FOM: Tychonoff theorem, Weak inaccessibles

Joe Shipman shipman at
Tue Feb 22 10:26:45 EST 2000


>> In particular, consider the Tychonoff theorem:  ``A product of
compact spaces is compact.''  Now, there is a very nice, short proof of
this result using nonstandard analysis.  (cf. Hurd and Loeb, or
Albeverio, et. al.). The first line of the proof applies AC in the
cartesian product form, but of course, in the case that the product
space is empty, the conclusion holds anyway.  That is, even if the
product of some compact spaces is empty, it is compact, because the
empty space is a compact space.  The remainder of the proof uses
infinitesimals, which are available in any set theory with the Boolean
Prime Filter theorem (BPF, or equivalently, BPI), to show that, when the
product of compact spaces is nonempty, it is compact.  Now, BPF is
strictly weaker than AC (cf. Rubin and Howard, and many others), so one
does not need, IMHO, the full strength of AC for the
Tychonoff Theorem.<<

The full Tychonoff theorem is equivalent to AC (J.L. Kelley, Fund. Math.
37 (1950), 75-76; probably also in Kelley's 1955 book "General
Topology").  You should check your argument carefully to make sure you
are not making some extra assumption on the underlying space or the
cardinality of the set indexing the product.

>>  I am a little confused by your comment above, though, that c could
simply be the \aleph_1'st fixed point of the aleph function.  How could
c=\aleph_c and c=\aleph_{\aleph_1}?  Either I misunderstood your comment
or this implies that
\aleph_1=\aleph_{\aleph_1}, it seems.  [I hope I'm not just getting
myself confused.  :-)] <<

The first fixed point of the aleph function is the limit of
aleph_0,aleph_aleph_0,aleph_aleph_aleph_0,....but this has countable
cofinality so can't be c.  The fixed points of the aleph function form a
proper class of ordinals and the aleph_oneth member of this class is the
first one which has uncountable cofinality and is therefore a candidate
for c.  A REGULAR fixed point of
the aleph function is a weak inaccessible, but within ZFC we can force
c=aleph_c if we don't insist on regularity.

-- JS

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