[FOM] 837: Tangible Incompleteness Restarted/11

Harvey Friedman hmflogic at gmail.com
Fri Oct 18 02:58:21 EDT 2019

Two topics in this posting:


I had forgotten about the EMBEDDING approach to invariance. EMBEDDINGS
here are partial functions from Q[0,1] into Q[0,1], and we use the
diagonal action as expected. I had worked with this some time ago. I
have upgraded the treatment, in the past few days, with a clearer
perspective and some new results.

I am busy reworking #109 on my website to a new #110. I will retain
the old news #109 as various things there may resurface.


We look at structures (Z,R), where N is the set of all integers, and R
containedin Z^k

Here are the natural statements from classical model theory that we
analyze with the lens of my Reverse Mathematics that I began to set up
in the early 1970s.

inclusion maximality is self explanatory. Point maximality means that
no new single point can be added and still stay in the class.

A. Let k >= 1. Every nonempty universal class of (Z,R) has an
inclusion maximal element. Provable in RCA_0.

B. Let k >= 1. Every universal class of (Z,R) has a point (inclusion)
maximal element containing any given element. Equivalent to Pi11-CA0
over RCA0 (both forms).

C. Let k >= 1. Every universal-existential class of (Z,R) has a point
(inclusion) maximal element containing any given element. Equivalent
to Pi11-CA0 over RCA0 (both forms).

D. :Let k >= 1. Every nonempty universal-existential class of (Z,R)
has a point (inclusion) maximal element. Equivalent to parameterless
Pi11-CA0 over RCA0 (both forms).

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 837th 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  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
836: Tangible Incompleteness Restarted/10  10/14/19  12:34PM

Harvey Friedman

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