THE CLASSICAL CONTINUUM WITHOUT POINTS

AbstractWe develop a point-free construction of the classical one-dimensional continuum, with an interval structure based on mereology and either a weak set theory or a logic of plural quantification. In some respects, this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of contemporary “actual infinity”. Also, in contrast to intuitionistic analysis, smooth infinitesimal analysis, and Eret Bishop’s (1967) constructivism, we follow classical analysis in allowing partitioning of our “gunky line” into mutually exclusive and exhaustive disjoint parts, thereby demonstrating the independence of “indecomposability” from a nonpunctiform conception. It is surprising that such simple axioms as ours already imply the Archimedean property and the interval analogue of Dedekind completeness (least-upper-bound principle), and that they determine an isomorphism with the Dedekind–Cantor structure of ℝ as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

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The Classical Continuum without Points

10.1093/oso/9780198712749.003.0002 ◽

2018 ◽

Author(s):

Geoffrey Hellman ◽

Stewart Shapiro

Keyword(s):

Set Theory ◽

Real Line ◽

Current Practice ◽

Classical Analysis ◽

One Dimensional ◽

The Real ◽

Actual Infinity ◽

Archimedean Property ◽

Weak Set Theory

This chapter develops a “semi-Aristotelian” account of a one-dimensional continuum. Unlike Aristotle, it makes significant use of actual infinity, in line with current practice. Like Aristotle, this account does not recognize points, at least not as parts of regions in the space. The formal background is classical mereology together with a weak set theory. The chapter proves an Archimedean property, and establishes an isomorphism with the Dedekind–Cantor structure of the real line. It also compares the present framework to other point-free accounts, establishing consistency relative to classical analysis.

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Regions-based Two-dimensional Continua: The Euclidean Case

10.1093/oso/9780198712749.003.0005 ◽

2018 ◽

Author(s):

Geoffrey Hellman ◽

Stewart Shapiro

Keyword(s):

Mathematical Induction ◽

Dimensional Case ◽

Two Dimensional ◽

Entire Space ◽

Euclidean Case ◽

Free Basis ◽

One Dimensional ◽

Archimedean Property ◽

The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

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The consistency problem for positive comprehension principles

Journal of Symbolic Logic ◽

10.1017/s0022481200041165 ◽

1989 ◽

Vol 54 (4) ◽

pp. 1401-1418 ◽

Cited By ~ 7

Author(s):

M. Forti ◽

R. Hinnion

Keyword(s):

Set Theory ◽

Difficult Problem ◽

Large Cardinal ◽

Positive Theory ◽

Construction Principle ◽

Short Summary ◽

Free Construction ◽

Unpublished Thesis ◽

Consistency Problem ◽

Original Construction

Since Gilmore showed that some theory with a positive comprehension scheme is consistent when the axiom of extensionality is dropped and inconsistent with it (see [1] and [2]), the problem of the consistency of various positive comprehension schemes has been investigated. We give here a short classification, which shows clearly the importance of the axiom of extensionality and of the abstraction operator in these consistency problems. The most difficult problem was to show the consistency of the comprehension scheme for positive formulas, with extensionality but without abstraction operator. In his unpublished thesis, Set theory in which the axiom of foundation fails [3], Malitz solved partially this problem but he needed to assume the existence of some unusual kind of large cardinal; as his original construction is very interesting and his thesis is unpublished, we give a short summary of it. M. Forti solved the problem completely by working in ZF with a free-construction principle (sometimes called an anti-foundation axiom), instead of ZF with the axiom of foundation, as Malitz did.This permits one to obtain the consistency of this positive theory, relative to ZF. In his general investigations about “topological set theories” (to be published), E. Weydert has independently proved the same result. The authors are grateful to the Mathematisches Forshungsinstitut Oberwolfach for giving them the opportunity of discussing these subjects and meeting E. Weydert during the meeting “New Foundations”, March 1–7, 1987.

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Making the hyperreal line both saturated and complete

Journal of Symbolic Logic ◽

10.2307/2275069 ◽

1991 ◽

Vol 56 (3) ◽

pp. 1016-1025 ◽

Cited By ~ 8

Author(s):

H. Jerome Keisler ◽

James H. Schmerl

Keyword(s):

Set Theory ◽

Nonempty Intersection ◽

Finite Intersection ◽

Second Order Arithmetic ◽

Ordered Field ◽

Finite Intersection Property ◽

Regular Cardinals ◽

Hyperreal Numbers ◽

Order Arithmetic ◽

Internal Sets

AbstractIn a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if κ < λ are uncountable regular cardinals and βα < λ whenever α < κ and β < λ then there is a κ-saturated nonstandard universe in which the hyperreal numbers have the λ-Bolzano-Weierstrass property. The result also applies to certain fragments of set theory and second order arithmetic.

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Constructing Banaschewski compactification without Dedekind completeness axiom

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204305168 ◽

2004 ◽

Vol 2004 (69) ◽

pp. 3799-3816

Author(s):

S. K. Acharyya ◽

K. C. Chattopadhyay ◽

Partha Pratim Ghosh

Keyword(s):

Topological Space ◽

Continuous Functions ◽

Topological Spaces ◽

Open Problems ◽

Structure Space ◽

Ordered Field ◽

Completeness Axiom ◽

Dedekind Completeness ◽

Stone Theorem ◽

Stone’S Theorem

The main aim of this paper is to provide a construction of the Banaschewski compactification of a zero-dimensional Hausdorff topological space as a structure space of a ring of ordered field-valued continuous functions on the space, and thereby exhibit the independence of the construction from any completeness axiom for an ordered field. In the process of describing this construction we have generalized the classical versions of M. H. Stone's theorem, the Banach-Stone theorem, and the Gelfand-Kolmogoroff theorem. The paper is concluded with a conjecture of a split in the class of all zero-dimensional but not strongly zero-dimensional Hausdorff topological spaces into three classes that are labeled by inequalities between three compactifications ofX, namely, the Stone-Čech compactificationβX, the Banaschewski compactificationβ0X, and the structure space𝔐X,Fof the lattice-ordered commutative ringℭ(X,F)of all continuous functions onXtaking values in the ordered fieldF, equipped with its order topology. Some open problems are also stated.

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Objects as Limits of Experience and the Notion of Horizon in Mathematical Theories

Phainomenon ◽

10.2478/phainomenon-2012-0019 ◽

2012 ◽

Vol 25 (1) ◽

pp. 131-153

Author(s):

Stathis Livadas

Keyword(s):

Set Theory ◽

The Other ◽

Actual Infinity ◽

Human Perceptions ◽

Subjective Origin

Abstract The present work is an attempt to bring attention to the application of several key ideas of Husserl ‘s Krisis in the construction of certain mathematical theories that claim to be altemative nonstandard versions of the standard Zermelo-Fraenkel set theory. In general, these theories refute, at least semantically, the platonistic context of the Cantorian system and to one or the other degree are motivated by the notions of the lifeworld as the pregiven holistic field of experience and that of horizon as the boundary of human perceptions and the de facto constraint in reaching limit-idealizations. Moreover, 1 try to give convincing reasons for the existence of an ultimate constitutional ‘vacuum’ of a subjective origin that is formally refiected in the application of a notion of actual infinity in dealing generally with the mathematical infinite.

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L’infini entre deux bouts. Dualités, univers algébriques, esquisses, diagrammes

Filozofski vestnik ◽

10.3986/fv.41.2.09 ◽

2020 ◽

Vol 41 (2) ◽

Author(s):

René Guitart

Keyword(s):

Set Theory ◽

Mathematical Theory ◽

Mathematical Thinking ◽

Mathematical Work ◽

Mathematical Activity ◽

Actual Infinity ◽

Zeno's Paradoxes ◽

Mathematical Problems ◽

Technical Details ◽

The One

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From this perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises.

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Equilibrium in Nash’s mind

10.31124/advance.12301079.v1 ◽

2020 ◽

Author(s):

Vasil Penchev

Keyword(s):

Game Theory ◽

Nash Equilibrium ◽

Human Brain ◽

Set Theory ◽

Donald Capps ◽

Emergent Properties ◽

Normal Brain ◽

Actual Infinity ◽

The Game Theory

Donald Capps suggested the hypothesis that “the Nash equilibrium is descriptive of the normal brain, whereas the game theory formulated by John van Neumann, which Nash’s theory challenges, is descriptive of the schizophrenic brain”. The paper offers arguments in its favor. They are from psychiatry, game theory, set theory, philosophy and theology. The Nash equilibrium corresponds to wholeness, stable emergent properties as well as to representing actual infinity on a material, limited and finite organ as a human brain.

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