[FOM] 314: Pi01 Incompleteness/Digraphs

Harvey Friedman friedman at math.ohio-state.edu
Sat Dec 22 04:12:35 EST 2007

In #313, we did not use the right kind of iterated images of binary
relations R. 

When we fixed this, we noticed that it is best to restate everything in
graph theoretic terms.



Here digraphs will not have multiple edges, but can have loops.

Let G be a digraph. We write V(G) for the set of vertices of G, and E(G) for
the set of edges of G. Edges of G are ordered pairs of vertices of G.

We say that G is a digraph on V(G).

Let S containedin V(G). We say that S is independent if and only if there is
no edge with vertices from S. We write S' = V(G)\S.

We write GS for {y: (therexists x in S)((x,y) in E(G))}.

We use interval notation. All intervals are discrete. We use [1,inf) for the
set of all positive integers.

Let G be a digraph on [1,inf)^k. We say that G is upwards if and only if

(x,y) in E(G) implies max(x) < max(y).

The following is an immediate consequence of a well known theorem from graph

THEOREM 1.1. Every upwards digraph G on [1,inf)^k has an independent set S,
where GS = S'. Furthermore, S is unique.

Let x,y in [1,inf)^k. We say that x,y are order equivalent if and only if
for all 1 <= i,j <= k,

x_i < x_j iff y_i < y_j.

Let G be a digraph with V(G) = [1,inf)^k. We say that G is order invariant
if and only if for all vertices x,y,z,w, if (x,y) and (z,w) are order
equivalent (as elements of [1,inf)^2k), then

(x,y) in E(G) iff (z,w) in E(G).

For n >= 1, the powers of n are the tuples whose coordinates are from
1,n,n^2,... .

PROPOSITION 1.2. Every upwards order invariant digraph G on [1,inf)^k has an
independent set S, where any power of (8k)! lying in some k element
independent subset of S', lies in some k element independent subset of GS,
and out of S+1. 

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

THEOREM 1.3. Theorem 1.1 is provable in RCA_0. Proposition 1.2 is provable
in MAH+ but not in MAH, assuming that MAH is consistent. Proposition 1 is
provably equivalent, over ACA, to CON(MAH). Proposition 1 is not provable in
any consistent subsystem of MAH. In particular, Proposition 1 is not
provable in ZFC, assuming ZFC is consistent. If we remove "and outside S+1",
then Proposition 1.2 is an immediate consequence of Theorem 1.1.

Here (8k)! is just a convenient expression.


The finite forms are obtained trivially by replacing [1,inf) with [1,n]. All
of the definitions are restated in the obvious way with [1,inf) replaced
throughout by [1,n]. Specifically,

PROPOSTION 2.2. Every upwards order invariant graph G on [1,n)^k has an
independent set S, where any power of (8k)! lying in some k element
independent subset of S', lies in some k element independent subset of GS,
and out of S+1.

Proposition 2.2 is explicitly Pi01.

All of the results read the same.

I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 314th 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
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

Harvey Friedman

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