[FOM] 836: Tangible Incompleteness Restarted/10

[Harvey Friedman]

MORE MATERIAL UNDER INVARIANT MAXIMALITY.

So far, we have these three major sections under Invariant Maximality:

2.2. Emulation Theory
2.3. Invariant Graph Theory
2.4. Invariant A...A Theory

Here are some further developments that should be included.

1. A very important addition that slipped through the cracks is the
use of embeddings as a special kind of invariance. Embeddings are
given by partial maps from Q into Q. We show that no f:Q into Q can be
used other than the identity. There is a lot of juicy material here,
some of course inaccessible to ZFC, and corresponding to Con(SRP)..

2. Use of intervals other than the Q[-n,n]. We use extended rational
intervals. This requires some minor modifications of the complete N
Tail Shift Invariance and the N Tail-Related Invariance. This cuts
across each of 2.2, 2.3, 2.4.

3. Maximality modifications:

A. Step Maximality. This was included in #108 but not in #109 on
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
Step Maximality cuts across each of 2.2, 2.3, 2.4.  So far the main
significance of Step Maximality is that it allows us to use intervals
without a right endpoint to get out of ZFC. Most notably, Q itself,
and in fact all of the [(a,infinity). We have step maximal emulators,
step maximal cliques, step maximal elements of A...A classes. It is
still unresolved whether step maximality is needed to get out of ZFC,
e.g, most notably for Q,  A bonus is that we can use N Tail Shift
Invariance and avoid "complete invariance".

B.Iterated Maximality. This turns out to be another way to get out of
ZFC in intervals like Q. For Emulation Theory, we define maximal
emulators of E_1 containedin ... containedin E_t containedin Q^k as
S_1 containedin... containedin S_t where each S_i is a maximal
emulator of E_i. We can equivalently define this directly without
passing through the ordinary t = 2 case.

For graphs, we use edge inclusion, as all of the t given graphs are
order invariant on Q^k. Each S_i is a maximal clique in G_i.

For A...A classes, we start with a tower of A...A classes, and find a
tower of respective elements, each maximal in the corresponding A...A
class.

A bonus is that we can use N Tail Shift Invariance and avoid "complete
invariance".

4. General modification. Duplication. This was included in #108 but
not in #109. Doesn't change the results. S is a duplicator of E if and
only if S,E are emulators of each other. Nice because whereas
emulation is transitive, duplication is an equivalence relation. Of
course maximal duplication is not symmetric.





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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 835th 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

Harvey Friedman