FOM: Sigma_1^1-AC error

CTSteel steelnet at
Wed Aug 4 19:08:36 EDT 1999

The errors in Bob Soare's statements about Sigma_1^1-AC versus Sigma_1^1-DC
are due to me (and perhaps also Harrington), not to Soare. Harvey proved that
Sigma_1^1-AC does not imply Sigma_1^1-DC. I confused this with some other
work, and this led to Soare's mis-attribution. Bob was careful enough to
ask me about it several times, so the fault is all mine.

Harrington did prove Delta_1^1-CA does not imply Sigma_1^1-AC. I believe his
proof had a priority argument in it, although to my mind its most ingenious
feature was its use of Sacks' "shiny little box" technique, which is based
on Kleene's Recursion Theorem.

John Steel

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