Set Theories
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.
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,
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,
Classifying Set Theories
Set theories may be classified using the following characteristic:
  • extensionality
  • well-foundedness
There are of course many others but these are arguably the best basis for a highest level of classification. I know little about non-extensional set theories, those of which I am aware are constructive set theories and on these I refer the reader to beeson80, though this may now be a bit dated.
Well-Founded Set Theories

Set theory as formalised by Frege was a "naive" set theory in which an unqualified principle of abstraction was available. This means that the extension of any property expressible in the formal language corresponded to that of some set, and this liberal ontological principle was shown by Russell's paradox to render the logical system inconsistent.

Two earliest response to this problem, published by Russell and Zermelo in 1908, both effectively realised consistency by constraining the ontology to be well-founded (though Russell's solution was more elaborate, its central principle was the avoidance of "vicious" circularity).

The dominant set theory since those early times is a derivative of Zermelo's system called Zermelo-Frankel set theory, particularly the version with a choice principle (ZFC).


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