The Peircean Continuum

The History of Continua ◽

10.1093/oso/9780198809647.003.0014 ◽

2020 ◽

pp. 328-346

Author(s):

Francisco Vargas ◽

Matthew E. Moore

Keyword(s):

Set Theory ◽

Proper Class ◽

Vital Importance ◽

Work In Progress

Charles Sanders Peirce’s views on continuity, the concept he lionized as “the master-key which … unlocks the arcana of philosophy”, are of vital importance for students of his philosophy, but have received much less attention from historians and philosophers of continuity. This is partly because Peirce’s mathematics of continuity was still very much a work in progress when he died over a century ago. In the first and principal section of this chapter, the first author summarizes the defining features of the theory of continuity that Peirce made the most progress on, and constructs a model for that theory as a proper class in Zermelo-Fraenkel Set Theory with Choice. This model provides a fuller mathematical vindication of Peirce’s conception than any reconstruction offered to date. The second section is an historical appendix, in which the second author briefly summarizes Peirce’s own attempts to put his conception into a rigorous form.

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CONSTRUCTION OF A MODEL FOR GÖDEL–BERNAYS SET THEORY FOR WHICH THE CLASS OF NATURAL NUMBERS IS A SET OF THE MODEL AND A PROPER CLASS IN THE THEORY

The Theory of Models ◽

10.1016/b978-0-7204-2233-7.50052-6 ◽

2014 ◽

pp. 436-437 ◽

Cited By ~ 1

Author(s):

PETR VOPĚNKA

Keyword(s):

Set Theory ◽

Proper Class ◽

Natural Numbers

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On constructing completions

Journal of Symbolic Logic ◽

10.2178/jsl/1122038923 ◽

2005 ◽

Vol 70 (3) ◽

pp. 969-978 ◽

Cited By ~ 12

Author(s):

Laura Crosilla ◽

Hajime Ishihara ◽

Peter Schuster

Keyword(s):

Set Theory ◽

Metric Space ◽

Ordered Set ◽

Proper Class ◽

Original Space ◽

Full Form ◽

Dedekind Cuts ◽

Complete Separable Metric Space ◽

Separable Metric Space

AbstractThe Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo–Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two–element coverings is used.In particular, the Dedekind reals form a set: whence we have also refined an earlier result by Aczel and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class: in particular, every complete separable metric space automatically is a set.

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A STRONG REFLECTION PRINCIPLE

The Review of Symbolic Logic ◽

10.1017/s1755020317000223 ◽

2017 ◽

Vol 10 (4) ◽

pp. 651-662 ◽

Cited By ~ 2

Author(s):

SAM ROBERTS

Keyword(s):

Set Theory ◽

Reflection Principle ◽

Higher Order ◽

Second Order ◽

Proper Class ◽

Strong Reflection ◽

Reflection Principles ◽

Extendible Cardinals ◽

Order Set

AbstractThis article introduces a new reflection principle. It is based on the idea that whatever is true in all entities of some kind is also true in a set-sized collection of them. Unlike standard reflection principles, it does not re-interpret parameters or predicates. This allows it to be both consistent in all higher-order languages and remarkably strong. For example, I show that in the language of second-order set theory with predicates for a satisfaction relation, it is consistent relative to the existence of a 2-extendible cardinal (Theorem 7.12) and implies the existence of a proper class of 1-extendible cardinals (Theorem 7.9).

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A guide to “Coding the universe” by Beller, Jensen, Welch

Journal of Symbolic Logic ◽

10.2307/2273986 ◽

1985 ◽

Vol 50 (4) ◽

pp. 1002-1019 ◽

Cited By ~ 2

Author(s):

Sy D. Friedman

Keyword(s):

Set Theory ◽

Negative Answer ◽

Large Cardinals ◽

Affirmative Answer ◽

Partial Ordering ◽

Proper Class ◽

Covering Lemma ◽

Singular Cardinals ◽

The Universe

In the wake of Silver's breakthrough on the Singular Cardinals Problem (Silver [74]) followed one of the landmark results in set theory, Jensen's Covering Lemma (Devlin-Jensen [74]): If 0# does not exist then for every uncountable x ⊆ ORD there exists a constructible Y ⊇ X, card(Y) = card(X). Thus it is fair to say that in the absence of large cardinals, V is “close to L”.It is natural to ask, as did Solovay, if we can fairly interpret the phrase “close to L” to mean “generic over L”. For example, if V = L[a], a ⊆ ω and if 0# does not exist then is V-generic over L for some partial ordering ∈ L? Notice that an affirmative answer implies that in the absence of 0#, no real can “code” a proper class of information.Jensen's Coding Theorem provides a negative answer to Solovay's question, in a striking way: Any class can be “coded” by a real without introducing 0#. More precisely, if A ⊆ ORD then there is a forcing definable over 〈L[A], A〉 such that ⊩ V = L[a], a ⊆ ω, A is definable from a. Moreover if 0# ∉ L[A] then ⊩ 0# does not exist. Now as any M ⊨ ZFC can be generically extended to a model of the form L[A] (without introducing 0#) we obtain: For any 〈M, A〉 ⊨ ZFC (that is, M ⊨ ZFC and M obeys Replacement for formulas mentioning A as a predicate) there is an 〈M, A〉-definable forcing such that ⊩ V = L[a], a ⊆ ω, 〈M, A〉 is definable from a. Moreover if 0# ∉ M then ⊩ 0# does not exist.

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Meridional Circulation, Turbulence, and Diffusion Processes in Stars

International Astronomical Union Colloquium ◽

10.1017/s0252921100080477 ◽

1976 ◽

Vol 32 ◽

pp. 109-116 ◽

Cited By ~ 1

Author(s):

S. Vauclair

Keyword(s):

Convection Zone ◽

Diffusion Processes ◽

Meridional Circulation ◽

Diffusion Time ◽

Work In Progress ◽

First Results ◽

Stellar Envelopes ◽

And Diffusion ◽

Turbulence And Diffusion ◽

Hr Diagram

This paper gives the first results of a work in progress, in collaboration with G. Michaud and G. Vauclair. It is a first attempt to compute the effects of meridional circulation and turbulence on diffusion processes in stellar envelopes. Computations have been made for a 2 Mʘstar, which lies in the Am - δ Scuti region of the HR diagram.Let us recall that in Am stars diffusion cannot occur between the two outer convection zones, contrary to what was assumed by Watson (1970, 1971) and Smith (1971), since they are linked by overshooting (Latour, 1972; Toomre et al., 1975). But diffusion may occur at the bottom of the second convection zone. According to Vauclair et al. (1974), the second convection zone, due to He II ionization, disappears after a time equal to the helium diffusion time, and then diffusion may happen at the bottom of the first convection zone, so that the arguments by Watson and Smith are preserved.

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