940: Stable Maximality/2

Sun Jul 24 19:19:48 EDT 2022

We now introduce the stronger Stable notion -- stronger than
Negatively Stable that we discussed in
https://cs.nyu.edu/pipermail/fom/2022-July/023510.html However it does
not give any added logical strength.

We have been using the convenient interval notation Q[(p,q)], for
intervals of rationals with endpoints p,q. It is also convenient to
use the interval notation Z[(p,q))]  for intervals of integers with
endpoints p,q. We allow p,q to be rationals, and not just integers.

See https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#118 for the Ross Program Lectures.

DEFINITION 1. S containedin Q[-n,n]^k is stable if and only
if for all p_1,...,p_k in Q[-n,n] and m_1,...,m_k in Z(max(p_1,...,p_k),n],
these 2^k equivalences hold:

(p_1 or m_1, ... , p_k or m_k) in S iff
(p_1 or m_1 + 1, ... , p_k or m_k + 1) in S

NEGATIVELY STABLE MAXIMAL EMULATOR.  NSME. Every subset of Q[-n,n]^k
has a negatively stable maximal emulator.

STABLE MAXIMAL EMULATOR.  SME. Every subset of Q[-n,n]^k
has a stable maximal emulator.

THEOREM. SME and NSME are each provably equivalent to Con(SRP) over WKL0.

This is precisely what is intended to be fully proved in the upcoming
Gent Lectures in September.

Again for NSME and for SME, we use

k = dimension
n = endpoint
r = size
d = degree

k and n arise with Q[-n,n]^k. In NSME, only 0,...,n-1 get moved, and
they get moved to 1,...,n, respectively..

However in SME, -n+1,...,n-1 get moved, and they get moved to
-n+2,...,n, respectively.

r is the bound on the cardinality of the given subset of Q[-n,n]^k.

The degree d is in connection with the emulators. Recall the
definition of e-emulators in
https://cs.nyu.edu/pipermail/fom/2022-July/023510.html
Emulators are the same as the 2-emulators.

We are allowed to leave off some of the /i, meaning we are quantifying
instead of fixing. We have proved the following using a bit more than Z_2.

THEOREM. SME(dim/2,end,size,deg).

I think it likely that this above cannot be proved in Z_2. Even with
NSME. Haven't
really tried this reversal.

THEOREM. SME(dim,end,size,deg/2), NSME(dim,end,size,deg/2),
SME(dim,end,size,deg), NSME(dim,end,size,deg) are provably equivalent
to Con(SRP) over WKL0.

***GAP STABILITY***

DEFINITION 1. S containedin Q[-n,n]^k is negatively gap stable if and only
if for all p_1,...,p_k in Q[-1,0) and m_1,...,m_k in {1,...,n-1},
these 2^k equivalences hold:

(p_1 or m_1, ... , p_k or m_k) in S iff
(p_1 or m_1 + 1 ,..., p_k or m_k + 1) in S

Now for the stronger gap stable notion.

DEFINITION 2. S containedin Q[-n,n]^k is gap stable if and only
if for all p_1,...,p_k in Q[-n,n] and m_1,...,m_k in Z[max(p_1,...,p_k)+1,n].
these 2^k equivalences hold:

(p_1 or m_1, ... , p_k or m_k) in S iff
(p_1 or m_1 + 1, ... , p_k or m_k + 1) in S

NEGATIVELY GAP STABLE MAXIMAL EMULATOR.  NGSME. Every subset of Q[-n,n]^k
has a negatively stable maximal emulator.

GAP STABLE MAXIMAL EMULATOR.  GSME. Every subset of Q[-n,n]^k
has a gap stable maximal emulator.

THEOREM. NGSME is provably equivalent to Con(WZ) over WKL0. GSME is
provably equivalent to Con(MAH) over WKL0.

Here WZ is weak Zermelo set theory. MAH is ZFC + Strongly Mahlo
cardinals of every order (scheme).

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

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 940th 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 #1-899 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/

900: Ultra Convergence/2  10/3/21 12:35AM
901: Remarks on Reverse Mathematics/6  10/4/21 5:55AM
902: Mathematical L and OD/RM  10/7/21  5:13AM
903: Foundations of Large Cardinals/1  10/12/21 12:58AM
904: Foundations of Large Cardinals/2  10/13/21 3:17PM
905: Foundations of Large Cardinals/3  10/13/21 3:17PM
906: Foundations of Large Cardinals/4  10/13/21  3:17PM
907: Base Theory Proposals for Third Order RM/1  10/13/21 10:22PM
908: Base Theory Proposals for Third Order RM/2  10/17/21 3:15PM
909: Base Theory Proposals for Third Order RM/3  10/17/21 3:25PM
910: Base Theory Proposals for Third Order RM/4  10/17/21 3:36PM
911: Ultra Convergence/3  1017/21  4:33PM
912: Base Theory Proposals for Third Order RM/5  10/18/21 7:22PM
913: Base Theory Proposals for Third Order RM/6  10/18/21 7:23PM
914: Base Theory Proposals for Third Order RM/7  10/20/21 12:39PM
915: Base Theory Proposals for Third Order RM/8  10/20/21 7:48PM
916: Tangible Incompleteness and Clique Construction/1  12/8/21   7:25PM
917: Proof Theory of Arithmetic/1  12/8/21  7:43PM
918: Tangible Incompleteness and Clique Construction/1  12/11/21  10:15PM
919: Proof Theory of Arithmetic/2  12/11/21  10:17PM
920: Polynomials and PA  1/7/22  4:35PM
921: Polynomials and PA/2  1/9/22  6:57 PM
922: WQO Games  1/10/22 5:32AM
923: Polynomials and PA/3  1/11/22  10:30 PM
924: Polynomials and PA/4  1/13/22  2:02 AM
925: Polynomials and PA/5  2/1/22  9::04PM
926: Polynomials and PA/6  2/1/22 11:20AM
927: Order Invariant Games/1  03/04/22  9:11AM
928: Order Invariant Games/2  03/7/22  4:22AM
929: Physical Infinity/randomness  3/21/22 02:14AM
930: Tangible Indiscernibles/1 05/07/22  7:46PM
931: Tangible Indiscernibles/2 5/14/22  1:34PM
932: Tangible Indiscernibles/3  5/14/22  1:34PM
933: Provable Functions of Set Theories/1 5/16/22  7/11AM
934: Provable Ordinals of Set Theories/1  5/17/22  8:35AM
935: Stable Maximality/Tangible Incompleteness/1  6/3/22  7:05PM
936: Stable Maximality/Tangible Incompleteness/2  6/4/22  11:31PM
937: Logic of Real Numbers/1  6/22/22  7:49AM
938: Logic of Real Functions/1 7/9/22  2:42AM
939: Stable Maximality/1  7/22/22  10:07AM

Harvey Friedman