[FOM] 319: Pi01 Incompleteness/Digraphs/#3
Harvey Friedman
friedman at math.ohio-state.edu
Tue Feb 12 10:36:45 EST 2008
We make an improvement on #315.
***********************************
1. INFINITE FORM.
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
theory.
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 vectors whose coordinates lie in
{1,n,n^2,...}.
The successor of V containedin [1,inf)^k is the set resulting from
raising all coordinates by 1.
PROPOSITION 1.2. Every upwards order invariant digraph G on [1,inf)^k
has an
independent set S such that the following holds. Every k element
independent subset of S'
has the same powers of (8k)! as some k element independent subset of
GS whose successor
has none.
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 delete "whose
successor has none",
then Proposition 1.2 is an immediate consequence of Theorem 1.1.
Here (8k)! is just a convenient expression.
2. FINITE FORM.
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 such that the following holds. Every k element
independent subset of S'
has the same powers of (8k)! as some k element independent subset of
GS whose successor
has none.
Proposition 2.2 is explicitly Pi01.
All of the results read the same.
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 319th 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
Harvey Friedman
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