[FOM] 312: Pi01 Incompleteness
Harvey Friedman
friedman at math.ohio-state.edu
Tue Nov 13 15:11:55 EST 2007
We return to Pi01 Incompleteness from ZFC. We take a fresh look at this with
some regularity.
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We use interval notation. All intervals are discrete. We use [1,inf) for the
set of all positive integers.
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.
We say that R containedin [1,inf)^t is order invariant if and only if for
all order equivalent x,y in [1,inf)^t,
x in R iff y in R.
For A containedin [1,inf)^k, we write A' = [1,inf)^k\A. We use A+1 for the
shift of A to the right by 1.
Let R containedin [1,inf)^p x [1,inf)^q. We say that R is strictly
dominating if and only if for all x,y,
R(x,y) implies max(x) < max(y).
We say that R is order invariant if and only if R is order invariant as a
subset of [1,in)^p+q.
Let A containedin [1,inf)^k, where k divides p. We write
RA = {y in [1,inf)^q: there exists x in A^p/k with R(x,y)}.
We say that A is free for R if and only if
RA intersect A = emptyset.
MAH = ZFC + {there exists a strongly n-Mahlo cardinal}_n. MAH+ = ZFC + "for
all n there exists a strongly n-Mahlo cardinal".
1. INFINITE FORMS.
We start with a special case of a famous theorem from graph theory (adapted
to hypergraphs).
THEOREM 1.1. For all k >= 1 and strictly dominating R containedin [1,inf)^3k
x [1,inf)^k, there exists A containedin [1,inf)^k such that RA = A'.
Furthermore, A is unique.
We can rewrite this using free sets.
THEOREM 1.2. For all k >= 1, every strictly dominating R containedin
[1,inf)^3k x [1,inf)^k has a free A containedin [1,inf)^k such that A'
containedin RA. Furthermore, A is unique.
To prepare us for the Propositions below, we can attempt to strengthen this
as follows.
THEOREM 1.3. The following statement is false. For all k >= 1, every
strictly dominating R containedin [1,inf)^3k x [1,inf)^k has a free A
containedin [1,inf)^k such that A' containedin (RA)\(A+1).
We can weaken Theorem 1.2 as follows.
THEOREM 1.4. For all k >= 1, every strictly dominating R containedin
[1,inf)^3k x [1,inf)^k has a free A containedin [1,inf)^k such that R(A')
containedin (RRA).
We can attempt to combine these variants as follows.
THEOREM 1.5. The following statement is false. For all k >= 1, every
strictly dominating R containedin [1,inf)^3k x [1,inf)^k has a free A
containedin [1,inf)^k such that R(A') containedin (RRA)\(A+1).
Theorem 1.5 can be "fixed" as follows.
PROPOSITION 1.6. For all k >= 1, every strictly dominating order invariant R
containedin [1,inf)^3k x [1,inf)^k has a free A containedin [1,inf)^k such
that R(A') intersect {k^8k,k^9k,...}^k containedin (RRA)\(A+1).
We can introduce a new parameter r as follows.
PROPOSITION 1.7. For all k,r >= 1, every strictly dominating order invariant
R containedin [1,inf)^rk x [1,inf)^k has a free A containedin [1,inf)^k such
that R(A') intersect {k^8rk,k^9rk,...}^k containedin (RRA)\(A+1).
THEOREM 1.8. Theorems 1.1 - 1.5 can be proved in RCA_0.
THEOREM 1.9. Theorems 1.6,1.7 is provable in MAH+. Theorems 1.6,1.7 are each
provably equivalent to CON(MAH) over ACA. Theorems 1.6,1.7 are each
unprovable in every consistent fragment of MAH. In particular, Theorems
1.5,1.6 are not provoable in ZFC provided ZFC is consistent.
2. FINITE FORMS.
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 1.6'. For all k,n >= 1, every strictly dominating order invariant
R containedin [1,n]^3k x [1,n]^k has a free A containedin [1,n]^k such that
R(A') intersect {k^8k,k^9k,...,k^nk}^k containedin (RRA)\(A+1).
PROPOSTION 1.7'. For all k,r,n >= 1, every strictly dominating order
invariant R containedin [1,n]^rk x [1,n]^k has a free A containedin [1,n]^k
such that R(A') intersect {k^8rk,k^9rk,...,k^nrk}^k containedin (RRA)\(A+1).
These Propositions are explicitly Pi01.
All of the results read the same.
**********************************
I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 312th 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
Harvey Friedman
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