# [FOM] 322: Pi01 Incompleteness/forward imaging 2

Harvey Friedman friedman at math.ohio-state.edu
Mon Mar 10 23:09:05 EDT 2008

```Here we rework posting #321 based on strictly dominating relations and
independent sets. WIth this infrastructure, the independent statements
are simpler.

1. INFINITE STATEMENTS.

We define N to be the set of all nonnegative integers.

Let R containedin N^4k and A containedin N^k. We define A' = N^k\A.

We say that R is upwards if and only if for all x in N^3k and y in
N^k, (x,y) in R implies max(x) < max(y).

We define

RA = {y in N^k: (therexists x in A^3)((x,y) in R)}.

We say that A is R independent if and only if A containedin N^k, and
A,RA are disjoint.

THEOREM 1.1. Every upwards R containedin N^4k has an independent A
such that RA = A'. Furthermore, A is unique.

We now use a restricted notion of forward image. Let R containedin
N^4k and A containedin N^k. Define

R*A = {y in N^k: (therexists x in A^3)((x,y) in R and min(y) not=
min(x)+1)}.

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, x
in R iff y in R.

The k dimensional powers of n are the elements of {1,n,n^2,...}^k.

PROPOSITION 1.2. Every upwards order invariant R containedin N^4k has
an independent A such that RRA and R*(A') have the same k dimensional
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 1.2 is provably equivalent, over ACA, to CON(MAH).
Proposition 1.2 is not provable in any consistent subsystem of MAH. In
particular, Proposition 1.2 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 that is sufficiently large.

We can use Z+ instead of N.

2. FINITE STATEMENTS.

Here we replace N by [0,n], where n is an arbitrary nonnegative
integer. Here are the details.

Let R containedin [0,n]^4k and A containedin [0,n]^k. We define A' =
[0,n]^k\A.

We say that A is R independent if and only if A containedin [0,n]^k,
and A,RA are disjoint.

THEOREM 2.1. Every upwards R containedin [0,n]^4k has an independent A
such that RA = A'. Furthermore, A is unique.

We now use a restricted notion of forward image. Let R containedin
[0,n]^4k and A containedin [0,n]^k. Define

R*A = {y in [0,n]^k: (therexists x in A^3)((x,y) in R and min(y) not=
min(x)+1)}.

We say that R containedin [0,n]^k is order invariant if and only if
for all order equivalent x,y in [0,n]^k, x in R iff y in R.

PROPOSITION 2.2. Every upwards order invariant R containedin [0,n]^4k
has an independent A such that RRA and R*(A') have the same k
dimensional 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 2.3. Theorem 2.1 is provable in EFA. Proposition 2.2 is
provable in MAH+ but not in MAH, assuming that MAH is consistent.
Proposition 2.2 is provably equivalent, over ACA, to CON(MAH).
Proposition 2.2 is not provable in any consistent subsystem of MAH. In
particular, Proposition 2.2 is not provable in ZFC, assuming ZFC is
consistent. If we delete * then Proposition 2.2 becomes a weakened
form of Theorem 2.1.

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.

We can use [1,n] instead of [0,n].

3. GENERALIZED FORWARD IMAGING.

Note that we have relied on a specific notion of forward imaging in
Propositions 1.2 and 2.2. It is natural to consider what happens when
we allow R* to be any of a collection of various notions of forward
imaging.

TEMPLATE. Every upwards order invariant R containedin N^4k has an
independent A such that RRA and R#(A') have the same k dimensional
powers of (8k)!.

TEMPLATE. Every upwards order invariant R containedin [0,n]^4k has an
independent A such that RRA and R#(A') have the same k dimensional
powers of (8k)!.

The forward imaging notion R* that we have used is particularly
simple. This suggests that we investigate

R(P;A) = {y in N^k: (therexists x in A^3)((x,y) in R and
P(min(x),min(y),max(x),max(y))}

where P is a Presburger relation. The Templates need to be adjusted to
take into account the complexity of P as a Presburger relation - e.g.,
the number of symbols in its definition in (N,<,0,1,+).

TEMPLATE 3.1. Let P containedin N^4 be a Presburger relation of
complexity at most c. Every upwards order invariant R containedin N^4k
has an independent A such that RRA and R(P;A') have the same k
dimensional powers of (8kc)!.

TEMPLATE 3.2. Let P containedin N^4 be a Presburger relation of
complexity at most c. Every upwards order invariant R containedin
[0,n]^4k has an independent A such that RRA and R(P;A') have the same
k dimensional powers of (8kc)!.

We conjecture that there is a decision procedure for determining the
truth value of each instance of these Templates. Also, each instance
of these Templates is either provable in RCA_0, refutable in RCA_0, or
provably equivalent to CON(MAH) over ACA.

**********************************
manuscripts. This is the 322nd 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

Harvey Friedman
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