Implicit definability and infinitary languages

1968 ◽  
Vol 33 (3) ◽  
pp. 446-451 ◽  
Author(s):  
Kenneth Kunen
Keyword(s):  
Set Theory ◽  
Recursive Enumerability ◽  
Weak Set Theory

In this paper we define the notions of invariant implicit definability (i.i.d.) and semi-invariant implicit definability (s.i.i.d.) on ε-models, A, of a certain weak set theory. These notions are intended to be the analogs of recursiveness and recursive enumerability, respectively. Following Barwise, with each A is associated an infinitary language whose formulas are elements of A.

scholarly journals The Classical Continuum without Points

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.

scholarly journals THE CLASSICAL CONTINUUM WITHOUT POINTS

2013 ◽  
Vol 6 (3) ◽  
pp. 488-512 ◽  
Author(s):  
GEOFFREY HELLMAN ◽  
STEWART SHAPIRO
Keyword(s):  
Set Theory ◽  
Classical Analysis ◽  
One Dimensional ◽  
Actual Infinity ◽  
Ordered Field ◽  
Free Construction ◽  
Archimedean Property ◽  
Topological Models ◽  
Dedekind Completeness ◽  
Weak Set Theory

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.

A Course on Set Theory

2009 ◽  
Author(s):  
Ernest Schimmerling
Keyword(s):  
Set Theory

Set Theory

2016 ◽  
Author(s):  
Daniel W. Cunningham
Keyword(s):  
Set Theory
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