930: Tangible Indiscernibles/1

Sat May 7 19:46:01 EDT 2022

I am now returning to Tangible Incompleteness after taking off an
unexpectedly inordinate amount of time to do some work in the foundations
of recursion theory and computational complexity. See

https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
118: String Replacement Systems

The idea here is to make good on the not ready for prime time earlier
posting

https://cs.nyu.edu/pipermail/fom/2021-September/022861.html
Generating r.e. sets

Comments welcome, especially since I am not too familiar with this kind of
classical material of Post and Thue and Chomsky and others.

My last FOM posting on the state of the art of Tangible Incompleteness
(infinite statements) is

https://cs.nyu.edu/pipermail/fom/2021-September/022870.html
Update on Tangible Incompleteness

Note that the statements can be rephrased to take the following forms:

Every order invariant subset of Q[0,n]^2k contains a maximal square.
Every order invariant subset of Q[0,n]^2k contains a maximal square with
"indiscernibles" 1,2,...n.

I have always been very happy that I can state in advance what the
"indiscernibles" are. It turns out that I was too happy.

Various core mathematical sets also have "indiscernibles" of various kinds,
but usually we cannot fix what the "indiscernibles" are in advance. Hence I
DID NOT USE indiscernibles in core mathematical sets as a major connection
between our Tangible Incompleteness and core mathematics.

HAVING MY CAKE AND EATING IT TOO

So the expositional breakthrough here is this.

1. We first develop some theory of "indiscernibles" for core mathematical
sets. (Other kinds of core mathematical objects will be considered later,
such as functions).
2. We then consider these statements:

Every order invariant subset of Q[0,1]^2k contains a maximal square.
Every order invariant subset of Q[0,1]^2k contains a maximal square with
"indiscernibles" of every finite size containing 1.
Every order invariant subset of Q[0,1]^2k contains a maximal square with
"indiscernibles" 1,1/2,,...,1/n.

I have my cake which is 1 and the middle statement above. Then I eat it too
with the third statement above. Both of these statements are equivalent to
Con(SRP) over WKL_0.

WHAT KIND OF INDISCERNIBLES ARE WE USING?

1. Every section of the set below an indiscernible c is order invariant
over the indiscernibles >= c.

2. Every section of the set below an indiscernible c is order invariant
over the indiscernibles > c.

3. Every section of the set below the first indiscernible is order
invariant over all of the indiscernibles.

4. Every section of the set below the first indisnerible c is
order invariant over all indiscernibles > c.

Most common for core mathematical sets is 2,4 above. Sometimes 1,3 occur.

In Tangible Incompleteness 1,3 give Con(SRP), 2 gives Con(MAH), and 4 given
Con(WZ), where WZ is weak Zermelo set theory.

DETAILS WILL BE PRESENTED IN THE NEXT POSTING Tangible Indiscernibles/2.

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

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 930th 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 *

Harvey Friedman
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