friedman at math.ohio-state.edu
Wed Apr 8 04:48:24 EDT 1998
This is a correction to the 14th in a series of positive self contained
postings to fom covering a wide range of topics in f.o.m. Previous postings
1:Foundational Completeness 11/3/97, 10:13AM, 10:26AM.
3:Simplicity 11/14/97 10:10AM.
4:Simplicity 11/14/97 4:25PM
5:Constructions 11/15/97 5:24PM
6:Undefinability/Nonstandard Models 11/16/97 12:04AM
7.Undefinability/Nonstandard Models 11/17/97 12:31AM
8.Schemes 11/17/97 12:30AM
9:Nonstandard Arithmetic 11/18/97 11:53AM
10:Pathology 12/8/97 12:37AM
11:F.O.M. & Math Logic 12/14/97 5:47AM
12:Finite trees/large cardinals 3/11/98 11:36AM
13:Min recursion/Provably recursive functions 3/20/98 4:45AM
14:New characterizations of the provable ordinals 4/8/98 2:09AM
A complete archiving of fom, message by message, is available at
Also, my series of positive postings (only) is archived at
1. (1st paragraph, "We present..."). Last sentence should read: We denote
the least ordinal greater than all provable ordinals of T by o(T).
2. The 2nd and 3rd paragraphs should be replaced by the following.
This concept is very robust. For instance, we may consider arithmetically
defined relations instead of primitively recursively defined relations. If
T contains ACA_0 then we get the same provable ordinals. In fact, we can
use Sigma-1-1 defined relations, and if T contains ATR_0 then again we get
the same provable ordinals.
For "good" theories T, o(T) is a recursive ordinal. However, there are
consistent r.e. theories T containing RCA_0 for which o(T) is Church-Kleene
omega_1. The situation is clarified by the following:
THEOREM. Let T be a Sigma-1-1 theory containing ACA_0. Then o(T) is a
recursive ordinal if and only if every Pi-1-1 theorem of T is true.
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