852: Order Theoretic Maximality/1

Harvey Friedman hmflogic at gmail.com
Thu Apr 30 19:17:01 EDT 2020

1. Two Typos in Product Inequality Theory/1,

1. TYPOS and missing definition

I wrote

PROPOSITION 3. Let t be a piecewise poly term in variables p,q,r.
There is a dot:A^2 into A, N containedin A containedin Q, such that
for all p,r <= q f< j from A,
t(p,q,p dot q) <= t(p,q,r) <= t(ush(p,q,r)) = t(j dot p,j dot q,j dot r).

PROPOSITION 7. Let t be a piecewise poly term in variables p,q,r and n
>> d,k. There is a dot:{-kn,...,kn}^2 into {-kn,...,kn} such that for
all r <= max(p,q) in GEN(k;0,n,...,kn),
t(p,q,p dot q) <= t(p,q,r) <= t(ush/n(p,q,r)) <= t(jk dot p,jk dot q,jk dot r).

This should read

PROPOSITION 3. Let t be a piecewise poly term in variables p,q,r.
There is a dot:A^2 into A, N containedin A containedin Q, such that
for all p,r <= q < j from A,
t(p,q,p dot q) <= t(p,q,r) <= t(ush(p,q,r)) = t(j dot p,j dot q,j dot r).

PROPOSITION 7. Let t be a piecewise poly term in variables p,q,r and n
>> d,k. There is a dot:{-kn,...,kn}^2 into {-kn,...,kn} such that for
all p,r, <= q < j <= k from GEN(k;0,n,...,kn),
t(p,q,p dot q) <= t(p,q,r) <= t(ush/n(p,q,r)) <= t(jn dot p,jn dot q,jn dot r).

MISSING DEFINITION. ush/n(p) is p if p < 0; p+n if p >= 0.


We now realize that there is a thematically prior subject to our
INVARIANT MAXIMALTIY that begs to be developed. Our INVARIANT
MAXIMALITY (Chapter 2 of the Tangible Incompleteness book) falls
nicely under the more general ORDER THEORETIC MAXIMALITY.

We haven't really done much but identify Order Theoretic Maximality
for future investigation. But we think that it is an important line of

At our absolutely highest level of generality, we have GENERAL

GENERAL MAXIMALITY TEMPLATE. Given two nice families of sets. Is there
a maximal element of the first that lies in the second?

We will postpone discussion on FOM at this highest level of
generality, which we expect to have a lot of interesting points of
contact with various branches of mathematics.

Instead we go to the next level of generality down from General
Maximality to ORDER THEORETIC MAXIMALITY. The next level down from
there is our already reasonably well developed INVARIANT MAXIMALITY.

We make sure that the reader of the Tangible Incompleteness book is
frequently informed that they can go directly to Invariant Maximality,
which is self contained, without reading about Order Theoretic
Maximality. Then they can come back to Order Theoretic Maximality to
understand the wider context which has many fundamental open issues.

So what is Order Theoretic Maximality?

DEFINITION 2.1. A rational interval is an interval of rationals whose
endpoints lie in Q union {-infinity,infpnity}.we use I,J for trational
intervals. An order theoretic subset of J^k is a set of the form {x in
J^k: phi) where phi is a boolean combination of inequalities xi < xj
and c < xi and xi < c, where c is a constant from J.

DEFINITION 2.2. An order theoretic subset of POW(J^k) is a set of the
form {A containedin J^k: (forall p1,...,pn in J)(phi)} where phi is a
Boolean combination of inequalities pi < pj and c < pi and pi < c, and
memberships (u1,...,uk) in A, where each ui is either a variable among
p1,...,pn or a constant from J.

ORDER THEORETIC MAXIMALITY TEMPLATE. Given two order theoretic subsets
of POW(J^k).,is there a maximal element of the first that lies in the

In this Template, the first order theoretic subset of POW(J^k) is
called the Source, and the second order theoretic subset of POW(J^k)
is called the Target.

In Invariant Maximality, we work only with J = Q[-n,n]. For the
Sources, we use three categories listed below (with variants); and for
the Targets we use the set of all S containedin Q[-n,n]^k such that S
is completely invariant with respect to ush/N. These Targets are order
theoretic since each J intersect N = {0,...,n} which is finite.

Category 1. {S: S^2 containedin E}, where E is an order invariant
subset of Q[-n,n]^k.

Category 2. {S: S is an emulator of E}, where A is a finite subset of
Q[-n,n]^k. ,

Category 3. {S: S is a clique in G}, where G is an order invariant
graph on Q[-n,n]^k}.

In a suitable sense,Category absorbs Categories 2 and 3.

In Categories 1,3 we can take E,G to be order theoretic, and still the
set of all such S will be an order theoretic subset of POW(Q[-n,n]^k).
However we would need to adjust the Target using a modification of
ush/N based on the parameters used for E,G. This more general form of
Invariant Maximality will go through nicely as long as the right
endpoint n is not allowed in the Sources. Or at least the Targets need
to be seriously reconsidered.

>From Invariant Maximality we see that a decision procedure for the
Order Theoretic Maximality Template cannot be verified in SRP.Of
course, this Template may be non recursive. But Invariant Maximality
indicates the possibility that there are significant fragments of the
Template that correspond to large cardinals.


My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 851st 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
837: Tangible Incompleteness Restarted/11 10/18/20  02:58AM
838: New Tangible Incompleteness/1 1/11/20 1:04PM
839: New Tangible Incompleteness/2 1/13/20 1:10 PM
840: New Tangible Incompleteness/3 1/14/20 4:50PM
841: New Tangible Incompleteness/4 1/15/20 1:58PM
842: Gromov's "most powerful language" and set theory  2/8/20  2:53AM
843: Brand New Tangible Incompleteness/1 3/22/20 10:50PM
844: Brand New Tangible Incompleteness/2 3/24/20  12:37AM
845: Brand New Tangible Incompleteness/3 3/28/20 7:25AM
846: Brand New Tangible Incompleteness/4 4/1/20 12:32 AM
847: Brand New Tangible Incompleteness/5 4/9/20 1 34AM
848. Set Equation Theory/1 4/15 11:45PM
849. Set Equation Theory/2 4/16/20 4:50PM
850: Set Equation Theory/3 4/26/20 12:06AM
851: Product Inequality Theory/1 4/29/20 12:08AM

Harvey Friedman

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