864: Structural Mapping Theory/2

Harvey Friedman hmflogic at gmail.com
Sun Feb 7 01:07:05 EST 2021

MAJOR DIGRESSION. So what happened to my 54 year Tangible
Incompleteness Project? There have been some conceptual advances that
are being written up in a yet to be released revised version of #111
on my downloadable manuscript page. I hope to put this up within a
week. Aside from newer better more thematic examples of Incompleteness
that are implicitly and explicitly Pi01 concerning invariant
maximality, there is also emerging ways to systematically push large
cardinal theory down into the natural numbers. More later...

The framework for this SMAT (structural mapping theory) is now
becoming far clearer, and certainly will evolve some, but it has
advanced enough for me to make this report.

So about SMAT. I'm trying to get SMAT into shape so it is clear what
results in SMAT look like. At some point I was able to make it clear
what results in RM = Reverse Mathematics look like and everybody
pretty much knew where and how to look. For SRM = Strict Reverse
Mathematics, major spade work has not been completed and publicized,
and it isn't yet so clear what results in SRM will look like. But that
will come.

We seek fundamental natural constructions in algebra and category
where we start with an arbitrary set D and build the structure based
on D. Particularly well known examples of this are the following.

1. The free Abelian group with D as the set of generators.
2. The free group with D as the set of generators.
3. The vector space of formal finite sums k_1d_1 + ... + k_nd_n, d's
from D, k's from a field K.
4. The semigroup of functions from D into D under composition.
5. The semigroup of one-one functions from D into D under composition.
6. The group of bijections from D onto D under compositions.

In each case we seek a D so that the associated structure has a non
surjective BLANK preserving map (into itself). Of particular interest
is preserving solvability of finite sets of equations, but ther eare
many other important preserving notions, like the weaker preserving
equations, and the stronger logic preserving - i.e., elementary
embedding (preserving first order properties). THere are others like
preserving universalized sets of equations, and also preserving
universal solvable and solvable universal. Also see absolutely
preserving as discussed in

[1]  https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/

CRUCIAL POINT. 1-3 are totally different than 4,5. In 1,3, it is
trivial that there are non surjective highly preserving mappings.
Specifically, the ones induced by a non surjective one-one map from
the generating set D into itself. This is related to the fact that in
1-3, the structure has the same cardinality as D (assuming D
infinite), whereas in 4,5, the structure has cardinality 2^|D|.

Our work on case 4 spawned the idea of SMAT. We are now looking at 5,6
and are getting rather excited as we prepare to get seriously

So how do we properly look at what is going on here?

THE SEMIGROUP in 4 can be viewed as the group of HOMOMORPHISMS of D
with no structure at all. THE SEMIGROUP in 5 can be viewed as the
semigroup of MONOMORPHISMS of D with no structure at all. THE GROUP in
6 can be viewed as the group of AUTOMORPHISMS of D with no structure
at all.

So this suggests that at least for this corner of SMAT, we often want
to FIRST identify a natural structure to each set D, and THEN form the
semigroup of homomorphisms of that structure, or the semigroup of
monomorphisms of that structure, or the group of automorphisms of that

In [1] we assume the axiom of choice throughout. We prove there that
ASSUMING the axiom of choice, If there is a non surjective solvable
equation preserving mapping on DD then |D| must be at least close to a
very rare kind of esoteric large cardinal of cofinality omega.
HOWEVER, if we do not assume the axiom of choice, then for all we
know, the following might hold:

For all sufficiently large D there is a non surjective equation
preserving mapping or even elementary embedding on DD.

And what about the semigroup of all functions from V into V? With the
axiom of choice, we know that all solvable equation preserving
mappings on this VV are surjective. However, without the axiom of
choice, the existence of a non surjective solvable equation preserving
mapping on VV is equivaelnt to the famous Reinhardt axiom j:V into V.


My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 864th 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
852: Order Theoretic Maximality/1 4/30/20 7:17PM
853: Embedded Maximality (revisited)/1 5/3/20 10:19PM
854: Lower R Invariant Maximal Sets/1:  5/14/20 11:32PM
855: Lower Equivalent and Stable Maximal Sets/1  5/17/20 4:25PM
856: Finite Increasing reducers/1 6/18/20 4 17PM :
857: Finite Increasing reducers/2 6/16/20 6:30PM
858: Mathematical Representations of Ordinals/1 6/18/20 3:30AM
859. Incompleteness by Effectivization/1  6/19/20 1132PM :
860: Unary Regressive Growth/1  8/120  9:50PM
861: Simplified Axioms for Class Theory  9/16/20  9:17PM
862: Symmetric Semigroups  2/2/21  9:11 PM
863: Structural Mapping Theory/1  2/4/21  11:36PM

Harvey Friedman

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