865: Structural Mapping Theory/3

martdowd at aol.com martdowd at aol.com
Thu Feb 11 17:01:57 EST 2021

Harvey Friedman writes

picking a specific set of counting numbers 0,1,2,... as von Neumann
ordinals, something horrendously ghastly, artificial, repulsive, and
wrong headed to most mathematicians

 There is another point of view.  Set theory is a "gift of nature" which allows us to do profound and elegant things like, as a mere exercise, give a formal definition of the natural numbers.
> So what is 1/0? The straightforward answer is that it is undefined. This
> leads to what is called FREE LOGIC which allows undefined terms.
Obviously introductory texts on fields will not resort to such drastic measures,  I only looked at one example, Lidl and Niederreitersection 1.2.  They use the notation "a^-1", only assuming that "a" is nonzero.  This suggests using two-sorted logic, one sort being the entire field and the other the non-zero elements.  Then "/" would be a function on sort 1 x sort 2.

Martin Dowd
-----Original Message-----Niederreiter
From: Harvey Friedman <hmflogic at gmail.com>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Wed, Feb 10, 2021 8:57 pm
Subject: 865: Structural Mapping Theory/3

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. Building Structures out of Structures


Set theoretic foundations of mathematics is special in that it is
based on a system of RIGID objects that do not gain their properties
from contexts. This has well known drawbacks like, e.g., actually
picking a specific set of counting numbers 0,1,2,... as von Neumann
ordinals, something horrendously ghastly, artificial, repulsive, and
wrong headed to most mathematicians. It is so bad that mathematics
professors rather quickly disavow it and say that we know what they
are and they are fluid and don't need that kind of definition, or that
these numbers only exist in a context of a structure consisting of
points with a relation among them, like a linearly ordered set where
we only care about isomorphisms between such, etcetera. You know the
drill. Of course some student raises their hand and asks what are
these objects living in these structures, where do they come from? And
the prof answers - we don't care, can come from anywhere. It nicely
rhymes, doesn't it? And it you are uncomfortable with such a thing,
then you don't really want to be a mathematician.

I don't believe there is any kind of even remotely philosophically
coherent ultimate foundational scheme that fails to be based on a
fully rigid conceptual framework. Where entities have the properties
they enjoy in light of what exactly they are, not the context in which
they are in. This ultimate foundation is, at least today, known as set
theory, with its rigid (intended to be) completely objective entities.
There is an arguably problematic extension to class theory, and this
raises a lot of unresolved questions in my mind. I'm not quite sure
what my attitude towards class theory is at this point, but there is
no question that it does offer some rather striking advantages laying
ultimate foundations for certain structures as proper classes which
are somehow quite a bit simpler to conceptualize than putting
cardinality restrictions down. That's a complex topic for another day.

So set theory starts with say emptyset and {emptyset} and this tells
us a lot. These are the most incredibly boring and lifeless
mathematical objects you can think of, enough to put a dragon to
sleep. Yet their purity, their, objectivity, their clarity, and their
rigidity, is really something to revel in.

What really makes set theoretic foundations so promising as the
ultimate foundations is the notion of hereditarily finite set - this
is the result of continuing from this boring beginning emptyset,
{emptyset}, by combining sets. Most conveniently, by taking singletons
and pairwise unions. This generates a remarkably elegant and powerful
universe of sets - the hereditarily finite sets - that have remarkably
simple and powerful properties, a tiny number of which FORMALLY DERIVE
the others. There is a lot more to be said about this, and I have said
some elsewhere, but will revisit it at a later posting.

Of course, as we know, when we really take the lid off, and throw away
any limiting axiom (that forces every set to be "finite"), then we are
into general infinitary set theory, and NO LONGER IS IT THE CASE that
a tiny number of principles logically derives the rest. E.g., power
set and union aren't derivable from more basic principles, as THEY ARE
in the hereditarily finite universe we have been talking about. BUT
there are some striking ways in which ZF and other systems are built
canonically from such finite set theories. These creates a rather
incredibly satisfying powerful edifice for an ULTIMATE FOUNDATION.

NOW NOBODY wants to have mathematics do away with its widely use
categorical language and its special attention to mathematical objects
requiring contexts to be useful and emphasizing the relationships
between structures rather than individuals in isolation and so forth.
But IT MUST BE DISCIPLINED by ultimate foundations.

Fortunately, almost entirely, this is fairly straightforward to
achieve, at least through the stuff generated by Eilenberg and Mac
Lane. For some specifics that we will need, see 4, although that will
be in a later posting. The situation seems to be far less clear cut,
and rather murky, with some more radical ideas that try to really try
to do away with the usual distinction between isomorphic and
identical. Such ideas must be discipline by our Ultimate Foundations.
Any attempt to circumvent that process requires a major
PHILOSOPHICALLY COHERENT development that needs to be fully fleshed
out and scrutinized.


There is a very natural useful and well studied concept of "structure"
that is superbly flexible and workable coming naturally out of
Ultimate Foundations. This is Many Sorted Structures. It's not that
they are part of the Ultimate Axioms we use. It's just that they are
extremely natural and directly usable.

Actually not quite. With say real numbers, we have division, and
division by zero is something that our mothers told us not to do. So
what is 1/0? The straightforward answer is that it is undefined. This
leads to what is called FREE LOGIC which allows undefined terms. There
are a few brands of this and some of them involve subtle philosophical
issues about the nature of existence and reference and so forth. I AM
NOT TALKING ABOUT ANY OF THAT HERE. I am thinking of a very
straightforward well worked out ISSUELESS version of FREE LOGIC. It
has obvious foundations.

It uses uparrows for "undefined" and downarrows for "defined" and =
for "both sides are defined and the same" and wigglequals for "both
sides are either = or both sides are  undefined". The uparrows and the
downarrows are used as suffixes to terms and form statements. It is
required that constants are always defined. Terms are defined if and
only if ALL of its subterms are defined. Relations at terms always
make sense, even if some of those terms are not defined. If at least
one of the terms is undefined, then the relation is automatically
considered to be FALSE at those terms. Quantification is normal and
quantifies over existing objects (what else could there be?).

So e.g.,
1/0 uparrows    is true
1/0 downarrows    is false
1/0 = 1/0    is false
1/0 wigglequals 1/0      is true
1/0 < 1/0    is false
1/0 <= 1/0    is false
-1 < 1/0    is false

The nice extension of predicate logic for this is well known, with
full syntax and semantics and completeness of Hilbert and Gentzen type
systems (references????).

A many sorted structure consists of at least one and at most finitely
many named sorts, each of which are sets. (The class possibility is
important and will be discussed later). There are zero or more, but
only finitely many, named constants, named relations, and named
functions. Equality on each sort is taken as given, but not across
sorts. Sorts may overlap but that is not reflected in the syntax.

Each named constant is drawn from a sort and that sort is part of its
name. Of course it may be an element of several of the sorts, but one
has to be picked for its name.

Each named relation has an arity k >= 1, and is on a list of k sorts.
It has to be a subset of the Cartesian product of these k sorts. k and
the names of those k+1 sorts are part of the name of the relation.

Each named function has an arity k >= 1, and is from a list of k sorts
into a single sort. It has to be a partial function from the Cartesian
product of the k sorts into the single sort.

I give a simple argument below that set theory is SUPREME in the sense
that it demands that everything else be ultimately interpreted in set
theory IN THE BACKGROUND, but not in any practical sense of course.
Normally there set theorists will have no issues with what you do, and
probably admire it, and feel comfortable letting you work on a leash,
provided there are no issues about how to generally routinely
interpret what you are doing in the standard ways set theoretically.
HOWEVER, once that becomes murky, then a big issue arises that needs
to be addressed. And the SUPREME COURT is set theory. Our attitude
towards INSURRECTIONS is actually rather positie. We seek not to
imprison you, or throw you into internment camps, but rather HELP YOU
SEE how to hopefully routinely adjust what you are doing so that it is
clear there are no longer any issues to give a proper set theoretic
interpretation. Sometimes this requires the experience and expertise
of a standard classical foundations expert who knows the incredible
sometimes hidden power of set theory to do all sorts of things its
founders never considered. Statements by various cultural heros, dead
or alive, that set theory foundations is insufficient for this or that
or cannot forge tools naturally to do this or that are HIGHLY SUSPECT,
and generally come from people not properly schooled in many decades
of classical set theoretic foundational thought.

The reason that set theory is THE SUPREME COMMANDER, and THE SUPREME
COURT, is that it is founded on RIGIDITY. The emptyset and say
{emptyset} and the like, the most lifeless boring things imaginable,
have an immutable unchanging forever existence where my emptyset and
my {emptyset} are the same as yours, and your newborn baby's also. If
my roof collapses or I get fired or a get a promotion, then my
emptyset and {emptyset} is not going to be affected at all. There is
no RELATIVITY in set theory.

There is a mathematical fact that is at the heat of the support for
this RIGIDITY. And this is that for sets under membership, all
automorphisms are the identity. Yes I know this makes things go
against the grain of a lot of practical mathematical thinking that
facilitates our making mathematical advances such as no x^n + y^n =
z^n, n >= 3. No major problem in interpreting that fact set
theoretically, but things get seriously anti set theoretic in spirit
when rigorously documenting the proof. That's fine with the SUPREMES.
If that facilitates real mathematical progress and rigor, then all the
better. But don't forget who the SUPREMES are.

This RIGIDITY has spawned a way of thinking of its own, quite
different than what core mathematics relishes in, that has given us
INTEREST. And probably only VERY CLUMSY hints of what is to come even
by 2100, a blink of an eye. It is preposterous to believe that these
VERY CLUMSY but incredible advances would be at all possible without
the vast technologies that have been developed from the RIGID POINT OF


The SUPREME COURT has ruled that all structures are to ultimately be
interpreted as a MANY SORTED RELATIONAL STRUCTURE. The Supremes have
not yet ruled on the validity of the notion of class, generally
considered to be set like entities which are too big to be sets.
Proper classes are flagged as under investigation, something in limbo.
There have been no rulings against them. The Court much prefers you
pick a suitable cardinal lambda and work within the cumulative
hierarchy below lambda rather than think you are working with all
sets. The Supremes also urge you to consider using large cardinal
hypotheses, of course labeling them as such, as they seem to be
becoming so useful and INHERENTLY NECESSARY in a growing list of basic
normal type rich situations.

Many sorted relational structures some with one or more sorts, but at
most finitely many sorts. The sorts have names, and are sets, possibly
overlapping. There are also named constants, relations, and functions.
There are finitely many of these too. Each constant is given a unique
sort and is an element of that sort. Even if it is an element of more
than one sort, it is still assigned a particular sort. The named
relations are assigned an arity n, and n sorts S_1,...,S_n. R is
assigned a subset of S_1 x ... x S_n. The named functions are also
assigned an arity n, and n+1 sorts S_1,...,S_n+1. F is assigned a
function from S_1 x ... x S_n into S_n+1.

The signature of a many sorted structure consists of its shape, which
is the list of named sorts, the list of named constants and their
assigned sorts, the list of named relations and their assigned sorts
for their arguments, and the list of named functions and their
assigned sorts for their arguments and values. Thus the signature is a
finite object.

For each sort S there are variables v_i^S, i >= 1, that range over S.
Terms and their sorts are built up by induction. The atomic formulas
are the equations between terms of the same sort, and primitive
relations at the right number of terms of the right sorts. Formulas
are built up in the usual way through the connectives not, and, or, if
then, if and only if, and quantification over each sort.


My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 865th 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

Harvey Friedman
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