[FOM] 832: Tangible Incompleteness Restarted/6

Harvey Friedman hmflogic at gmail.com
Wed Oct 2 13:15:57 EDT 2019

```We now get into A...A Classes. We also have the new section 3, a major

1. BOOLEAN RELATION THEORY - my website

2. INVARIANT MAXIMALITY
2.1. Invariant Emulation Theory - #3 and here
2.1.1. N, Z+, Q, Q[(a,b)], order equivalent, maximal emulator,
invariance - #3 and here
2.1.2.Upper Shift, N (sub)tail, N (sub)tail shift, N
tail-related - #3 and here
2.1.3. Invariant Maximal Emulation - #3
2.1.4. Emulation usability of shifts - #4
2.2. Invariant Graph Theory - #3
2.2.1. Graphs, order invariant graphs, cliques - #3
2.2.2. Invariant Maximal Cliques - #3
2.2.3. Clique usability of shifts - #4
2.3. A...A Classes - later
2.3.1. Positive, of S containedin Q[-n,n]^k, Q^k - later
2.3.2. Of S containedin Q[-n,n]^k, Q^k - later

3. INDUCTIVE DOMAINS - in development
3.1. Inductive Domains in Q
3.2. Invariantly Inductive Domains in Q

4. SEQUENTIAL CONSTRUCTIONS - in development

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

2.3. A...A Classes

2.3.1. Positive, of S containedin Q[-n,n]^k, Q^k

DEFINITION 2.3.1. An A...A class X of S containedin Q[-n,n]^k (S
containedin Q^k) is the class of all S containedin Q[-n,n]^k (S
containedin Q^k) satisfying a sentence beginning with zero or more
universal quantifiers ranging over Q[-n,n] (Q) followed by a
propositional combination of formulas v < w, v in S. A maximal element
of X is an S containedin Q[-n,n]^k (S containedin Q^k) such that no S
Ul {x} lies in X.

Of course there is the stronger notion of maximality that says that we
cannot add any number of new elements to any of the components all at
once and stay in the class. This notion turns out to be useless for
our independent statements - except of course when it is equivalent to
the single new element version above. We do have equivalence for the
following special kind of A...A class:

DEFINITION 2.3.2. A positive A...A class of S containedin Q[-n,n]^k (S
containedin Q^k) is given by a sentence where the propositional part
uses only conjunction and disjunction (and no negation, implication,
equivalence).

THEOREM 2.3.1. Every subset of every element of a positive A...A class
is an element. The single new element definition of maximal element is
the same as the superset definition of maximal element, for positive
A...A classes of S containedin Q[-n,n]^k (S containedin Q^k).

THEOREM 2.3.2. The set of emulators of any given E containedin
Q[-n,n]^k (Q^k) and the set of cliques of any given order invariant
graph on Q[-n,n]^k (Q^k) are positive A...A classes of S containedin
Q]-n,n]^k (Q^k),

PROPOSITION 2.3.3. Every positive A...A class of S containedin
Q[-n,n]^k has a stable maximal element.

PROPOSITION 2.3.4. Every positive A...A class of S containedin Q^k has
a stable maximal element.

PROPOSITION 2.3.5. Every positive A...A class of S containedin
Q[-n,n]^k has a stable step maximal element.

PROPOSITION 2.3.6. Every positive A...A class of S containedin Q^k has
a stable step maximal element.

THEOREM 2.3.7. Propositions 2.3.3, 2.3.5, 2.3.6 are provably
equivalent to Con(SRP) over WKL_0. The forward direction is provable
in RCA_0. RCA_0 + Con(SRP) proves Proposition 2.3.4.

We do not know if Proposition 2.3.4 is provable in RCA_0.

Step maximality is from TANGIBLE MATHEMATICAL INCOMPLETENESS OF ZFC
August 16, 2018, #106, 66 pages

and we should have mentioned Proposition 2.3.3 for emulators and
cliques. In that paper we used Q[0,n] instead of Q]-n,n], and so step
maximality is here augmented using the initial segment <= 0. In any
case recall that we promise to integrate the above paper with the
present development with special case to make sure the reader can
easily navigate.

2.3.2. Of S containedin Q[-n,n]^k, Q^k

THEOREM 2.3.2.1. There is a nonempty A...A class of S containedin
Q[-n,n] whose unique element {n} is not stable.

PROPOSITiON 2.3.2.2. Every A...A class of S containedin Q^k has a
stable maximal element.

PROPOSITION 2.3.2.3. Every A...A class of S containedin Q^k has a
stable step maximal element.

THEOREM 2.3.2.4. Proposition 2.3.2.3 is provably equivalent to
Con(SRP) over WKL_0. The forward direction is provable in RCA_0.
Proposition 2.3.2.2 is provable in RCA_0 + Con(SRP).

We do not know if Proposition 2.3.2.2 is provable in RCA_0.

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 832nd 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

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  10/2/19  9/29/19  2:54AM

Harvey Friedman
```