# [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

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).

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

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.

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

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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