[FOM] 333: Single Quantifier Comprehension/more
Harvey Friedman
friedman at math.ohio-state.edu
Mon Jan 26 15:54:53 EST 2009
This is an update to #325: Single Quantifier Comprehension, http://www.cs.nyu.edu/pipermail/fom/2008-April/012802.html
We refer the reader to #325. This is intended as an update, although
it is reasonably self contained.
Elementary foundation is the weakened form of Foundation, which
asserts that there are no epsilon cycles.
SINGLE QUANTIFIER COMPREHENSION
1. Single quantifier schemes, with one free unary function symbol,
true in the hereditarily finite sets. This axiomatizes ZF\infinity
with elementary Foundation.
2. Extended single quantifier comprehension schemes, including the
third quantifier E! (there exists unique), true in the hereditarily
finite sets, with one free binary relation symbol. Also axiomatizes ZF
\infinity with elementary foundation. The advantage of this is that we
do not need function schemes.
DOUBLE QUANTIFIER COMPREHENSION
3. EA comprehension schemes, with one free binary relation symbol,
true in the hereditarily finite sets. This axiomatizes ZF\infinity
with elementary foundation.
4. I think AE is the same, but I haven't written a sketch yet.
MULTIPLE QUANTIFIER COMPREHENSION
5. E...EA comprehension schemes, with any free relation and function
symbols, true in the hereditarily finite sets. This axiomatizes ZF
\infinity with elementary foundation.
6. AA comprehension formulas don't have this property.
So ZF (with elementary foundation) is axiomatized exactly with any of
1,2,3,5 with the addition of INFINITY.
All of these classes 1-4 are decidable, and quite robust. E.g., we can
replace "true in the hereditarily finite sets" by "provably true in a
weak theory of hereditarily finite sets", and also by "consistent with
a weak fragment of ZF without Infinity".
So another message from this work is that
1) every single quantifier comprehension scheme (with function
schemes) is either provable or refutable in ZF.
In fact,
2) every single quantifier comprehension scheme (with function
schemes) is either provable in ZF without Infinity or refutable in a
weak fragment of ZF without Infinity.
Also, the above is true even if we use elementary foundation instead
of foundation. And we can go somewhat further than a single quantifier.
There is an AA comprehension formula true in the hereditarily finite
sets, and demonstrably so in weak systems, but which is refutable in
weak systems with Infinity added. To give a counterexample to 1), you
have to, and can, go somewhat higher complexity.
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 332nd 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
Harvey Friedman
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