Non Well-Founded Sets
Overview
Non well-founded sets are discussed, primarily to provide context for the presentation of work on interpretations of set theory based on infinitary comprehension.
After some discussion of the motivation for non well-founded set theories, a crude classification of non well-founded sets is used to delimit the scope of interpretations based on "infinitary comprehension".
Introduction
After some discussion of the motivation for non well-founded set theories, a crude classification of non well-founded sets is used to delimit the scope of interpretations based on "infinitary comprehension".
Prologue

Sets are fundamental to modern mathematics, but the most general conception of set proved to be incoherent and was quickly replaced (for most mathematical work) by the narower, coherent, conception of sets which arises from the constraint of well-foundedness.

This constraint is however pragmatically onerous, and research has continued on broader conceptions of set. One particular source of motivation for non well-founded sets can be found in the pragmatics of practical formalisation of mathematics, and it is this domain which guides the work I am engaged in on infinitary comprehension.

Kinds of Non Well-Founded Theories

I will mention here just three broad kinds of non well-founded set theory, only one of which is considered further.

  • non-extensional
  • based on comprehension (sets as extensions of properties)
  • sets as arbitrary properties or graphs
  • sets as rules or functions
  • based on other conceptions of set

Note that these are not offered as exclusive categories. Most often non-extensional theories are constructive and sets are treated as effective rules. Comprehension is pervasive, but there are counter-examples. My own primary concern is with conceptions of set in which are integrated the well-founded and the non well-founded sets, the conception of set as rule and of graph, remaining within the general conveption of sets as extensions of properties.

These are further considered in turn.

Kinds of Set Theory
Non-Extensional Set Theories

As far as I am aware these are mostly constructive, and therefore do not provide a foundation for mathematics as it is known and practised but as constructivists would like it to be,

They are mentioned here since they are I think usually not well-founded, but I shall say no more about them because they are not appropriate as foundations for the kind of mathematics which interests me.

Set Theories based on Comprehension

Frege's infamous foundation for mathematics effectively admitted unrestricted comprehension, i.e. it allowed that to any property of sets (expressible in his system) there is an individual set with the same extension. This gave rise to paradox, but nevertheless provides a compelling, if over-generous, conception of what kind of thing a set might be.


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