890: Update on Tangible Incompleteness

Sat Sep 18 09:50:09 EDT 2021

1. TWO LEAD STATEMENTS FOR MATHEMATICIANS - no change
2. FIRST NONDETERMINISTIC ALGORITHM - Pi01 incompleteness
3. SECOND NONDETERMINISTIC ALGORITHM - Pi02 incompleteness

1. TWO LEAD STATEMENTS FOR MATHEMATICIANS - no change

We have only changed the names.

SECTION INVARIANT MAXIMAL SQUARES/1. Every order invariant subset of
Q[0,n]^2k has a maximal square whose
<1 sections are order invariant over Z[1,n].

SECTION INVARIANT MAXIMAL SQUARES/2.  Every order invariant subset of
Q[0,n]^2k has a maximal square
whose <i sections, i < n, are order invariant over Z[i,n].

2. FIRST NONDETERMINISTIC ALGORITHM - Pi01 incompleteness

For the remainder of this posting, all intervals lie in N, the set of
nonnegative integers.

We use four parameters t,k,n,r, where t >> k,n,r. Specifically,
we will require that  t >= (8knr)!!. These double
factorials (factorial of factorial) are safe but crude overkill
quantities that I expect to vastly reduce when things settle down.

We work entirely in the space [0,nt]^k. We also use an order invariant
irreflexive symmetric relation R on [0,nt]^k.

We now present the nondeterministic algorithm ALG(t,k,n,r,R)/1.

DEFINITION 2.1. Let A containedin [0,nt]^k. fld(A) is the set of all
coordinates of elements of A.

DEFINITION 2.2. Let x be in [0,nt]^k. ush(x;t) results from adding t
to all coordinates of x that are >= t if this stays in [0,nt]^k;  x
otherwise. ush is read "upper shift".

This nondeterministic algorithm is in r stages and produces sets A_1
containedin ... containedin A_r containedin [0,nt]^k, and sets
B_1,...,B_r containedin [0,nt]^k.

We initialize by setting A_1 = emptyset. B_1 = {0,t,2t,...,nt}^k.

Suppose A_i and B_i have been constructed.

case 1 .B_i = emptyset. Set A_i+1 = A_i and "refresh" B. I.e., B_i+1 =
fld(A_i)^k.

case 2. B_i not= emptyset. We "process" some element of B_i. I.e.,
choose any u in B_i, and choose any x in N^k not R related to u,
max(x) <= max(u). Set A_i+1 = A_i union {x,ush(x;t)}, B_i+1 =
B_i\{u}.

Note that A starts off empty and B starts off with substantially many
elements. Then B is reduced nondeterministically, one by one, until it
becomes empty, while A gains at most two elements at a time (there may
be lots of duplications). When B becomes empty B is refreshed with
lots of elements built out of the A at that point. The A's never lose
elements. Obviously this process can be carried out, if only by the
mindless rule of always processing u by throwing u and ush(u;t)
into A - recall that R is irreflexive so that u is not R related to u,
so this is a legal processing of u.

This execution of ALG(t,k,n,r,R)/1 is considered SUCCESSFUL if and
only if any two distinct elements of A_r are R related.

THEOREM 2.1. (EFA) If we delete the term ush(x;t) in case 2 above,
then ALG(t,k,n,r,R)/1 can be successfully executed, even staying
entirely within {0,t,2t,...,nt}^k. Furthermore, we do not need that R
is order invariant.

PROPOSITION 2.2. ALG(t,k,n,r,R)/1 can be successfully executed.

Note that Proposition 2.2 is explicitly Pi01.

THEOREM 2.3. Proposition 2.2 is provably equivalent to Con(SRP) over EFA.

3. SECOND NONDETERMINISTIC ALGORITHM - Pi01 incompleteness

We use three parameters t,k,n and an order invariant symmetric reflexive
relation R on [0,nt]^k.

ALG(t,k,n,R)/2.is a simple modification of ALG(t,k,n,r,R)/1.
Firstly, the length of this algorithm is not determined in advance (it
was previously set at r).

ALG(t,k,n,R)/2 uses a more selective refresh. ALG(t,k,n,R)/2 terminates
when the refresh hits emptyset.

As before, the first refresh is by default considered to be B_1 =
{0,t,2t,...,nt}^k. Each succeeding refresh B_i+1 is the usual
fld(A_i)^k but with only the k-tuples whose max is smaller than the
largest integer appearing in the previous refresh B_i. Thus the
largest
number appearing in these successive refreshes are now strictly decreasing.

As before, the execution of ALG(t,k,n,R)/2 is considered SUCCESSFUL if and
only if we achieve termination and any two distinct elements of every
A is R related.

PROPOSITION 3.1. For all k,n there exists r such that the following
holds. For all t >= r and order invariant irreflexive symmetric
binary relations R on [0,nt]^k, ALG(t,k,n,R)/2 can be successfully
executed  in at most r steps.

Using the decision procedure for Presburger, the second sentence above
is decidable. Hence Proposition 3.1 is explicitly Pi02 using the
decision procedure for Presburger.

THEOREM 3.2. Proposition 3.1 is provably equivalent to 1-Con(SRP) over
EFA. The least witness functions for Proposition 3.1 with fixed k =
1,2,3,... are cofinal in the provably recursive functions of SRP.

##########################################

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 890th 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 #1-799 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

800: Beyond Perfectly Natural/6  4/3/18  8:37PM
801: Big Foundational Issues/1  4/4/18  12:15AM
802: Systematic f.o.m./1  4/4/18  1:06AM
803: Perfectly Natural/7  4/11/18  1:02AM
804: Beyond Perfectly Natural/8  4/12/18  11:23PM
805: Beyond Perfectly Natural/9  4/20/18  10:47PM
806: Beyond Perfectly Natural/10  4/22/18  9:06PM
807: Beyond Perfectly Natural/11  4/29/18  9:19PM
808: Big Foundational Issues/2  5/1/18  12:24AM
809: Goedel's Second Reworked/1  5/20/18  3:47PM
810: Goedel's Second Reworked/2  5/23/18  10:59AM
811: Big Foundational Issues/3  5/23/18  10:06PM
812: Goedel's Second Reworked/3  5/24/18  9:57AM
813: Beyond Perfectly Natural/12  05/29/18  6:22AM
814: Beyond Perfectly Natural/13  6/3/18  2:05PM
815: Beyond Perfectly Natural/14  6/5/18  9:41PM
816: Beyond Perfectly Natural/15  6/8/18  1:20AM
817: Beyond Perfectly Natural/16  Jun 13 01:08:40
818: Beyond Perfectly Natural/17  6/13/18  4:16PM
819: Sugared ZFC Formalization/1  6/13/18  6:42PM
820: Sugared ZFC Formalization/2  6/14/18  6:45PM
821: Beyond Perfectly Natural/18  6/17/18  1:11AM
822: Tangible Incompleteness/1  7/14/18  10:56PM
823: Tangible Incompleteness/2  7/17/18  10:54PM
824: Tangible Incompleteness/3  7/18/18  11:13PM
825: Tangible Incompleteness/4  7/20/18  12:37AM
826: Tangible Incompleteness/5  7/26/18  11:37PM
827: Tangible Incompleteness Restarted/1  9/23/19  11:19PM
828: Tangible Incompleteness Restarted/2  9/23/19  11:19PM
829: Tangible Incompleteness Restarted/3  9/23/19  11:20PM
830: Tangible Incompleteness Restarted/4  9/26/19  1:17 PM
831: Tangible Incompleteness Restarted/5  9/29/19  2:54AM
832: Tangible Incompleteness Restarted/6  10/2/19  1:15PM
833: Tangible Incompleteness Restarted/7  10/5/19  2:34PM
834: Tangible Incompleteness Restarted/8  10/10/19  5:02PM
835: Tangible Incompleteness Restarted/9  10/13/19  4:50AM
836: Tangible Incompleteness Restarted/10  10/14/19  12:34PM
837: Tangible Incompleteness Restarted/11 10/18/20  02:58AM
838: New Tangible Incompleteness/1 1/11/20 1:04PM
839: New Tangible Incompleteness/2 1/13/20 1:10 PM
840: New Tangible Incompleteness/3 1/14/20 4:50PM
841: New Tangible Incompleteness/4 1/15/20 1:58PM
842: Gromov's "most powerful language" and set theory  2/8/20  2:53AM
843: Brand New Tangible Incompleteness/1 3/22/20 10:50PM
844: Brand New Tangible Incompleteness/2 3/24/20  12:37AM
845: Brand New Tangible Incompleteness/3 3/28/20 7:25AM
846: Brand New Tangible Incompleteness/4 4/1/20 12:32 AM
847: Brand New Tangible Incompleteness/5 4/9/20 1 34AM
848. Set Equation Theory/1 4/15 11:45PM
849. Set Equation Theory/2 4/16/20 4:50PM
850: Set Equation Theory/3 4/26/20 12:06AM
851: Product Inequality Theory/1 4/29/20 12:08AM
852: Order Theoretic Maximality/1 4/30/20 7:17PM
853: Embedded Maximality (revisited)/1 5/3/20 10:19PM
854: Lower R Invariant Maximal Sets/1:  5/14/20 11:32PM
855: Lower Equivalent and Stable Maximal Sets/1  5/17/20 4:25PM
856: Finite Increasing reducers/1 6/18/20 4 17PM :
857: Finite Increasing reducers/2 6/16/20 6:30PM
858: Mathematical Representations of Ordinals/1 6/18/20 3:30AM
859. Incompleteness by Effectivization/1  6/19/20 1132PM :
860: Unary Regressive Growth/1  8/120  9:50PM
861: Simplified Axioms for Class Theory  9/16/20  9:17PM
862: Symmetric Semigroups  2/2/21  9:11 PM
863: Structural Mapping Theory/1  2/4/21  11:36PM
864: Structural Mapping Theory/2  2/7/21  1:07AM
865: Structural Mapping Theory/3  2/10/21  11:57PM
866: Structural Mapping Theory/4  2/13/21  12:47AM
867: Structural Mapping Theory/5  2/14/21  11:27PM
868: Structural Mapping Theory/6  2/15/21  9:45PM
869: Structural Proof Theory/1  2/24/21  12:10AM
870: Structural Proof Theory/2  2/28/21  1:18AM
871: Structural Proof Theory/3  2/28/21  9:27PM
872: Structural Proof Theory/4  2/28/21  10:38PM
873: Structural Proof Theory/5  3/1/21  12:58PM
874: Structural Proof Theory/6  3/1/21  6:52PM
875: Structural Proof Theory/7  3/2/21  4:07AM
876: Structural Proof Theory/8  3/2/21  7:27AM
877: Structural Proof Theory/9  3/3/21  7:46PM
878: Structural Proof Theory/10  3/3/21  8:53PM
879: Structural Proof Theory/11  3/4/21  4:22AM
880: Tangible Updates/1  4/15/21 1:46AM
881: Some Logical Thresholds  4/29/21  11:49PM
882: Logical Strength Comparability  5/8/21 5:49PM
883: Tangible Incompleteness Lecture Plans  5/16/21 1:29:44
884: Low Strength Zoo/1  5/16/21 1:34:
885: Effective Forms  5/16/21 1:47AM
886: Concerning Natural/1   5/16/21  2:00AM
887: Updated Adventures  9/9/21 9:47AM  2021
888: New(?) kinds of questions  9/9/21 12:32PM
889: Generating r.e. sets  9/12/21  3:38PM

Harvey Friedman