Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF

Philosophia Mathematica 24 (1):50-59 (2016)
Abstract
Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted _aussonderung_ but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for "low" sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no set of _all_ ordinals is obtained thereby. This "recurrence problem" is discussed.
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DOI 10.1093/philmat/nku005
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Frege's Cardinals and Neo-Logicism.Roy T. Cook - 2016 - Philosophia Mathematica 24 (1):60-90.

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