FOM: iterative conception of set
holmes at catseye.idbsu.edu
Thu Feb 26 18:42:24 EST 1998
This posting is from M. Randall Holmes
(Neil Tennant wrote)
Set theory in the wake of the responses to the paradoxes seems to me
to be a melange of the intuitive and the conceptual. The purely
logical notion of set (class) as the extension of a concept had to
give way to the hybrid notion of sets as mathematical objects produced
by an iterative procedure, but also obeying certain laws of
abstraction. As a reducing theory, ZFC and its extensions do not
dictate any "style" that now clearly ought to be mimicked by any other
attempt at foundations.
The notion "set of ZFC" can be coded into higher order logic as
"isomorphism class of (pointed) well-founded extensional relations".
This notion is purely logical if the notion of set (class) as the
extension of a concept is taken as purely logical. It is a third-order
concept in higher order logic.
To get this notion to satisfy the axioms of ZFC, one needs to suppose
that one has "enough" objects. Nothing else is needed. One
doesn't even need any concept of iterative construction, though this
concept is certainly intuitively appealing.
The opinions expressed | --Sincerely, M. Randall Holmes
above are not the official | Math. Dept., Boise State Univ.
opinions of any person | holmes at math.idbsu.edu
or institution. | http://math.idbsu.edu/~holmes
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