[FOM] 344:Goedel's Second Revisited 2
Harvey Friedman
friedman at math.ohio-state.edu
Sun May 31 13:41:29 EDT 2009
Before going back to novel statements of Goedel's Second, I want to
discuss proofs of Goedel's Second, with its ordinary garden variety
statement (which of course is not formal, since one does not display
the sentence Con).
The question is: to what extent can we avoid making use of self
referential sentences?
There is a very clear way to prove a weak form of Goedel's Second
without use of self referential sentences.
However, I do not quite see how to give a proof of the usual Goedel's
Second without use of self referential sentences. I think that the
claim has been made that this can be done. What is the best effort
along these lines?
PROOF OF A WEAK FORM OF GOEDEL'S SECOND.
There is a weak form of Goedel's Second - namely that any theory T
subject to weak conditions does not prove its own 1-consistency.
We use bounded formulas, which will always be in the language L(0,S,
+,dot), and where all quantifiers are bounded to variables. I.e., we
use (forall x < y), (therexists x < y), where x,y are distinct
variables.
Sigma-0-1 formulas are obtained from bounded formulas by putting one
existential quantifier in front.
Here 1-Con(PA) asserts that
"every Sigma-0-1 sentence provable in PA is true".
This weak form of Goedel's Second has a particularly friendly proof.
This is because we can associate sensible 'quantitative information'
to reasonable formal systems, and show that T + 1-Con(T) has strictly
greater 'quantitative information'. Hence T and T + 1-Con(T) cannot be
logically equivalent. Therefore T does not prove 1-Con(T).
The quantitative information is the provably recursive functions of T.
These are the recursive functions f:N into N such that for some Turing
machine computing f, T proves that that Turing machines halts at all
inputs.
THEOREM 1.1. Let T be an r.e. theory with at least the symbols 0,S,
+,dot in the nonnegative integer sort, and perhaps with other symbols
in the nonnegative integer sort and in other sorts. Suppose T is 1-
consistent and extends EFA(0,S,+,dot). Then T + 1-Con(T) has strictly
more provably recursive functions than T. In fact, EFA + 1-Con(T) has
strictly more provably recursive functions than T.
Proof: EFA + 1-Con(T) has a provably recursive enumeration of the
provably recursive functions of T. Then diagonalize. QED
In the statement of Theorem 1.1, we have required that T extend
EFA(0,S,+,dot), because we want to be liberal about what is allowed as
a formalization of 1-Con(T). We made the same choice in #343, since
EFA(0,S,+,dot) easily allows for finite sequence coding. We can
alternatively use weak fragments of EFA(0,S,+,dot) if we append some
detailed description of what the formalization of 1-Con(T) is like. In
this direction, we can even use Q.
COROLLARY 1.2. Let T be an r.e. theory with at least the symbols 0,S,
+,dot in the nonnegative integer sort, and perhaps with other symbols
in the nonnegative integer sort and in other sorts. Suppose T is 1-
consistent and extends EFA(0,S,+,dot). Then T does not prove 1-Con(T).
Who is this proof due to? I have known it for some time. Also Saul
Kripke has known this for quite some time. But who else?
To make this proof fully rigorous, we probably want to use Hilbert-
Bernays style derivaability conditions. But here we don't need the
part that justifies the construction of self referential sentences.
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 343rd 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
Harvey Friedman
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