[FOM] 303: PA Completeness (restatement)

[Harvey Friedman]

This is a follow up to
http://www.cs.nyu.edu/pipermail/fom/2006-October/011044.html

I made the 

CONJECTURE. Every true sentence in 0,S,+,dot with at most two variables and
term complexity at most 2, is provable in PA (and even in EFA). Every
instance of every universally true scheme in 0,S,+,dot and the unary
schematic letter R, with at most two variables and term complexity at most
2, is provable in PA.

My intention is that the conjecture involve only finitely many formulas and
schemes, up to logical equivalence. However, in light of much work on
formulas with only two (or more) variables, this conjecture unintentionally
involves infinitely many formulas and schemes, up to logical equivalence.

So let me revise my view here.

1. I hereby restate the conjecture using sentences and sentence schemes with
two quantifiers. 
2. I am still very interested in the original conjecture as it stands,
although I would now prefer to call it a question.

RESTATED CONJECTURE. Every true sentence in 0,S,+,dot with at most two
quantifiers and term complexity at most 2, is provable in PA (and even in
EFA). Every instance of every true sentence scheme in 0,S,+,dot and one
unary schematic letter, with at most two quantifiers and term complexity at
most 2, is provable in PA.

I have come across a highly relevant result of Shih-Ping Tung:
 
Tung, Shih Ping 
Provability and decidability of arithmetical universal-existential
sentences. 
Bull. London Math. Soc. 18 (1986), no. 3, 241--247.
03B25 (03F30 03H15)

Excerpt from review:

"In this paper, it is shown that the set of AE arithmetical sentences true
in the standard model is decidable. ... The result is obtained by proving
that a weak subtheory of PA can prove all true AE sentences and refute all
false AE sentences." Reviewed by Franco Montagna

Thus for the sentence part of my Conjecture (restated above), we need only
consider sentences of the form (forall x,y)(A) in the language 0,S,+,dot,
where A is quantifier free.

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

I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 303rd 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

Harvey Friedman