[FOM] 346: Goedel's Second Revisited 3
Harvey Friedman
friedman at math.ohio-state.edu
Tue Jun 16 11:04:23 EDT 2009
NOTE: Posting 344: Thematic Pi01 Incompleteness 4 http://www.cs.nyu.edu/pipermail/fom/2009-June/013826.html
was misnumbered, and should be 355: Thematic Pi01 Incompleteness 4.
Recall that 1-Con asserts that "every provable Sigma01 sentence is
true".
In #344, http://www.cs.nyu.edu/pipermail/fom/2009-May/013778.html we
presented a proof of the weak form of Goedel's Second for 1-Con which
is far more straightforward than any proof known for Goedel's second.
The proper credit for this proof is unclear, but it has been
independently rediscovered by some people. See Avigad's http://www.cs.nyu.edu/pipermail/fom/2009-June/013785.html
There, it is stated that
> My notes give another proof of Goedel's second which doesn't use self
> reference explicitly, but, rather, shows how the self reference can be
> made implicit in a diagonalization argument. On his web page, Haim
> Gaifman has expository notes, "The easy way to Gödel's proof," which
> takes a different approach to the same end. (I wrote my notes after
> seeing an earlier version of his.)
Here we give a sufficient condition for Goedel's Second for 1-Con
whose hypotheses are incomparably simpler than any hypotheses (known
to be) sufficient for Goedel's Second. This suggests that there must
be a real limit to how simple any (fully rigorous) proof of Goedel's
Second can be, compared to the proof of Goedel's Second for 1-Con.
In fact, we can put it this way. The hypotheses for Goedel's Second
are subtly *intensional*, whereas the hypothesis for Goedel's Second
for 1-Con are *extensional*. This seems to be a huge difference, both
in terms of formulations, and consequently in terms of proofs.
We will work with only theories containing EFA(0,S,+,x) = exponential
function arithmetic for the language 0,S,+,x. In fact, we have
confined ourselves to extensions of EFA in our entire discussion of
Goedel's Second, and will continue this for a while; but later come
back and fine tune things to incorporate weaker theories.
Here are the hypotheses. We use SIG for the set of sentences of the form
(therexists x1,...,xk)(phi)
where x1,...,xk are variables of sort N and phi is a bounded formula
in 0,S,+,x in the usual sense (bounded quantifiers).
We use SIG(eq) for the set of sentences of the form
(therexists x1,...,xk)(psi)
where x1,...,xk are variables of sort N and psi is an equation between
terms in 0,S,+,x.
For n in N, write n* for the term S...S0, with n S's.
1. T is a many sorted theory, with a sort N, equipped with at least
0,S,+,x.
2. T is based on a finite alphabet of letters. A positive integer c.
3. T proves EFA(0,S,+,x). T is 1-consistent.
4. Formulas PROV(x), TRUE(x) with exactly the free variable x, where x
is of sort N.
5. Let A be in SIG(eq). There exists n <= 2^[c](lth(A)) such that
i. if P is any proof in T of A then there is a vector alpha of
witnesses for PROV(n) such that max(alpha) <= 2^[c](lth(P)).
ii. if alpha is any vector of witnesses for TRUE(n) then there is a
vector beta of witnesses for A such that max(beta) <= 2^[c](lth(alpha)).
8. 1-Con(T) is the sentence (forall x)(PROV(x) implies TRUE(x)).
THEOREM. T does not prove 1-Con(T).
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 346th 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.
Harvey Friedman
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