FOM: Answers to some questions of Shipman
Andrian-Richard-David.Mathias at univ-reunion.fr
Wed Mar 8 05:53:33 EST 2000
1. A simple "meta-equivalent" of the Axiom of Constructibility
V=L is "equivalent" to saying that there is a Sigma_1 formula, without
parameters, that defines a function G, assigning to each set x a set G(x),
such that the relation "y is a member of G(x)" is a well-ordering of the
entire universe. More precisely,
for any Sigma_1 formula G with no free variables, let (V=L)_G be the formula
that says that G is a function with domain the universe, and such that the
relation "y is a member of G(x)" defines a linear ordering of the universe
in which each non-empty set has a least element.
Let (V=L) be Goedel's formulation of the axiom of constructibility.
Then (i) there is a Sigma_1 formula H with no free variables such that ZF
proves that if (V=L) then (V=L)_H;
(ii) for any Sigma_1 formula G with no free variables, ZF proves that
if (V=L)_G then (V=L).
The assertion (V=L)_G is stronger than saying there is a Sigma_1 wellordering
of the universe, which does not imply that V=L: see a paper of Jech and
Harrington (from memory, JSL, around 1978).
2. If there is a real-valued measurable cardinal, there is in some set-generic
extension an elementary embedding of the original universe into some inner
model of the extension. That implies that the sharp of each real in the ground
model is in the extension; but set-generic extensions add no sharps, so they
are already in the ground model.
A R D Mathias
ardm at univ-reunion.fr
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