[FOM] 347:Goedel's Second Revisited 4

Harvey Friedman friedman at math.ohio-state.edu
Fri Jun 19 16:48:33 EDT 2009


QUESTION: Use a natural Goedel numbering of sentences of PA. Among the  
sentences of PA that are independent of PA, is the one with least  
Goedel number true?

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

We now return to the topic of Goedel's Second Revisited 1 http://www.cs.nyu.edu/pipermail/fom/2009-May/013753.html

We misstated the results of section 4 there. We need that T extends  
SEFA(0,S,+,dot). We will start with a restatement of section 4, in  
more general form, and then consider refinements.

We then break new ground by giving interpretation based forms of  
Goedel's Second. We already did this much earlier on the FOM assuming  
that the systems extend induction. But now we do this generally, and  
in a more striking way than in

307: Formalized Consistency Problem Solved  http://www.cs.nyu.edu/pipermail/fom/2007-January/011282.html

GOEDEL'S SECOND INCOMPLETENESS THEOREM REVISITED
by
Harvey M. Friedman
June 19, 2009

1. Introduction.
2. The Innovation.
3. Extensions of PA(0,S,+,dot) in L(0,S,+,dot).
4. Extensions of SEFA(0,S,+,dot).
5. Extensions of EFA(0,S,+,dot).
6. Extensions of EFA(0,S,+,dot) - interpretations.

4. EXTENSIONS OF SEFA(0,S,+,dot) in L(0,S,+,dot).

Let T be a many sorted theory with a sort for N (the N sort), which  
comes equipped with 0,S,+,dot. We have the condition in two equivalent  
forms.

CONDITION B1. phi is a Pi01 sentence in L(0,S,+,dot) such that EFA(0,S,
+,dot) + phi proves every inequation not(s = t) that is provable in T.

CONDITION B2. phi is a Pi01 sentence in L(0,S,+,dot) such that EFA(0,S, 
+,dot) + phi proves every Pi01 sentence that is provable in T.

THEOREM 4.1. Let T be a consistent r.e. many sorted theory with a sort  
for N, equipped with 0,S,+,dot. Every
standard consistency statement for T in L(0,S,+,dot), using any r.e.  
presentation of  T, obeys condition B1, B2.

Note that Theorem 4.1 is the same as Theorem 3.1 together with the
observation that standard consistency statements are in PI form.

THEOREM 4.2 (corrected). Let T be a consistent many sorted theory with  
a sort for N, equipped with 0,S,+,dot, where T proves SEFA(0,S,+,dot).  
No theorem of T obeys condition B1 or B2.

Recall that SEFA(0,S,+,dot) is bounded arithmetic with the coded forms  
of

(forall n)(2^n exists).
(forall n,m)(2^[n](m) exists).

Note that Theorem 4.2 applies far more generally than Theorem 3.2. The
cause for this is that condition B1 (B2) requires the sentence to be  
in PI
form.

5. EXTENSIONS OF EFA(0,S,+,dot).

With some care, we can use EFA and PFA instead of SEFA and EFA.

Let T be a many sorted theory with a sort for N (the N sort), which  
comes equipped with 0,S,+,dot. We have the condition in two equivalent  
forms.

CONDITION C1. phi is a Pi01 sentence in L(0,S,+,dot) such that PFA(0,S,
+,dot) + phi proves every inequation not(s = t) that is provable in T.

CONDITION C2. phi is a PI sentence in L(0,S,+,dot) such that EFA(0,S, 
+,dot) + phi proves every PI sentence that is provable in T.

THEOREM 5.1. Let T be a consistent r.e. many sorted theory with a sort  
for N, equipped with 0,S,+,dot. Every standard consistency statement  
for T in L(0,S,+,dot), using any r.e. presentation of
T, obeys condition C1, C2.

THEOREM 5.2. Let T be a consistent many sorted theory with a sort for  
N, equipped with 0,S,+,dot, where T proves EFA(0,S,+,dot). No theorem  
of T obeys condition C1 or C2.

Recall that EFA(0,S,+,dot) is bounded arithmetic with the coded form of

(forall n)(2^n exists).

PFA(0,S,+,dot) is bounded arithmetic, or ISigma0. Here PFA stands for  
"polynomial function arithmetic".

6. EXTENSIONS OF EFA(0,S,+,dot) - INTERPRETATIONS.

CONDITION D1. phi is a Pi01 sentence in L(0,S,+,dot) such that T is  
interpretable in PFA(0,S,+,dot) + phi.

CONDITION D2. phi is a Pi01 sentence in L(0,S,+,dot) such that T is  
interpretable in phi.

Obviously, condition D2 is simpler than condition D1. However, the  
consistency statement doesn't mean anything without the context of  
PFA(0,S,+,dot).

THEOREM 6.1. Let T be a consistent r.e. many sorted theory with a sort  
for N, equipped with 0,S,+,dot. Every standard consistency statement  
for T in L(0,S,+,dot), using any r.e. presentation of
T, obeys condition D1

THEOREM 6.2. Let T be a consistent many sorted theory with a sort for  
N, equipped with 0,S,+,dot, where T proves EFA(0,S,+,dot). No theorem  
of T obeys condition D1 or D2.


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

I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 347th 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
319: Pi01 Incompleteness/Digraphs/#3  2/12/08  9:21PM
320: Pi01 Incompleteness/#4  2/13/08  5:32PM
321: Pi01 Incompleteness/forward imaging  2/19/08  5:09PM
322: Pi01 Incompleteness/forward imaging 2  3/10/08  11:09PM
323: Pi01 Incompleteness/point deletion  3/17/08  2:18PM
324: Existential Comprehension  4/10/08  10:16PM
325: Single Quantifier Comprehension  4/14/08  11:07AM
326: Progress in Pi01 Incompleteness 1  10/22/08  11:58PM
327: Finite Independence/update  1/16/09  7:39PM
328: Polynomial Independence 1   1/16/09  7:39PM
329: Finite Decidability/Templating  1/16/09  7:01PM
330: Templating Pi01/Polynomial  1/17/09  7:25PM
331: Corrected Pi01/Templating  1/20/09  8:50PM
332: Preferred Model  1/22/09  7:28PM
333: Single Quantifier Comprehension/more  1/26/09  4:32PM
334: Progress in Pi01 Incompleteness 2   4/3/09  11:26PM
335: Undecidability/Euclidean geometry  4/27/09  1:12PM
336: Undecidability/Euclidean geometry/2  4/29/09  1:43PM
337: Undecidability/Euclidean geometry/3  5/3/09   6:54PM
338: Undecidability/Euclidean geometry/4  5/5/09   6:38PM
339: Undecidability/Euclidean geometry/5  5/7/09   2:25PM
340: Thematic Pi01 Incompleteness 1  5/13/09  5:56PM
341: Thematic Pi01 Incompleteness 2  5/21/09  7:25PM
342: Thematic Pi01 Incompleteness 3  5/23/09  7:48PM
343: Goedel's Second Revisited 1  5/27/09  6:07AM
344: Goedel's Second Revisited 2  6/1/09  9:21PM
345: Thematic Pi01 Incompleteness 4 6/15/09  1:15PM
appears misnumbered as 344.
346: Goedel's Second Revisited 3  6/16/09  11:04PM

Harvey Friedman



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