FOM: conservative extensions, correction and sequel

Stephen G Simpson simpson at
Sun Oct 11 12:33:56 EDT 1998

This is a sequel to my posting of 9 Oct 1998 23:24:08.

I said

 >   Sigma^1_2-DC_0 is conservative over Pi^1_1-CA_0 for Pi^1_3 sentences.

but this is wrong: it should be Sigma^1_2-AC_0, not Sigma^1_2-DC_0.

More generally, for each k, Sigma^1_{k+1}-AC_0 is conservative over
Pi^1_k-CA_0 for Pi^1_l sentences, l=min(k+2,4).  Also,
Sigma^1_infinity-DC_0 is conservative over Pi^1_infinity-CA_0 for
Pi^1_4 sentences.  AC = countable axiom of choice, DC = countable
dependent choice, CA = comprehension axiom.  All of these results are
proved in my forthcoming book.

Does anyone agree or disagree with my use of the term
"instrumentalism" to make philosophical sense out of conservative
extension theorems?  Are there any conservative extension results that
cannot be so interpreted?

-- Steve

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