[FOM] 792: Emulation Theory/Inductive Equations/2

[Harvey Friedman]

1. THEMATIC BREAKTHROUGH IN FINITE FORM FOR SRP
2. TEMPLATES FOR /INFINITE/SRP
3. TEMPLATES FOR INFINITE/HUGE
4. TEMPLATES FOR FINITE/SRP
5. IMMORTALITY

1. THEMATIC BREAKTHROUGH IN FINITE FORM FOR SRP

Recall the Inductive Upper Shift theorem from

Concrete Mathematical Incompleteness Status 2/8/18
http://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#98

which reads

INDUCTIVE UPPER SHIFT. IUS. Every order invariant R containedin Q^2k
has some S = S#\R<[S] containing ush(S).

Compare with these three:

FIUS/1. Every order invariant R containedin Q^2k has some finite S ==k
S#\R<[S] containing ush(S)|<=k.

FIUS/2. Every order invariant R containedin Q^2k has some finite S ==k
S#\R<[S], S containing ush(S)|<=k.

FIUS/3. For every order invariant R containedin Q^2k, there exists
finite S containedin Q^k|<=k where S,S#\R<[S] are ==k and contain
ush(S)|<=k.

Everything is self explanatory except of course for the equivalence
relation ==k on subsets of Q^k. I use a triple bar for ==.

DEFINITION 1.1. A ==k B if and only if A,B containedin Q^k and
(Q,<,A,0,...,k) and (Q,<,B,0,...,k) satisfy the same universal
sentences with at most k quantifiers.

Obviously FIUS/1-3 are explicitly Pi02 and become explicitly Pi01 if
we place a priori upper bounds on the numerators and denominators used
in S.

2. TEMPLATING FOR /INFINITE/SRP

As a preliminary warmup, we first Template the provable statement

TRIVIALITY. Every order invariant R containedin Q^2k has some S = S#\R<[S].

TEMPLATE 1. Let alpha be a formal Boolean combination of S,S#,R<[S].
For all order invariant R containedin Q^2k there exists S containedin
Q^k such that alpha = Q^k.

Of course, R<[S] is not the most fundamental relevant construct. R[S]
is. So we should go for the more ambitious

TEMPLATE 2. Let alpha be a formal Boolean combination of
S,S#,R[S],R<[S]. For all order invariant R containedin Q^2k there
exists S containedin Q^k such that alpha = Q^k.

CONJECTURE 2.1. Every instance of Template 2 is provable or refutable
in RCA_0.

TEMPLATE 3. Let alpha be a formal Boolean combination of
S,S#,R<[S],ush(S). For all order invariant R containedin Q^2k there
exists S containedin Q^k such that alpha = Q^k.

TEMPLATE 4. Let alpha be a formal Boolean combination of
S,S#,R[S],R<[S],ush(S). For all order invariant R containedin Q^2k
there exists S containedin Q^k such that alpha = Q^k.

CONJECTURE 2.2. Every instance of Template 4 is provable in WKL_0. +
Con(SRP) or refutable in RCA_0.

Partial results can be expected along the way that limit the
complexity of the finitely many formal Boolean combinations covered -
in various ways.

3. TEMPLATES FOR INFINITE/HUGE

Recall from https://u.osu.edu/friedman.8/files/2014/01/CMIstatus020818-ucpy18.pdf

IIUS. Every order invariant R containedin Q^2k has some S =_<=
S#\R<[S] that strongly contains its upper shift.

TEMPLATE 5. Let alpha,beta,gamma be formal Boolean combinations of
S,S#,R<[S],R[S],ush(S). For all order invariant R containedin Q^2k
there exists S containedin Q^k such that alpha = Q^k and beta =_<=
gamma and delta strongly contains epsilon.

More ambitious is:

TEMPLATE 6. Let
alpha,beta_1,...,beta_n,gamma_1,...,gamma_n,delta_1,...,delta_m,epsilon_1,...,epsilon_m
be formal Boolean combinations of S,S#,R<[S],R[S],ush(S). For all
order invariant R containedin Q^2k there exists S containedin Q^k such
that alpha = Q^k and each beta_i =_<= gamma_i and each delta_i
strongly contains epsilon_i.

CONJECTURE 3.1. Every instance of Template 6 is provable in WKL_0 +
Con(HUGE) or refutable in RCA_0.

Partial results can be expected along the way that limit the
complexity of the finitely many formal Boolean combinations covered -
in various ways.

4. TEMPLATING FOR FINITE/SRP

Recall from the above,

FIUS/3. For every order invariant R containedin Q^2k, there exists
finite S containedin Q^k|<=k where S,S#\R<[S] are ==k and contain
ush(S)|<=k.

TEMPLATE 7.  Let alpha,beta,gamma,delta be formal Boolean combinations
of S,S#,R<[S],R[S],ush(S),ush(S)|<=k. For all order invariant R
containedin Q^2k there exists finite S containedin Q^k such that alpha
= Q^k and beta ==k gamma.

More ambitious is:

TEMPLATE 8.  Let alpha,beta_1,...,beta_n,gamma_1,...,gamma_n be formal
Boolean combinations of S,S#,R<[S],R[S],ush(S),ush(S)|<=k. For all
order invariant R containedin Q^2k there exists finite S containedin
Q^k such that alpha = Q^k and each beta ==k gamma_i.

Partial results can be expected along the way that limit the
complexity of the finitely many formal Boolean combinations covered -
in various ways.

5. IMMORTALITY

It is clear that once the Concrete Mathematical Incompleteness book is
finished, or at least some key results are written up and pass
credibility tests, then we have an essentially never ending
enterprise, even without any further breakthroughs in terms of new
statements. These TEMPLATES insure this kind of Immortality. If at
some point these Templates get fully treated, there is no problem
making them yet more challenging....

Of course, history shows that there will in fact be lots of major
breakthroughs in terms of new statements, and yet more challenging
templates...

************************************************************************
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 792nd 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 #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
711: Large Cardinals and Continuations/21  9/18/16 10:42AM
712: PA Incompleteness/1  9/23/16  1:20AM
713: Foundations of Geometry/1  9/24/16  2:09PM
714: Foundations of Geometry/2  9/25/16  10:26PM
715: Foundations of Geometry/3  9/27/16  1:08AM
716: Foundations of Geometry/4  9/27/16  10:25PM
717: Foundations of Geometry/5  9/30/16  12:16AM
718: Foundations of Geometry/6  101/16  12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7  10/2/16  1:59PM
721: Large Cardinals and Emulations//23  10/4/16  2:35AM
722: Large Cardinals and Emulations/24  10/616  1:59AM
723: Philosophical Geometry/8  10/816  1:47AM
724: Philosophical Geometry/9  10/10/16  9:36AM
725: Philosophical Geometry/10  10/14/16  10:16PM
726: Philosophical Geometry/11  Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25  10/20/16  1:37PM
728: Philosophical Geometry/12  10/24/16  3:35PM
729: Consistency of Mathematics/1  10/25/16  1:25PM
730: Consistency of Mathematics/2  11/17/16  9:50PM
731: Large Cardinals and Emulations/26  11/21/16  5:40PM
732: Large Cardinals and Emulations/27  11/28/16  1:31AM
733: Large Cardinals and Emulations/28  12/6/16  1AM
734: Large Cardinals and Emulations/29  12/8/16  2:53PM
735: Philosophical Geometry/13  12/19/16  4:24PM
736: Philosophical Geometry/14  12/20/16  12:43PM
737: Philosophical Geometry/15  12/22/16  3:24PM
738: Philosophical Geometry/16  12/27/16  6:54PM
739: Philosophical Geometry/17  1/2/17  11:50PM
740: Philosophy of Incompleteness/2  1/7/16  8:33AM
741: Philosophy of Incompleteness/3  1/7/16  1:18PM
742: Philosophy of Incompleteness/4  1/8/16 3:45AM
743: Philosophy of Incompleteness/5  1/9/16  2:32PM
744: Philosophy of Incompleteness/6  1/10/16  1/10/16  12:15AM
745: Philosophy of Incompleteness/7  1/11/16  12:40AM
746: Philosophy of Incompleteness/8  1/12/17  3:54PM
747: PA Incompleteness/2  2/3/17 12:07PM
748: Large Cardinals and Emulations/30  2/15/17  2:19AM
749: Large Cardinals and Emulations/31  2/15/17  2:19AM
750: Large Cardinals and Emulations/32  2/15/17  2:20AM
751: Large Cardinals and Emulations/33  2/17/17 12:52AM
752: Emulation Theory for Pure Math/1  3/14/17  12:57AM
753: Emulation Theory for Math Logic  3/10/17  2:17AM
754: Large Cardinals and Emulations/34  3/12/17  12:34AM
755: Large Cardinals and Emulations/35  3/12/17  12:33AM
756: Large Cardinals and Emulations/36  3/24/17  8:03AM
757: Large Cardinals and Emulations/37  3/27/17  2:39AM
758: Large Cardinals and Emulations/38  4/10/17  1:11AM
759: Large Cardinals and Emulations/39  4/10/17  1:11AM
760: Large Cardinals and Emulations/40  4/13/17  11:53PM
761: Large Cardinals and Emulations/41  4/15/17  4:54PM
762: Baby Emulation Theory/Expositional  4/17/17  1:23AM
763: Large Cardinals and Emulations/42  5/817  2:18AM
764: Large Cardinals and Emulations/43  5/11/17  12:26AM
765: Large Cardinals and Emulations/44  5/14/17  6:03PM
766: Large Cardinals and Emulations/45  7/2/17  1:22PM
767: Impossible Counting 1  9/2/17  8:28AM
768: Theory Completions  9/4/17  9:13PM
769: Complexity of Integers 1  9/7/17  12:30AM
770: Algorithmic Unsolvability 1  10/13/17  1:55PM
771: Algorithmic Unsolvability 2  10/18/17  10/15/17  10:14PM
772: Algorithmic Unsolvability 3  Oct 19 02:41:32 EDT 2017
773: Goedel's Second: Proofs/1  Dec 18 20:31:25 EST 2017
774: Goedel's Second: Proofs/2  Dec 18 20:36:04 EST 2017
775: Goedel's Second: Proofs/3  Dec 19 00:48:45 EST 2017
776: Logically Natural Examples 1  12/21 01:00:40 EST 2017
777: Goedel's Second: Proofs/4  12/28/17  8:02PM
778: Goedel's Second: Proofs/5  12/30/17  2:40AM
779: End of Year Claims  12/31/17  8:03PM
780: One Dimensional Incompleteness/1  1/4/18  1:14AM
781: One Dimensional Incompleteness/2  1/6/18  11:25PM
782: Revolutionary Possibilities/1  1/12/18  11:26AM
783: Revolutionary Possibilities/2  1/20/18  9:43PM
784: Revolutionary Possibilities/3  1/21/18  2:59PM
785: Revolutionary Possibilities/4  1/22/18  12:38AM
786: Revolutionary Possibilities/5  1/24/18  12:15AM
787: Revolutionary Possibilities/6  1/25/18  4:09AM
788: Revolutionary Possibilities/7  2/1/18  2:18AM
789: Revolutionary Possibilities/8  2/1/18  9:02AM
790: Revolutionary Possibilities/9  2/2/18  3:07AM
791: Emulation Theory/Inductive Equations/1  2/8/18  11:52PM

Harvey Friedman