In this paper, we prove the following. If
, then there is a generic extension of
, the constructible universe, in which it is true that the set
of all constructible reals (here—subsets
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In this paper, we prove the following. If
, then there is a generic extension of
, the constructible universe, in which it is true that the set
of all constructible reals (here—subsets of
) is equal to the set
of all (lightface)
reals. The result was announced long ago by Leo Harrington, but its proof has never been published. Our methods are based on almost-disjoint forcing. To obtain a generic extension as required, we make use of a forcing notion of the form
in
, where
adds a generic collapse surjection
b from
onto
, whereas each
,
, is an almost-disjoint forcing notion in the
-version, that adjoins a subset
of
. The forcing notions involved are independent in the sense that no
-generic object can be added by the product of
and all
,
. This allows the definition of each constructible real by a
formula in a suitably constructed subextension of the
-generic extension. The subextension is generated by the surjection
b, sets
with
, and sets
with
. A special character of the construction of forcing notions
is
, which depends on a given
, obscures things with definability in the subextension enough for vice versa any
real to be constructible; here the method of
hidden invariance is applied. A discussion of possible further applications is added in the conclusive section.
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