[FOM] 309: Thematic PA Incompleteness
Harvey Friedman
friedman at math.ohio-state.edu
Mon Oct 22 10:56:27 EDT 2007
We present some PA incompleteness based on the following Theme.
HOW MANY OBJECTS OF A CERTAIN KIND ARE NEEDED IN ORDER TO GUARANTEE THAT AT
LEAST TWO ARE EQUIVALENT ON SOME LOCALE? WHAT CAN WE SAY ABOUT THIS NUMBER?
We conjecture that lots of interesting known mathematics can be formulated
in terms of this Theme.
For instance, here is how the usual Ramsey theorem fits in. We define [n] =
{1,2,...,n}.
HOW MANY FUNCTIONS FROM [n]^k INTO [m] ARE NEEDED IN ORDER TO GUARANTEE THAT
AT LEAST TWO ARE IDENTICAL ON SOME E^k, |k| = r? WHAT CAN WE SAY ABOUT THIS
NUMBER?
Let F(k,m,,r,n) be the required number. Obviously F(k,m,r,n) is weakly
decreasing in n. For any k,m,r, define
G(k,m,r) = inf_n G(k,m,r,n).
Then G can be effectively computed using the ordinary Ramsey theorem. The
algorithm can be proved to work easily within PA. However, no algorithm can
be shown to work within EFA = exponential function arithmetic.
***PA INCOMPLETENESS***
Let f,g:[n]^k into N and A containedin [n]^k. We say that f,g are order
equivalent over A if and only if for all x,y in A,
f(x) < f(y) if and only if g(x) < g(y).
Here we ask the following.
HOW MANY FUNCTIONS FROM [n]^k INTO N ARE NEEDED IN ORDER TO GUARANTEE THAT
AT LEAST TWO ARE ORDER EQUIVALENT ON SOME E^k, |k| = r? WHAT CAN WE SAY
ABOUT THIS NUMBER?
Let alpha(k,r,n) be the required number. Obviously alpha(k,r,n) is weakly
decreasing in n. Accordingly, we define, for any k,r,
beta(k,r) = inf_n alpha(k,r,n).
THEOREM 2.1. beta is effectively computable. However, there is no algorithm
which, provably in PA, computes beta, or even just beta with r = 4k.
THEOREM 2.2. For all r >= 4k, beta(k,r) = beta(k,4k).
THEOREM 2.3. Theorem 2.2 is provable in EFA + 1-CON(PA) but not in PA, or
even in PA + CON(PA). The realizations of the inf in the definition of
beta(k,r), even for r = 4k, is epsilon_0 recursive, but not < epsilon_0
recursive.
**********************************
I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 308th 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
Harvey Friedman
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