# [FOM] 261:Pi01/nicer

Harvey Friedman friedman at math.ohio-state.edu
Mon Dec 5 02:26:51 EST 2005

```I have made the Pi01 independent statement nicer, and put up the new version
at

http://www.math.ohio-state.edu/%7Efriedman/

two previous versions. This latest is dated December 4, 2005.

>From posting #260, the December 3 version (see website), we had

OLD THEOREM 1.4. For all k,n >= 1, every strictly dominating order invariant
R containedin [1,n]2k has an antichain A such that flog R[A\2^(8k)!-1]
contains flog A'.

OLD PROPOSITION A. For all k,n >= 1, every strictly dominating order
invariant R containedin [1,n]2k has an antichain A such that flog
RRRR[A\2^(8k)!-1] contains flog RRR[A'].

We now have

NEW THEOREM A(1,0). For all k,n >= 1 and t >= (8k)!, every strictly
dominating order invariant R containedin [1,n]2k has an antichain A such
that flog_t+1 R[A\{t}k] = flog_t+1 A'.

NEW PROPOSITION A(4,3). For all k,n >= 1 and t >= (8k)!, every strictly
dominating order invariant R containedin [1,n]2k has an antichain A such
that flog_t+1 RRRR[A\{t}k] = flog_t+1 RRR[A'].

*************************************

manuscripts. This is the 259th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-249 can be found at
http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM
archives, 6/15/05, 9:18PM.

250. Extreme Cardinals/Pi01  7/31/05  8:34PM
251. Embedding Axioms  8/1/05  10:40AM
252. Pi01 Revisited  10/25/05  10:35PM
253. Pi01 Progress  10/26/05  6:32AM
254. Pi01 Progress/more  11/10/05  4:37AM
255. Controlling Pi01  11/12  5:10PM
256. NAME:finite inclusion theory  11/21/05  2:34AM
257. FIT/more  11/22/05  5:34AM
258. Pi01/Simplification/Restatement  11/27/05  2:12AM
259. Pi01 pointer  11/30/05  10:36AM
260. Pi01/simplification  12/3/05  3:11PM

Harvey Friedman

```