867: Structural Mapping Theory/5

Harvey Friedman hmflogic at gmail.com
Sun Feb 14 23:27:16 EST 2021

SMAT (structural mapping theory) continues to clarify and evolve. The
starting point that generates the excitement is
[1]  https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
especially section 3.

1. Special Role of Set Theoretic Foundations - Ultimate Foundations
2. Many sorted structures.
3. Categories as many sorted structures -
4. Semigroups of Functions -
and continue here
5. Function Systems - next posting
6. Preservation of Mappings between Many Sorted Structures - next posting

4, STRUCTURES OF FUNCTIONS - continued from

Earlier, we associated to each set D the set of all functions f:D into
D satisfying a first order sentence (i.e., (D,f) satisfies the first
order sentence) where this forms a semigroup, and then ask for non
surjective preserving mappings from this semigroup into itself. If we
take all f:D into D then this is the symmetric semigroup we
investigated in our [1]. What if it is other various semigroups?

The first order properties of unary functions is a well studied and
somewhat tame subject. So this suggests that an interesting study of
the first order sets of f:D into D that form semigroups (or various
kinds of semigroups like groups) seems worthwhile, and we have a lot
of uniformity in the choice of infinite sets D here. Like what is the
computational complexity of the set of first order properties so that
we get a semigroup (or semigroup with certain properties) on the
infinite  this way? We are not thinking of allowing any parameters in
these first order definitions, but maybe we can profitably do that?

And if we get semigroups then of course we ask our usual like [1]
about whether for some D is there a non surjective such and such
PRESERVING mapping into itself of the semigroup?

In [1] we actually go a bit further and get close to characterizing
which D have the property that the associated semigroup (in [1], the
symmetric semigroup) has the non surjective PRESERVING mapping into
itself? Can we get more exact information there?

And in [1], for the symmetric semigroups, we dug in into equation
preserving (same as monomorphisms) and SOLVABLE EQUATION PRESERVING
(the main event) and ELEMENTARY EMBEDDING (an acquired taste logicians
love to work with). We also mention in passing the obvious hierarchy
EMBEDDING mentioning the expected results which should go in tandem
from Pi_1 elementary embedding (this is I2) to Pi_2 elementary
embedding, ..., to Pi_n elementary embedding.

EQUATION PRESERVING there was dispensed with without any serious
challenge to ZFC, just regular simple math. That needs to be revisited
in this more general setting for theories of  unary functions from D
into D.

Also neglected in [1] is the proper class case of D = V = class of all
sets. As I told you, my thinking about proper classes is not fixed so
I do not automatically work with them. But here we have this:

In the investigation in [1], the nonsurjective statement for the
symmetric semigroup of class functions f:V into V, with solvable
equation preserving, implies the usual famous Reinhardt axiom called
j:V into V. And therefore over NBG + AxC we can refute my [1] for this
symmetric semigroup based on V by Kunen's inconsistency of Reinhardt
(and a lot less). HOWEVER, if you are in the mood to be working in NBG
without AxC, then j:V into V is still alive and kicking in early 2021,
and our [1] is equivaelnt over NBG to a STRONG FORM of Reinhardt.

Stepping back for the moment, you have to swallow quite a bit of
exotica to even seriously talk of the symmetric semigroup of functions
f:V into V under composition. FIrst of all, the elements of this
symmetric semigroup are themselves proper classes, and so the domain
of the semigroup is a hyperclass (class of classes). We are asking for
a nonsurjective solvable equation preserving mapping on the semigroup
into itself, and so we are asking for an object which is a
hyperhyperclass. So NBG is nowhere near strong enough to really be
honestly talking about this.

So if someone asks you what good are proper classes, hyperclasses, and
hperhyperclasses, you can tell them about this situation and also
start talking about Reinhardt's axioms on steroids.

BUT THERE is another axiom, this time even making sense over ZF, which
seems to be a much more profound extension of Reinhardt's j:V into V.
Of course, these as well as Reinhardt are very inconsistent with AxC.
This is called the Berkeley Axiom. It is actually an idea of Woodin,
who might have been willing to have it called Woodin's axiom IF HE
HADN"T ALREADY had the so called Woodin Cardinal - something
incredibly puny compared to what we are talking about here. Last I
heard, But maybe not because Woodin EXPECTS TO SHOW his Berkeley axiom
is INCONSISTENT over just ZF.

Berkeley Axiom: Every transitive set under epsilon of sufficiently
large cardinality carries a nontrivial elementary embedding into
itself. It doesn't make any difference if we replace "nontrivial" with
"nonsurjective" in this axiom.

So it would seem that

1) There is a nonsurjective solvable equation preserving mapping of
every sufficient large symmetric semigroup
2) There is a nonsurjective elementary embedding of every sufficiently
large symmetric semigroup
3) Berkeley Axiom

are all provably equivalent over ZF.


My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 867th 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
864: Structural Mapping Theory/2  2/7/21  1:07AM
865: Structural Mapping Theory/3  2/10/21  11:57PM
866: Structural Mapping Theory/4  2/13/21  12:47AM

Harvey Friedman

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