[FOM] Historical Queries on AC
joeshipman at aol.com
Tue Jan 8 09:51:09 EST 2008
1) In 1938, Tarski (Fund. Math. vol. 30) showed that AC follows from
the axiom that there is a Universe containing any set (in other words,
that arbitrarily large inaccessible cardinals exist). Of course, the
consistency (rather than the truth) of AC doesn't need the full
Universes axiom, just one inaccessible limit of inaccessibles (because
that set will satisfy ZF and the Universes axiom).
Was this published before or after Godel's 1938 PNAS paper proving the
consistency of GCH and AC?
If Tarski's paper came first, there is a sense in which he, not Godel,
was the first mathematician to provide a consistency proof of AC
acceptable to modern mathematicians (because the Tarski "Universes
axiom" is freely used by modern mathematicians in algebraic geometry
and other core mathematical areas).
2) When was the first proof that GCH implies AC published, and by whom?
I know that Lindenbaum and Tarski announced the result in 1926, but the
proof commonly cited comes from a 1947 paper of Sierpinski.
3) Sierpinski's proof is is stronger than just "GCH-->AC", it actually
shows that for a set A to be well-orderable one needs only that there
are no intermediate cardinals anywhere in the sequence A < P(A) <
P(P(A)) < P(P(P(A))) < P(P(P(P(A)))). Has anyone improved this to
require a smaller set of no-intermediate-cardinal assumptions?
4) Analogously to 3), what is the best known result on how many levels
of Universes above A are necessary in order to well-order A?
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