941: New Stable Maximality/1

Fri Jul 29 20:44:21 EDT 2022

I finished my lectures at the 2022 Ross Program for Gifted High School
students. The seven lectures, in edited form, are now at

https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#118

**********************

WE START OVER !!!!

I discovered another major simplification in the way I do all this. I
will even repeat earlier stuff to make this self contained.

DEFINITION 1. Q is the set of all rationals. N is the set of all
nonnegative integers. X* is the set of all finite sequences from X.
Q[p,q] is Q intersect [p,q]

DEFINITION 2. S is an emulator of E containedin Q[-n,n]^k if and only
if S containedin Q[-n,n]^k where the concatenation of any list of two
elements of S is order equivalent to the concatenation of some list of
two elements of E. S containedin Q[-n,n]^k is a maximal emulator of E
containedin Q[-n,n]^k if and only if S is an emulator of E containedin
Q[-n,n]^k which is not a proper subset of any emulator of E
containedin Q[-n,n]^k.

DEFINITION 3 The N tail successor of x in Q* is obtained by adding 1
to all coordinates in N that are greater than all coordinates not in
N. We write NTS(x).

DEFINITION 4. S containedin Q[-n,n]^k is stable if and only if for all
x in Q[-n,n-1]^k, x in S iff NTS(x) in S.

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

THEOREM 1. SME is provably equivalent to Con(SRP) over WKL0.

SME IS OUR LEAD STATEMENT IN STABLE MAXIMALITY.

Recall the high school definition:

HIGH SCHOOL DEFINITION. S containedin Q[-1,1]^2 is negatively stable
if and only if for all -1 <= p < 0,
i. (0,0) in S iff (1,1) in S
ii. (p,0) in S iff (p,1) in S
iii. (0,p) in S iff (1,p) in S

THEOREM 2. S containedin Q[-1,1]^2 is negatively stable if and only if
S containedin Q[-1,1]^2 is stable.

So we have complete compatibility with the Gifted High School
development https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#118

DEFINITION 5. The N gap tail shift of x in Q^k is obtained by adding 1
to all coordinates in N that are are at least 1 greater than all
coordinates not in N. We write GNTS(x).

DEFINITION 6. S containedin Q[-n,n]^k is gap stable if and only if for
all x in Q[-n,n-1]^k, x in S iff NTS(x) in S.

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

THEOREM 3. GSME is provably equivalent to Con(MAH) over WKL0.

DEFINITION 7. S containedin Q[-n,n]^k i s negatively stable if and
only if for all x in (Q[-n,0) U N[0,n-1])^k, x in S iff NTS(x) in S.

DEFINITION 8. S containedin Q[-n,n]^k i s negatively gap stable if and
only if for all x in (Q[-n,0) U N[1,n-1])^k, x in S iff GNTS(x) in S.

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

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

THEOREM 4. NSME is provably equivalent to Con(SRP) over WKL0. NGSME is
provably equivalent to Con(WZ) over WKL0.

Here MAH is ZFC + {there exists a strongly k-Mahlo cardinal}_n. WZ is
weak Zermelo, which is Zermelo with separation only for bounded
formulas.

***ADMISSIBLE SUCCESSOR FUNCTION STABILITY***

We now present a very simple general theory that shows that what we
have done is in various senses best possible.

DEFINITION 9. x,y in Q* are N order equivalent if and only if lth(x) =
lth(y) and for all 1 <= i,j <= lth(x), x_i < x_j iff y_i < y_j. and
x_i in N iff y_i in N.

DEFINITION 10. A successor function on Q* is an f:Q* into Q* such that
f(x) is obtained from x by adding 1 to some coordinates of x lying in
N (none or all). An admissible successor function on Q* is a successor
function f:Q* into Q* where for all x in Q*, ,xf(x) and f(x)ff(x) are
N order equivalent.

Note that our N Tail Successor (NTS) is an admissible successor function on Q*.

DEFINITION 11. S containedin Q[-n,n]^k is admissibly stable if and
only if for all admissible successor functions f:Q* into Q* and x in
Q[-n,n-1]^k, x in S iff f(x) in S.

ADMISSIBLE STABILE MAXIMAL EMULATOR. ASME. Every subset of Q[-n,n]^k
has an admissibly stable maximal emulator.

THEOREM 5. ASME is provably equivalent to Con(SRP) over WKL0.

FOUR PARAMETERS

DEFINITION 12. A d-emulator of E containedin Q[-n,n]^k is an S
containedin Q[-n,n]^k such that the concatenation of any list of d
elements of S is order equivalent to the concatenation of some list of
d elements of E.

SME, GSME, NSME, NGSME, ASME all have a four parameter form. The four
parameters are fixed nonnegative integers, and also the choice of
quantifying over all nonnegative integers.

The dimension k as in Q[-n,n]^k
the endpoint n as in Q[-n,n]^k
The size r as in the cardinality of the E containedin Q[-n,n]^k
The degree d as in d-emulator, where emulators are 2-emulators

We write SME(dim/k,end/n,size/r,deg/d). And we can leave off none to
all of the k,n,r,d. Just dim means quantification over all k. Just end
means quantification over all n. Just size means quantification over
all r, which is manifested in "all (finite) subsets of Q[0n,n]^k".
Just deg means quantification over all d.

The first focus of the upcoming Gent lectures will be on
SME(dim,end,size,deg/2) and SME(dim,end,size,deg) both of which are
shown to be provably equivalent to Con(SRP) over WKL0.

I will look to analyzing SME(dim/3,end/2,size,deg/2) and variants,
with the idea that they are independent of ZFC.

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

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 941st 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
940: Stable Maximality/2  7/24/2 19:19:48 EDT 2022

Harvey Friedman