FOM: Re: GCH.
Jan Mycielski
jmyciel at euclid.Colorado.EDU
Sat Jun 24 17:49:47 EDT 2000
AXIOMS WHICH IMPLY GCH (ADDITIONAL REMARKS)
I wanted to add some remarks to my FOM note of June 22.
1. The axioms A and C which I proposed there are consequences of
V = L. (This is immediate since V = L implies GCH.). Thus if ZF is
consistent ZFC + A + C is also consistent. But the intention of A, B and C
(recall that C implies B) is that those axioms should be also consistent
with all natural large cardinal axioms. As well known (D. Scott), V = L
does not have this property.
2. Let alpha be any regular cardinal and T be the topology in
P(alpha) (= {0, 1}^alpha) which is obtained from the usual product
topology by closing the class of open sets under intersections of fewer
than alpha sets. A subset of P(alpha) is T-meager iff it is a union of at
most alpha T-nowhere dense sets. A subset of P(alpha) is said to have the
T-property of Baire iff it differs from an open set only by a T-meager
set.
The axiom A has two natural companions:
A*. For every regular cardinal alpha every OD subset of P(alpha)
which is of power larger than alpha has a non-empty T-perfect subset.
A**. Every OD subset of P(alpha) has the T-property of Baire.
Those axioms do not appear to imply A, since both are restricted
to regular alpha. Of course, unlike A, they are inconsistent with V = OD.
PROBLEM. Is the theory ZFC + A + A* + A** + C consistent?
Jan Mycielski
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