[FOM] Least class that ...

Richard Haney rfhaney at yahoo.com
Mon Jan 30 22:19:39 EST 2006

A. S. Troelstra & D. van Dalen in * Constructionism in Mathematics: An
Introduction *, Vol. 1, seem to make frequent use of the idea of the
least class containing some basic elements and closed under some sort
of (say, generalized iteration of) operations.  This sort of notion
seems to be used routinely in many other contexts as well in
mathematics.  A typical definition I've seen for "the least class that
..." is the set-theoretic intersection of all classes containing the
basic elements and closed under the operations.

This sort of definition seems to assume some "axiom of infinity" of
sorts (especially when unbounded constructions are involved) and to
also be impredicative.

Does anyone know of a generally useful, alternative type of definition
(for "the least class that ..." or an alternative concept) that will
satisfy finitists and predicativists as well?  Is such an idea of "the
least class that ..." really needed in metamathematics?

Richard Haney

Institutionally nonaffiliated
Research interests in empirical and other issues in the foundations of
mathematics relevant to "real-world" applicability

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