# [FOM] 326: Progress in Pi01 Incompleteness 1

Harvey Friedman friedman at math.ohio-state.edu
Wed Oct 22 23:58:54 EDT 2008

PROGRESS IN Pi01 INCOMPLETENESS 1
DRAFT
Harvey M. Friedman
October 22, 2008

We define N to be the set of all nonnegative integers. For any set A
containedin V, we write A' = V\A.

Let R containedin N^4k. We say that R is upwards if and only if for
all x in N^3k and y in N^k, (x,y) in R implies max(x) < max(y).

We define RA for A containedin N^k only, by

RA = {y in N^k: (therexists x in A^3)((x,y) in R)}.

THEOREM 1.1. For all k >= 1 and upwards R containedin N^4k, some RA is
A'. Furthermore, A is unique.

We want to work with very concrete R containedin N^4k.

We say that x,y in N^k are order equivalent if and only if for all 1
<= i,j <= k, x_i < x_j iff y_i < y_j.

We say that R containedin N^k is order invariant if and only if for
all order equivalent x,y in N^k, x in R iff y in R.

We have the following concrete form of Theorem 1.1. Here EXP means
"the characteristic function is computable in exponential time".

THEOREM 1.2. For all k >= 1 and upwards order invariant R containedin
N^4k, some RA is A'. Furthermore, A is unique and lies in EXP.

We can weaken Theorem 1.2 as follows.

THEOREM 1.3. For all k >= 1 and upwards order invariant R containedin
N^4k, some RRA is R(A'). Furthermore, we can require that A lies in EXP.

But not like this.

THEOREM 1.4. The following is false. For all k >= 1 and upwards order
invariant R containedin N^4k, some RRA is R(A') union A+2.

However!!

PROPOSITION 1.5. For all k >= 1 and upwards order invariant R
containedin N^4k, some RRA is a subset of R(A') union A+2 with the
same k dimensional powers of (8k)!. Furthermore, we can require that A
lies in EXP.

It is clear that Proposition 1.5 is explicitly arithmetical, with the
help of EXP. By an infinite tree argument, we can see that Proposition
1.5 is implicitly Pi01. We can easily make Proposition 1.5 explicitly
Pi01 as follows.

All of the definitions above - A', R is upwards, RA, R is order
invariant - can be restated for A containedin [0,r]^k and R
containedin [0,r]^4k.

PROPOSITION 1.6. For all k,r >= 1 and upwards order invariant R
containedin [0,r]^4k, some RRA is a subset of R(A') union A+2 with the
same k dimensional powers of (8k)!.

MAH = ZFC + {there exists a strongly n-Mahlo cardinal}_n. MAH+ = ZFC +
"for all n there exists a strongly n-Mahlo cardinal".

THEOREM 1.7. Theorems 1.1 - 1.4 are provable in RCA_0. Propositions
1.5, 1.6 are provable in MAH+ but not in MAH, assuming that MAH is
consistent. Propositions 1.5, 1.6 are provably equivalent, over ACA,
to CON(MAH). Propositions 1.5, 1.6 are not provable in any consistent
subsystem of MAH. In particular, Propositions 1.5, 1.6 are not
provable in ZFC, assuming ZFC is consistent. If we delete "union A+2"
then Propositions 1.5, 1.6 become weakened forms of Theorems 1.1, 1.2,
1.3. These results hold even if we remove the second sentence of
Proposition 1.5.

The 4 in "4k" can be extended to any higher number in the obvious way
without changing the results. We have not investigated the
independence status when 4 is replaced by 2 or 3. Probably 3 will
still give the same results, but 2 is not enough for independence.

We can use Z+ instead of N. Then we can use A+1 instead of A+2.

We can weaken Proposition 1.6 by replacing r with various definite
functions of k. In particular, if we use iterated exponentials, then
we get the consistency of: impredicative theory of types, or, roughly,
Zermelo set theory (with bounded separation). I think that using the
Ackermann function should get Con(ZFC). As we climb up the < epsilon_0
recursive functions, we should be going up through the consistency of
MAH. Details to be worked out later.

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

manuscripts. This is the 326th 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. NOTE: The title of #269 has been corrected
from the original.

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
261. Pi01/nicer  12/5/05  2:26AM
262. Correction/Restatement  12/9/05  10:13AM
263. Pi01/digraphs 1  1/13/06  1:11AM
264. Pi01/digraphs 2  1/27/06  11:34AM
265. Pi01/digraphs 2/more  1/28/06  2:46PM
266. Pi01/digraphs/unifying 2/4/06  5:27AM
267. Pi01/digraphs/progress  2/8/06  2:44AM
268. Finite to Infinite 1  2/22/06  9:01AM
269. Pi01,Pi00/digraphs  2/25/06  3:09AM
270. Finite to Infinite/Restatement  2/25/06  8:25PM
271. Clarification of Smith Article  3/22/06  5:58PM
272. Sigma01/optimal  3/24/06  1:45PM
273: Sigma01/optimal/size  3/28/06  12:57PM
274: Subcubic Graph Numbers  4/1/06  11:23AM
275: Kruskal Theorem/Impredicativity  4/2/06  12:16PM
276: Higman/Kruskal/impredicativity  4/4/06  6:31AM
277: Strict Predicativity  4/5/06  1:58PM
278: Ultra/Strict/Predicativity/Higman  4/8/06  1:33AM
279: Subcubic graph numbers/restated  4/8/06  3:14AN
280: Generating large caridnals/self embedding axioms  5/2/06  4:55AM
281: Linear Self Embedding Axioms  5/5/06  2:32AM
282: Adventures in Pi01 Independence  5/7/06
283: A theory of indiscernibles  5/7/06  6:42PM
284: Godel's Second  5/9/06  10:02AM
285: Godel's Second/more  5/10/06  5:55PM
286: Godel's Second/still more  5/11/06  2:05PM
287: More Pi01 adventures  5/18/06  9:19AM
288: Discrete ordered rings and large cardinals  6/1/06  11:28AM
289: Integer Thresholds in FFF  6/6/06  10:23PM
290: Independently Free Minds/Collectively Random Agents  6/12/06
11:01AM
291: Independently Free Minds/Collectively Random Agents (more)  6/13/06
5:01PM
292: Concept Calculus 1  6/17/06  5:26PM
293: Concept Calculus 2  6/20/06  6:27PM
294: Concept Calculus 3  6/25/06  5:15PM
295: Concept Calculus 4  7/3/06  2:34AM
296: Order Calculus  7/7/06  12:13PM
297: Order Calculus/restatement  7/11/06  12:16PM
298: Concept Calculus 5  7/14/06  5:40AM
299: Order Calculus/simplification  7/23/06  7:38PM
300: Exotic Prefix Theory   9/14/06   7:11AM
301: Exotic Prefix Theory (correction)  9/14/06  6:09PM
302: PA Completeness  10/29/06  2:38AM
303: PA Completeness (restatement)  10/30/06  11:53AM
304: PA Completeness/strategy 11/4/06  10:57AM
305: Proofs of Godel's Second  12/21/06  11:31AM
306: Godel's Second/more  12/23/06  7:39PM
307: Formalized Consistency Problem Solved  1/14/07  6:24PM
308: Large Large Cardinals  7/05/07  5:01AM
309: Thematic PA Incompleteness  10/22/07  10:56AM
310: Thematic PA Incompleteness 2  11/6/07  5:31AM
311: Thematic PA Incompleteness 3  11/8/07  8:35AM
312: Pi01 Incompleteness  11/13/07  3:11PM
313: Pi01 Incompleteness  12/19/07  8:00AM
314: Pi01 Incompleteness/Digraphs  12/22/07  4:12AM
315: Pi01 Incompleteness/Digraphs/#2  1/16/08  7:32AM
316: Shift Theorems  1/24/08  12:36PM
317: Polynomials and PA  1/29/08  10:29PM
318: Polynomials and PA #2  2/4/08  12:07AM
319: Pi01 Incompleteness/Digraphs/#3  2/12/08  9:21PM
320: Pi01 Incompleteness/#4  2/13/08  5:32PM
321: Pi01 Incompleteness/forward imaging  2/19/08  5:09PM
322: Pi01 Incompleteness/forward imaging 2  3/10/08  11:09PM
323: Pi01 Incompleteness/point deletion  3/17/08  2:18PM
324: Existential Comprehension  4/10/08  10:16PM
325: Single Quantifier Comprehension  4/14/08  11:07AM

Harvey Friedman