866: Structural Mapping Theory/4
Harvey Friedman
hmflogic at gmail.com
Sat Feb 13 00:47:11 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/
#113
especially section 3.
1. Special Role of Set Theoretic Foundations - Ultimate Foundations
https://cs.nyu.edu/pipermail/fom/2021-February/022480.html
2. Many sorted structures.
https://cs.nyu.edu/pipermail/fom/2021-February/022480.html
3. Categories as many sorted structures - here
4. Semigroups of Functions - here
3. CATEGORIES AS MANY SORTED STRUCTURES
NOBODY IS SUGGESTING HERE THAT ANYBODY CHANGE THEIR MATHEMATICAL
NOTATION. I AM ONLY TALKING ABOUT ULTIMATE FOUNDATIONS WHICH YOU DO
NOT HAVE TO TAKE INTO CONSIDERATION WHEN YOU DO YOUR MATHEMATICS. YOU
CAN LET FOUNDATIONALISTS LIKE ME WORRY ABOUT ALL THIS.
Putting categories under the rigid sweeping *unusually rigorous and
specific* umbrella of many sorted structures is rather typical of how
this is routinely done throughout mathematics, and does involve our
use of FREE LOGIC that we explained in SMAT/3.
A category is a many sorted structure of the form (OBJ,MORPH; ; ; dom,
cdm, id, comp). The items before the semicolon are the list of sorts.
After the semicolon, there is listed the constants, followed by a
semicolon, then the relations, followed by a semicolon, and finally
the functions, followed by ). There are only FINITELY many things
overall that are listed. There must be at least one sort listed.
In the case here, there are no constants, no relations, and four
functions. Constants always denote. Functions are always partial
functions (they may or may not be everywhere defined). The constants,
relations, and functions have to come with the relevant sorts. HERE WE
have omitted them in order to not shock the reader. Here is the full
presentation.
(OBJ,MORPH; ; ; dom(MORPH into OBJ), cdm(MORPH into OBJ), id(OBJ into
MORPH), comp(MORPH x MORPH into MORPH)
To be a category, this many sorted structure must obey a few
fundamental sentences.
a. For all morphisms x, dom(x) defined, and cdm(x) defined.
b. For all objects x, id(x) defined.
c. For all objects x, dom(id(x)) = cdm(id(x)) = x.
d. For all morphisms x,y, comp(x,y) defined if and only if cdm(x) = dom(y).
e. For all morphisms x,y,z, if cdm(x) = dom(y) then comp(x,y) is
defined and dom(comp(x,y)) = dom(x) and cdm(comp(xy)) = cdm(y).
f. For all morphisms x,y,z, comp(comp(x,y),z) =* comp(x,comp(y,z)).
g. For all objects x and morphisms y, if dom(y) = x then comp(id(x),y) = y.
h. For all objects x and morphisms y, if cdm(y) = x then comp(y,id(x)) = y.
Notice we have used "defined" and = and =* as explained in SMAT/3 with
FREE LOGIC.
There are some redundancies here. b logically follows from g.
NOTE: Unless stated otherwise, we demand that the sorts of a many
sorted structure be SETS. The class theoretic version is of course
allowing the sorts to be CLASSES. The above is therefore the
definition of a SMALL CATEGORY. Under this terminology, a CATEGORY is
a many sorted CLASS structure, which is the same as a many sorted
structure, but where we allow the sorts to be PROPER CLASSES, and also
allow the relations and functions to be PROPER CLASSES. Since the
functions in a many sorted structure are only PARTIAL functions, even
if the sorts are PROPER CLASSES, the relations and functions may all
be SETS. Constants are always sets because they are elements of sorts,
and any ELEMENT of a CLASS is a set.
NOTE: My attitude towards classes is fluid.
4, STRUCTURES OF FUNCTIONS
We now come to fitting the initial result that started all this
commotion, into this framework. I am referring of course to
[1] https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#113
especially section 3.
We could simply define SMAT in the following FAR TOO GENERAL WAY:
*) SMAT is the investigation of which many sorted structures (set or
classlike) have a non surjective mapping preserving certain basic
relationships.
Aside from not being even remotely specific concerning what
preservation is being sought after, there is no indication of what
kinds of many sorted structures should be looked at for this to be
interesting.
Now it may well be that *) is actually quite interesting and fruitful
in many contexts far more broad than anything that I have in mind
presently, or anything even remotely suggested by [1]. But at this
stage, of course we need to do a hell of a lot better than *).
We will start very close to home. Home is the semigroup of all f:D
into D under composition. This is the symmetric semigroup on D.
DEFINITION. Let phi be a first order theory of a single unary function
f:D into D. Let SG(phi,D) be the semigroup generated by taking all f:D
into D obeying phi.
QUESTION: Is there a D such that there is a non surjective solvable
equation preserving mapping of SG(phi,D)?
Clearly yes from [1] using large cardinal hypothesis I2. And for this
to always be the case for any phi, that is equivalent to I2 according
to [1]. But for particular phi, it is far from clear what the status
of this question is.
##########################################
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 866th 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
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
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
Harvey Friedman
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