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Title:A non-well-founded set theory (GST)
Author(s):Harnish, Stephen Hostetler
Doctoral Committee Chair(s):Takeuti, Gaisi
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Computer Science
Abstract:This thesis introduces a new set theory referred to as the graph-isomorphism set theory (GST). GST does not satisfy the foundation axiom. Peter Aczel has presented several non-well-founded (NWF) set theories within a unified framework. For any bisimulation $\sim,$ he defined the axioms AFA$\sp\sim$: an accessible pointed graph (apg) is an exact picture iff it is $\sim$-extensional. When the $\sim$ are regular bisimulations whose definitions are absolute for full systems, then the resulting set theories AFA$\sim$ + ZFC$\sp-$ (ZFC minus foundation) are known to be consistent relative to ZFC. Regularity assures that the AFA$\sp\sim$ set theories satisfy extensionality.
Isomorphism between apgs (denoted by $\cong)$ is a bisimulation. However, this bisimulation is not regular. Now consider the axiom AFA$\sp\cong.$ It implies the following criterion for set equality: two sets are equal iff their canonical graphs are isomorphic. GST is the theory AFA$\sp\cong$ + ZFC$\sp{{-}{-}}$ where ZFC$\sp{{-}{-}}$ is a modification of ZFC that excludes foundation and the standard statement of extensionality. More specifically, ZFC$\sp{{-}{-}}$'s axioms of separation, extensionality, infinity, and choice add requirements that certain sets are noncyclic. These modified axioms of GST are equivalent to the standard ones (within ZFC).
Chapter 2 of this thesis presents GST's axioms and basic properties. GST implies the negations of extensionality and foundation. This chapter ends with two surprising cardinality results that show that some finite sets have infinite power sets in GST. Chapter 3 demonstrates that the GST universe has an inner $\in$-model of ZFC. Finally, Chapter 4 employs the concept of iterated bisimulation collapse to prove that GST is consistent relative to ZFC.
Most recent applications of NWF sets employ the solution lemma, which is a consequence of Aczel's particular anti-foundation axiom named AFA. Chapter 1 sketches how bisimulation collapses and canonical decorations lead to analogs of the solution lemma. These analogs suggest wider use of anti-foundation axioms besides AFA. This chapter ends with applications of GST to geometry and the theory of languages.
Issue Date:1996
Rights Information:Copyright 1996 Harnish, Stephen Hostetler
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9625142
OCLC Identifier:(UMI)AAI9625142

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