160 research outputs found
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
Numbers and functions in Hilbert's finitism
David Hilbert's finitistic standpoint is a conception
of elementary number theory designed to answer the intuitionist doubts
regarding the security and certainty of mathematics. Hilbert was
unfortunately not exact in delineating what that viewpoint was, and
Hilbert himself changed his usage of the term through the 1920s and 30s.
The purpose of this paper is to outline what the main problems are in
understanding Hilbert and Bernays on this issue, based on some
publications by them which have so far received little attention, and on
a number of philosophical reconstructions of the viewpoint (in
particular, by Hand, Kitcher, and Tait)
Zeno-machines And The Metaphysics Of Time
This paper aims to explore the nature of Zeno-machines by examining their conceptual coherence, from the perspective of contemporary theories on the passage of time. More specifically, it will analyse the following questions: Are Zeno-machines and supertasks coherent if we adopt the eternalist theory of time? What conclusions can be drawn from choosing the eternalist thesis, or the presentist thesis, when examining Zeno-machines? To this end, an overview of the opposing theories of time is provided, alongside the usual objections to Zeno-machines and their theoretical foundations from Zeno's dichotomy paradox.17216116
Naming the largest number: Exploring the boundary between mathematics and the philosophy of mathematics
What is the largest number accessible to the human imagination? The question
is neither entirely mathematical nor entirely philosophical. Mathematical
formulations of the problem fall into two classes: those that fail to fully
capture the spirit of the problem, and those that turn it back into a
philosophical problem
Constructing categories and setoids of setoids in type theory
In this paper we consider the problem of building rich categories of setoids,
in standard intensional Martin-L\"of type theory (MLTT), and in particular how
to handle the problem of equality on objects in this context. Any
(proof-irrelevant) family F of setoids over a setoid A gives rise to a category
C(A, F) of setoids with objects A. We may regard the family F as a setoid of
setoids, and a crucial issue in this article is to construct rich or large
enough such families. Depending on closure conditions of F, the category C(A,
F) has corresponding categorical constructions. We exemplify this with finite
limits. A very large family F may be obtained from Aczel's model construction
of CZF in type theory. It is proved that the category so obtained is isomorphic
to the internal category of sets in this model. Set theory can thus establish
(categorical) properties of C(A, F) which may be used in type theory. We also
show that Aczel's model construction may be extended to include the elements of
any setoid as atoms or urelements. As a byproduct we obtain a natural extension
of CZF, adding atoms. This extension, CZFU, is validated by the extended model.
The main theorems of the paper have been checked in the proof assistant Coq
which is based on MLTT. A possible application of this development is to
integrate set-theoretic and type-theoretic reasoning in proof assistants.Comment: 14 page
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
Halfway Up To the Mathematical Infinity I: On the Ontological & Epistemic Sustainability of Georg Cantor’s Transfinite Design
Georg Cantor was the genuine discoverer of the Mathematical Infinity, and whatever he claimed, suggested, or even surmised should be taken seriously -- albeit not necessary at its face value. Because alongside his exquisite in beauty ordinal construction and his fundamental powerset description of the continuum, Cantor has also left to us his obsessive presumption that the universe of sets should be subjected to laws similar to those governing the set of natural numbers, including the universal principles of cardinal comparability and well-ordering -- and implying an ordinal re-creation of the continuum. During the last hundred years, the mainstream set-theoretical research -- all insights and adjustments due to Kurt G\"odel's revolutionary insights and discoveries notwithstanding -- has compliantly centered its efforts on ad hoc axiomatizations of Cantor's intuitive transfinite design. We demonstrate here that the ontological and epistemic sustainability} of this design has been irremediably compromised by the underlying peremptory, Reductionist mindset of the XIXth century's ideology of science
Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic
This paper sets out a predicative response to the Russell-Myhill paradox of
propositions within the framework of Church's intensional logic. A predicative
response places restrictions on the full comprehension schema, which asserts
that every formula determines a higher-order entity. In addition to motivating
the restriction on the comprehension schema from intuitions about the stability
of reference, this paper contains a consistency proof for the predicative
response to the Russell-Myhill paradox. The models used to establish this
consistency also model other axioms of Church's intensional logic that have
been criticized by Parsons and Klement: this, it turns out, is due to resources
which also permit an interpretation of a fragment of Gallin's intensional
logic. Finally, the relation between the predicative response to the
Russell-Myhill paradox of propositions and the Russell paradox of sets is
discussed, and it is shown that the predicative conception of set induced by
this predicative intensional logic allows one to respond to the Wehmeier
problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi
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