FOM: 46:Tamism

Wed Jul 14 06:25:20 EDT 1999

This is the 46th in a series of self contained postings to fom covering a
wide range of topics in f.o.m. Previous ones are:

1:Foundational Completeness   11/3/97, 10:13AM, 10:26AM.
2:Axioms  11/6/97.
3:Simplicity  11/14/97 10:10AM.
4:Simplicity  11/14/97  4:25PM
5:Constructions  11/15/97  5:24PM
6:Undefinability/Nonstandard Models   11/16/97  12:04AM
7.Undefinability/Nonstandard Models   11/17/97  12:31AM
8.Schemes 11/17/97    12:30AM
9:Nonstandard Arithmetic 11/18/97  11:53AM
10:Pathology   12/8/97   12:37AM
11:F.O.M. & Math Logic  12/14/97 5:47AM
12:Finite trees/large cardinals  3/11/98  11:36AM
13:Min recursion/Provably recursive functions  3/20/98  4:45AM
14:New characterizations of the provable ordinals  4/8/98  2:09AM
14':Errata  4/8/98  9:48AM
15:Structural Independence results and provable ordinals  4/16/98
10:53PM
16:Logical Equations, etc.  4/17/98  1:25PM
16':Errata  4/28/98  10:28AM
17:Very Strong Borel statements  4/26/98  8:06PM
18:Binary Functions and Large Cardinals  4/30/98  12:03PM
19:Long Sequences  7/31/98  9:42AM
20:Proof Theoretic Degrees  8/2/98  9:37PM
21:Long Sequences/Update  10/13/98  3:18AM
22:Finite Trees/Impredicativity  10/20/98  10:13AM
23:Q-Systems and Proof Theoretic Ordinals  11/6/98  3:01AM
24:Predicatively Unfeasible Integers  11/10/98  10:44PM
25:Long Walks  11/16/98  7:05AM
26:Optimized functions/Large Cardinals  1/13/99  12:53PM
27:Finite Trees/Impredicativity:Sketches  1/13/99  12:54PM
28:Optimized Functions/Large Cardinals:more  1/27/99  4:37AM
28':Restatement  1/28/99  5:49AM
29:Large Cardinals/where are we? I  2/22/99  6:11AM
30:Large Cardinals/where are we? II  2/23/99  6:15AM
31:First Free Sets/Large Cardinals  2/27/99  1:43AM
32:Greedy Constructions/Large Cardinals  3/2/99  11:21PM
33:A Variant  3/4/99  1:52PM
34:Walks in N^k  3/7/99  1:43PM
35:Special AE Sentences  3/18/99  4:56AM
35':Restatement  3/21/99  2:20PM
36:Adjacent Ramsey Theory  3/23/99  1:00AM
37:Adjacent Ramsey Theory/more  5:45AM  3/25/99
38:Existential Properties of Numerical Functions  3/26/99  2:21PM
39:Large Cardinals/synthesis  4/7/99  11:43AM
40:Enormous Integers in Algebraic Geometry  5/17/99 11:07AM
41:Strong Philosophical Indiscernibles
42:Mythical Trees  5/25/99  5:11PM
43:More Enormous Integers/AlgGeom  5/25/99  6:00PM
44:Indiscernible Primes  5/27/99  12:53 PM
45:Result #1/Program A  7/14/99  11:07AM

1. TAMISM

Number theory, finite algebra, and finite combinatorics provide a wealth of
theorems that could be subject to the reverse mathematics idea but which
are readily provable in RCA_0 and so cannot be analyzed under the present
framework.

Incidentally, this remark is believed to apply to many modern developments;
e.g., we expect that the Faltings and Wiles theorems can be proved in
RCA_0, but nobody seems to have verified this.

Basically, there is almost no reverse mathematics of arithmetical sentences
from standard mathematics. [There are Kruskal and related theorems, and the
new 1998 ³perfect² finite forms - another story for another day.]

We attack this by proposing reverse *arithmetic*. This involves a reduction
(or alteration) of the primitives, weakening of the axioms, and a
restricting (or refocusing) of the subject matter analyzed. This is
discussed in my talk at the Boulder meeting; there will be a paper for its
Proceedings. In this posting, I will confine myself to what I call
**severe** reverse arithmetic** and a doctrine I call "tamism."

An interesting and devastating suggestion is:

"that meta-mathematical phenomena uncovered by reverse math is an artifact
of choice of base theory; the base theory
injects a math logic slant that is foreign to the spirit of math; the
present way of formalizing math is not sufficiently tied to the math itself
so that mathematically irrelevant issues come up such as what formulas
should induction be applied to; the base theory does not consist entirely
of "essential" math; math might be formalized in a more subtle and relevant
way so that a consistency proof can be given in weak fragments of
arithmetic such as EFA (exponential function arithmetic)."

We call this interesting position "tamism".

The case for tamism is bolstered by the success of the program of creating
so called "tame" systems which formalize much important math in a quite
different way, which do allow consistency proofs of the kind that Hilbert
envisioned. Two of the most famous examples are that of Presburger
arithmetic and real closed fields. By the way, I am involved in the tame
systems and structures program (some aspects jointly with Chris Miller).
NOTE: I now claim that the metamathematics of real closed fields can be
carried out in EFA, including a consistency proof. See forthcoming posting.

Tamism in the extreme would contend that all (interesting) mathematics can
be cast in a single demonstrably consistent tame system, thus skirting
virtually the entire mainline development of mathematical logic of the 20th
century, and in particular reverse mathematics as currently practiced.

It is imperative that we look for absolute refutation of tamism that leave
no wiggle room. DOWN WITH TAMISM!!

In an embodiment of reverse mathematics, the level of necessity of the
mathematics formalized in the base theory is what we call its ³severity².

Current reverse mathematics is distinctly not severe on the choice of
axioms for the base theory RCA_0. My original formulation of the axioms for
RCA_0 is considerably more severe than the logically equivalent formulation
in Simpson¹s book. Simpsonís presentation is of course more convenient.

However, even my original axioms for RCA_0 are not severe enough. There is
always wiggle room about what is absolutely essential about infinitary
objects. And the severity level of the choice of theorems analyzed in
reverse mathematics is also subject to wiggle room; e.g., "arbitrary
continuous functions on I are too general for modern mathematics?!"

So for an absolute refutation of tamism, we look to finite mathematics -
especially number theory and finite combinatorics.

2. REFUTING TAMISM. SEVERE REVERSE ARITHMETIC

The base theory for severe reverse arithmetic will be the axioms of
discrete ordered (commutative) rings (with unit) in the language
0,1,+,-,mult,=,<.

NOTE: In order to develop a smooth theory of reverse arithemetic - as
opposed to **severe** reverse arithmetic - we may well need a somewhat
bigger base theory. It took some time for it to be clear that my RCA_0 was
the "right" base theory for the present (first) incarnation of reverse
mathematics. It's not too strong and not too weak. Further incarnations of
reverse mathematics will involve weaker base theories (or base theories
with different primitives). Similarly, the "right" base theory for the
first incarnation of reverse arithmetic - not too strong and not too weak -
will take some experimentation. This matter will be discussed in the
Proceedings of the Boulder meeting.

We are going to refer to EFA (exponential function arithmetic) as a
benchmark, with 0,S,+,mult,=,<, base 2 exp, successor axioms, defining
equations, definition of <, and induction with respect to all bounded
formulas in the language.

EFA can easily be reformulated in the context of discrete ordered rings.

We need to find some essential facts about the integers (and associated
finite objects) - universally respected and loved - which when added to the
axioms of discrete ordered rings imply EFA. With EFA in hand, we can go
further in many directions to get even stronger systems, burying tamism
further.

1. Axioms for the discrete ordered ring, (Z,0,1,+,-,mult,=,<).
2. Axioms for vectors of finite length from Z, equality, length, and
picking out coordinates: Every integer is a vector of length 1. Vectors of
the same length are equal if and only if their coordinates are equal. There
is a zero vector of every length. Vectors of the same length can be added,
substracted, and multiplied by scalars. We can append any integer to any
vector and obtain a vector of one higher length. We can delete the last
coordinate of any vector of length > 1 and obtain a vector of one smaller
length. We can change any coordinate of any vector and obtain a vector of
the same length.
3. Axioms for linear transformations from vectors of one length into
vectors of another length: The constantly zero functions are linear
transformations. Linear transformations of the same dimensions can be added
and subtracted. Linear transformations can be multiplied by scalars. Linear
transformations with matching dimensions can be composed. For any linear
transformation, there is a unique linear transformation which agrees with
it at all unit basis vectors except one, where its value is reassigned.
4. Axioms for finite sets of vectors, all of the same dimension: Every
singleton is a finite set of vectors. Finite sets of vectors are closed
under pairwise union, pairwise intersection, set theoretic difference, and
Cartesian product. The translation by a vector of any finite set of vectors
is a finite set of vectors.
5. Axioms for the unit basis vectors: Every finite set of vectors in a
given dimension is the forward image of the set of unit basis vectors in
some dimension under a linear transformation on that dimension. The forward
image of the set of unit basis vectors under a linear transformation is a
finite set of vectors. (NOTE: This forward image is the set of rows in the
matrix of the linear transformation).
6. Any finite set of nonzero integers has a common multiple.
7. For all integers n >= 1, {k: 1 <= k <= n} and {k^2: 1 <= k <= n} exist
as finite sets.
8. Every nonempty finite set of integers has a least element.

THEOREM 1. EFA and 1-8 are interpretable in each other, and prove the same
sentences in the language of ordered rings.

Dimitracopoulos has proved the MRDP theorem in EFA, and this allows some
nice simple additions to 1-8 that climb up to Simga_1 induction. E.g.,

9. Axioms for polynomnials (as funtions) from vectors of a fixed length
into integers. Constant and projection functions are polynomials.
Polynomials can be added, substracted, and multiplied.
10. Every polynomial takes on a value of least magnitude.

THEOREM 2. Sigma_1 induction and 1-10 are interpretable in each other, and
prove the same sentences in the language of ordered rings.

There are much simpler ways to extend the discrete ordered ring axioms to
get strength if we are willing to be less severe. This is discussed in my
talk at the Boulder meeting, June 13-17, 1999. It will be posted on my
website soon, and there will be a manuscript submitted for the conference
proceedings.