming, by revisiting former formalizations of meta-programming, and by defining Reflective Predicate Logic, a conservative extension of first-order logic, which provides a simple formalization of meta-programming
On Small Types in Univalent Foundations
We investigate predicative aspects of constructive univalent foundations. By
predicative and constructive, we respectively mean that we do not assume
Voevodsky's propositional resizing axioms or excluded middle. Our work
complements existing work on predicative mathematics by exploring what cannot
be done predicatively in univalent foundations. Our first main result is that
nontrivial (directed or bounded) complete posets are necessarily large. That
is, if such a nontrivial poset is small, then weak propositional resizing
holds. It is possible to derive full propositional resizing if we strengthen
nontriviality to positivity. The distinction between nontriviality and
positivity is analogous to the distinction between nonemptiness and
inhabitedness. Moreover, we prove that locally small, nontrivial (directed or
bounded) complete posets necessarily lack decidable equality. We prove our
results for a general class of posets, which includes e.g. directed complete
posets, bounded complete posets, sup-lattices and frames. Secondly, we discuss
the unavailability of Zorn's lemma, Tarski's greatest fixed point theorem and
Pataraia's lemma in our predicative setting, and prove the ordinal of ordinals
in a univalent universe to have small suprema in the presence of set quotients.
The latter also leads us to investigate the inter-definability and interaction
of type universes of propositional truncations and set quotients, as well as a
set replacement principle. Thirdly, we clarify, in our predicative setting, the
relation between the traditional definition of sup-lattice that requires
suprema for all subsets and our definition that asks for suprema of all small
families.Comment: Extended version of arXiv:2102.08812. v2: Revised and expanded
following referee report
Recommended from our members
A defence of predicativism as a philosophy of mathematics
A specification of a mathematical object is impredicative if it essentially involves quantification over a domain which includes the object being specified (or sets which contain that object, or similar). The basic worry is that we have no non-circular way of
understanding such a specification. Predicativism is the view that mathematics should be limited to the study of objects which can be specified predicatively.
There are two parts to predicativism. One is the criticism of the impredicative aspects of classical mathematics. The other is the
positive project, begun by Weyl in Das Kontinuum (1918), to reconstruct as much as possible of classical mathematics on the basis of a predicatively acceptable set theory, which accepts only countably infinite objects. This is a revisionary project, and certain parts of mathematics will not be saved.
Chapter 2 contains an account of the historical background to the predicativist project. The rigorization of analysis led to Dedekind's and Cantor's theories of the real numbers, which relied on the new notion of abitrary infinite sets; this became a central part of modern classical set theory. Criticism began with Kronecker; continued in the debate about the acceptability of Zermelo's Axiom of Choice; and was somewhat clarified by Poincaré and Russell. In the
light of this, chapter 3 examines the formulation of, and motivations behind the predicativist position.
Chapter 4 begins the critical task by detailing the epistemological problems with the classical account of the continuum. Explanations of classicism which appeal to second-order logic, set theory, and
primitive intuition are examined and are found wanting.
Chapter 5 aims to dispell the worry that predicativism might collapses into mathematical intuitionism. I assess some of the arguments for intuitionism, especially the Dummettian argument from indefinite
extensibility. I argue that the natural numbers are not indefinitely extensible, and that, although the continuum is, we can nonetheless make some sense of classical quantification over it. We need not reject the Law of Excluded Middle.
Chapter 6 begins the positive work by outlining a predicatively acceptable account of mathematical objects which justifies the Vicious Circle Principle. Chapter 7 explores the appropriate shape of formalized predicative mathematics, and the question of just how much mathematics is predicatively acceptable.
My conclusion is that all of the mathematics which we need can be predicativistically justified, and that such mathematics is
particularly transparent to reason. This calls into question one currently prevalent view of the nature of mathematics, on which
mathematics is justified by quasi-empirical means.Supported by the Arts and Humanities Research Council [grant number 111315]
Domain Theory in Constructive and Predicative Univalent Foundations
We develop domain theory in constructive univalent foundations without
Voevodsky's resizing axioms. In previous work in this direction, we constructed
the Scott model of PCF and proved its computational adequacy, based on directed
complete posets (dcpos). Here we further consider algebraic and continuous
dcpos, and construct Scott's model of the untyped
-calculus. A common approach to deal with size issues in a predicative
foundation is to work with information systems or abstract bases or formal
topologies rather than dcpos, and approximable relations rather than Scott
continuous functions. Here we instead accept that dcpos may be large and work
with type universes to account for this. For instance, in the Scott model of
PCF, the dcpos have carriers in the second universe and suprema
of directed families with indexing type in the first universe .
Seeing a poset as a category in the usual way, we can say that these dcpos are
large, but locally small, and have small filtered colimits. In the case of
algebraic dcpos, in order to deal with size issues, we proceed mimicking the
definition of accessible category. With such a definition, our construction of
Scott's again gives a large, locally small, algebraic dcpo with
small directed suprema.Comment: A shorter version of this paper will appear in the proceedings of CSL
2021, volume 183 of LIPIc
Infinity
This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks
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