SETS CONSTRUCTIVE USING L
C. C.
CHANG1
Introduction. At the end of 1966,1 started thinking about using the infinitary
languages LKK, K a regular infinite cardinal, to form a hierarchy much in the same
way Godel [5] formed the ramified heirarchy of sets from the usual first-order
language L^. At that time I was interested in locating more precisely the Hanf
number of L^^ , and I had just proved some results about the behavior of the
ramified hierarchy with respect to the infinitary languages LXK under the assumption
of existence of large cardinals. (See Chang [1] and Theorem IX of this paper.)
Therefore, I thought it was significant to look more closely at the hierarchy of
/c-constructible sets. At the very beginning, the concept and notion of /c-constructi-
bility seemed difficult to handle, as one was always afraid of saying something non-
sensical like, for example, that an infinitary formula is absolute. Soon after,
however, I realized that practically all of the Godel results on the ramified hierarchy
have natural translations to the hierarchy of /c-constructible sets. The most impor-
tant of these is, of course, the isomorphism theorem which has an analog given
here in Theorem V. From this one can deduce a version of the GCH, Theorem VI,
assuming that all sets are /c-constructible. Furthermore, it turns out that the con-
struction of Scott [9] using measurable cardinals also generalizes, thus yielding the
present Theorem VIII. The solution of these easy problems led to other problems
about /c-constructible sets. Many of these have been solved by other people since
the Set Theory conference at UCLA.
In this paper, I shall first give the basic notions and definitions, then state the
list of fairly standard results taken from the ramified hierarchy but given here for
1 The reserach and preparation of this paper was partially supported by NSF Grants #5600,
#8827, and partially by a Fulbright Grant.
1
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