Foundation Systems
Overview
Sets are the simplest way of combining things into collections. Nevertheless there are many different theories of sets, which differ mainly but not exclusively in their conception of what sets there are.
This document is concerned with methods for approaching the resolution of the continuum hypothesis is "classical" set theory. It is more a philosophical than a technical contribution, from an author with insufficient relevant technical competence to make a substantive technical contribution.
Set Theories
Sets are the simplest way of combining things into collections. Nevertheless there are many different theories of sets, which differ mainly but not exclusively in their conception of what sets there are.
Introduction
The only feature (so far as I am aware) common to all set theories is the primacy of the binary relation of membership. Some assert that extensionality is essential, but others devise set theories which are not extensional. Most professional set theorists an other mathematicians work within theories whose ontology is confined to pure well-founded sets, but a minority work work with non-well-founded and/or impure ontologies,
Semantic Approaches to the Resolution of the Continuum Hypothesis
This document is concerned with methods for approaching the resolution of the continuum hypothesis is "classical" set theory. It is more a philosophical than a technical contribution, from an author with insufficient relevant technical competence to make a substantive technical contribution.
Some Technical Background
A brief statement of the problem and some relevant facts.
Non-Semantic Approaches to the Resolution of CH
When set theorists talk of how CH might be resolved what they generally seem to be looking for is some conjecture which can be proven to entail either CH or its negation and which is intuitively more plausible than CH (or its negation). I begin by discussing some of the disadvantages of this approach.
Semantic Approaches to the Resolution of CH
A semantic approach is outlined, consisting first refining our understanding of the meaning of CH and then exploiting intuitions specific to that refined semantics.
Refining the Meaning of CH
"The" meaning of CH is refined. The method is to chose a set of models of ZFC and to take the refined "CH" to be the claim that CH is true in the nominated set of models. The set chosen is the set of "standard" models.
Exploiting Semantic Refinements
Some ideas on how the extra information available in refined statements of CH can be exploited in determining whether CH is true.

up quick index © RBJ

privacy policy

Created:2002-10-15

$Id: index.xml,v 1.1 2004/03/11 10:44:05 rbj Exp $

V