[FOM] 321: Pi01 Incompleteness/forward imaging
Harvey Friedman
friedman at math.ohio-state.edu
Tue Feb 19 17:12:16 EST 2008
1. INFINITE FORM.
We define N to be the set of all nonnegative integers.
We will use three different notions of: forward image of R on A. Here
it is required that for some k >= 1, R containedin N^4k, and A
containedin N^k. Specifically,
RA = {w in N^k: (therexists x,y,z in A)(R(x,y,z,w))}.
R<A = {w in N^k: (therexists x,y,z in A)(R(x,y,z,w) and
max(x),max(y),max(z) < max(w))}.
R*A = {w in N^k: (therexists x,y,z in A)(R(x,y,z,w) and
max(x),max(y),max(z) not= max(w)-1)}.
For A containedin N^k, define A' = N^k\A.
NOTE: Typographically, in R<A, the < is a subscript.
THEOREM 1.1. For all k >= 1 and R containedin N^4k, some R<A is A'.
Furthermore, A is unique.
We say that x,y in N^k are order equivalent if and only if for all 1
<= i,j <= k, x_i < x_j iff y_i < y_j. We say that R containedin N^k is
order invariant if and only if for all order equivalent x,y in N^k,
R(x) iff R(y).
For n >= 1, the powers of n are the tuples whose coordinates are drawn
from {1,n,n^2,...}.
PROPOSITION 1.2. For all k >= 1 and order invariant R containedin
N^4k, some R*R<A is a subset of R(A') with the same powers of (8k)!.
MAH = ZFC + {there exists a strongly n-Mahlo cardinal}_n. MAH+ = ZFC +
"for all n there exists a strongly n-Mahlo cardinal".
THEOREM 1.3. Theorem 1.1 is provable in RCA_0. Proposition 1.2 is
provable in MAH+ but not in MAH, assuming that MAH is consistent.
Proposition 1 is provably equivalent, over ACA, to CON(MAH).
Proposition 1 is not provable in any consistent subsystem of MAH. In
particular, Proposition 1 is not provable in ZFC, assuming ZFC is
consistent. If we delete * then Proposition 1.2 becomes a weakened
form of Theorem 1.1.
The 4 in "4k" can be extended to any higher number in the obvious way
without changing the results. We have not investigated the
independence status when 4 is replaced by 2 or 3. Probably 3 will
still give the same results, but 2 is not enough for independence.
Here (8k)! is just a convenient expression.
We can use Z+ instead of N.
2. FINITE FORM.
The finite form is obtained trivially by replacing N with [0,n]. All
of the definitions are restated in the obvious way with N replaced
throughout by [0,n]. Specifically,
PROPOSTION 2.2. For all k,n >= 1 and order invariant R containedin
[0,n]^4k, some R*R<A is a subset of R(A') with the same powers of (8k)!.
Note that Proposition 2.2 is explicitly Pi01.
All of the results read the same.
We can use [1,n] instead of [0,n].
3. GENERALIZED FORWARD IMAGING.
Note that all three of the forward imaging notions that we have used
can be unified in the following way.
Let # containedin N^2. We define #-imaging as follows.
For all k >= 1 and R containedin N^4k, we define R# the function
mapping subsets of N^k into subsets of N^k, given by
R#A = {w: (therexists x,y,z in A)(R(x,y,z,w) and #(max(x),max(w)),
#(max(y),max(w)), #(max(z),max(w)))}.
Note that the three forward image construction that we have used, RA,
R<A, R*A, can be viewed as R#A, R##A, R###A, where
#(n,m) iff n = n.
##(n,m) iff n < m.
###(n,m) iff n not= m-1.
It is reasonable to require that # containedin N^2 be a Presburger
relation. I.e., definable in (N,+). More restrictive requirements also
make sense.
This leads to the following Template.
TEMPLATE. Let #,##,### be Presburger subsets of N^2. For all k >= 1
and order invariant R containedin N^4k, some R#R##A is a subset of
R###(A') with the same powers of (8k)!.
TEMPLATE. Let #,##,### be Presburger subsets of N^2. For all k,n >= 1
and order invariant R containedin [0,n]^4k, some R#R##A is a subset of
R###(A') with the same powers of (8k)!.
We intend to determine the status of all instances of these Templates.
We conjecture that every instance is either provable in RCA_0,
refutable in RCA_0, or provably equivalent to CON(MAH) over RCA_0. The
latter occurs according to THeorems 1.3 and 2.3.
Obviously, further Templating can be done along these lines. E.g., we
can consider using
R#1 R#2 ... R#p A is a subset of R#p+1 R#p+2 ... R#p+q A' with the
same powers of (8k)!
where p,q >= 0 and the #'s are Presburger subsets of N^2. We make the
extended conjecture.
**********************************
I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 321st 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
Harvey Friedman
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