# FOM: 80:Boolean Relation Theory

Harvey Friedman friedman at math.ohio-state.edu
Fri Mar 10 09:41:28 EST 2000

```This posting is self contained. The most recent relevant informal
disucssion on FOM was on Mon, 28 Feb 2000.

The last technical research update on this topic in an FOM posting was made
on Fri, 18 Feb 2000 10:34. Here we make small modifications in the setup,
and also state a first classfication theorem, together with a first
dimensionality theorem and a first finiteness theorem.

NOTE: The effect of Boolean relation theory on the history of mathematical
thought (whatever that proves to be!!) does not depend on establishing the
strong conjectures 12,13,14. of section 5. Conjectures 12,13 will almost
certainly eventually be
proved, but not for a while (14 is less certain). In the meantime, it is
hoped that the partial results establish
the main points and the validity of the subject, which cuts across
virtually all areas of mathematics. They also strongly suggest the truth of
at least Conjectures12,13.

PRACTICAL GOAL

A practical goal is to get this field developed by many people over the
entire mathematical landscape - not just discrete mathematics with no
structure as it is now - so that it gets an official AMS classification
number that is as broad as possible. E.g., reverse mathematics now has AMS
classification number 03B30. See http://www.ams.org/mathweb/msc2000/

I am convinced that the field of Boolean relation theory is potentially big
enough, natural enough, attractive enough, and deep enough, to support this
goal - even if it had nothing whatsoever to do with metamathematical
issues. Interviews with well known specialists in combinatorics provide
support for this.

Any feedback will be greatly appreciated.

BOOLEAN RELATION THEORY REPORT
March 10, 2000

CAUTION!!! The dust has not settled yet. We are still trying to get a sharp
improvement on the classification of 10'''), section 6, in Theorem 15. So
this has postponed the detailed checking and writing up of the proofs,
leaving the possibility of **serious errors**. Of course, the actual
classification claimed in section 6 has been written down in detail.
Nevertheless, feedback on these matters is so important that I am moving

PROMISE!!! Manuscripts with proofs will exist by the end of the calendar
year even if no further progress is made (most unlikely).

1. GENERAL CONTEXT.

Boolean relation theory is a relatively concrete subject, but we begin with
the most general formulation of the subject.

A multivariate function is a function of several variables whose domain is
of the form A^n, where A is a set and n >= 1. We do not put any restriction
on the range. An important special case is where n = 1. The arity of the
function is taken to be n (in the degenerate case where A is empty, the
arity is taken to be 1).

Let f be a multivariate function. For any set E we define fE to be the set
of all values of f at tuples from E. This is the same as the usual forward
image of f on E^n, where the arity of f is n. Nonetheless, the notation fE
is very convenient.

Let V be any set of multivariate functions whatsoever, and K be any family
of sets whatsoever. In typical cases, V will be a set of multivariate
functions all with a common domain underlying domain A (i.e., each
multivariate function has domain a Cartesian power of A), and K will be a
family of subsets of A. In fact, in many typical cases, all of the elements
of Y will be of arity 1 and have the same domain, and K will be a family of
subsets of that domain.

One can find suitable pairs (V,K) for Boolean relation theory throughout
the whole of mathematics. It is premature for me to comment much further on
this outside the discrete mathematics setting discussed below. I just
cannot give the matter any justice at this time.

A Boolean expression in k >= 1 set variables X_1,...,X_k is an expression
involving X_1,...,X_k and the set theoretic operations of union,
intersection, and complement.

A Boolean equation in X_1,...,X_k is an equation s = t, where s,t are
Boolean expressions in X_1,...,X_k.

We will be looking at Boolean equations only in the context of pairs (V,K).
In this context, the set consisting of all elements of K together with all
values of all elements of V constitute the universal set for the purposes
of taking complements. (Recall that the complement of a set is a proper
class, and not a set, and thus complementation should always be taken
relative to a universal set).

The general theory of Boolean functions and equations is a well developed
subject. For example, see the treatises

Roman Sikorski, Boolean Algebras, 3rd ed., Springer Verlag, 1967.
Sergiu Rudeanu, Boolean Functions and Equations, North-Holland, 1974.

We provide some basic information about Boolean equations. Firstly, there
are exactly 2^2^k Boolean expressions in k variables up to formal Boolean
equivalence; i.e., derivable equality from the equational axioms of Boolean
algebra. Secondly, there are also exactly 2^2^k Boolean equations in k
variables up to formal Boolean equivalence; i.e., up to logical equivalence
from the equational axioms of Boolean algebra. Thirdly, every Boolean
expression has a canonical conjunctive normal form that is provably equal
under the equational axioms of Boolean algebra. Fourthly, every Boolean
equation has the canoncial normal form t = 0, where t is a conjunctive
normal form and 0 abbreviates the intersection of a variable with its
complement. Fifthly, every finite set of Boolean equations can be combined
into a single equation which is provably equivalent from the equational
axioms of Boolean algebra.

Finally, we give the most useful and intelligible presentation of a Boolean
equation in variables X_1,...,X_k. An elementary inclusion in X_1,...,X_k
is an inclusion of the form

X_i1 intersect ... intersect X_ip containedin X_j1 union ... union X_jq

where 0 <= p,q <= k and 1 <= i1 < ... < ip <= k and j1 < ... < jq <= k and
the i's are distinct from the j's. If p = 0 then the left side is written U
for the universal set. If q = 0 then the right side is written emptyset for
the empty set.

We can write any Boolean equation in X_1,...,X_k as a set of elementary
inclusions, where it is provable in the equational axioms of Boolean
algebra (with 0,1 for emptyset,universal set) that the equation is
equivalent to the set of elementary inclusions.

This representation is not unique. However, there is an appropriate notion
of reduced set of elementary inclusions which is unique for any given
Boolean equation.

2. MAIN CLASSIFICATION PROBLEMS.

The most basic classification problem of Boolean relation theory is the
equational classification problem, which is given as follows.

Let (V,K) be given as in the previous section. Let n,m >= 1 and E be a
Boolean equation in m(n+1) variables. Determine whether the following
statement holds:

1) For all f_1,...,f_n in V there exists A_1,...,A_m in K such that E holds
of the A's and their images under the f's.

I.e.,

2) For all f_1,...,f_n in V there exists A_1,...,A_m in K such that E holds
of A_1,...,A_m,f_1A_1,...,f_1A_m,...,f_nA_1,...,f_nA_m.

As we shall see in section 3, this classication problem turns out to be
interesting even in the case n = m = 1, for some very basic (V,K).

It is also interesting to go beyond Boolean equations. A first
step in this direction is to consider sets S of Boolean equations and
inequations in m(n+1) variables. This includes the case of proper
inclusions, which is an equation and an inequation.

More generally, we can consider all propositional combinations of Boolean
equations. A Boolean inequation is a simple example of a propositional
combination of a single Boolean equation. WIth propositional combinations
of Boolean equations, one can write things like

if A intersect B is containedin C union D then (C intersect D = emptyset or
E is properly containedin F).

There is another way of going beyond Boolean equations. Recall the notation
fE for the forward image of the multivariate function f on the single
dimensional set E. We can allow, say, f[A' union B], or f applied to any
Boolean expression. Again, this is the multivariate forward image
construction as in fE. And of course, there is nothing to stop us from
considering the following general class of generalized Boolean expressions:

i) each set variable is a gBe;
ii) if s,t are gBe's then so are s union t, s intersect t, s';
iii) if s is a gBe and f is a multivariate function letter then f[s] is a gBe.

And finally we consider the generalized Boolean expression statements,
gBes, given as follows:

i) if s,t are gBe then s = t is a gBes;
ii) if phi,psi are gBes's then notphi, phi and psi, phi or psi, phi implies
psi, are gBes's.

And we can extend the equational classification problem of Boolean relation
theory in the obvious way by replacing Boolean equations with generalized
Boolean equation statements. Will that provide enough work for
mathematicians and logicians - necessarily working together - for the next
Millenium?

3. ONE DISCRETE FUNCTION AND ONE DISCRETE SET.

Let N be the set of all nonnegative integers. Throughout the remainder of
this abstract, we assume that K is the family K_0 of all infinite subsets
of N. Also V will always be a set of multivariate functions from N into N.
Thus the universal set for taking complements will always be N.

Let us first concentrate on the class V_0 of all multivariate functions
from N into N. What can we say about the equational classification problem
for (V_0,K_0)?

Let us concentrate on the situation with a single function and a single set.

There are sixteen Boolean equations, up to Boolean equivalence, to use for
the following statement:

3) For all f in V_0 there exists A in K_0 such that the given Boolean
equation holds of A,fA.

In fact, all sixteen of these problems are trivial in the following sense.
Any of these statements hold if and only if it holds with A = N.

But what about the Boolean inequations?

This is quite an interesting story. Consider the completely trivial looking
inequation

fA not= N.

The statement

4) For all f in V_0 there exists A in K_0 such that fA not= N.

is a theorem, proved using infinite Ramsey theory. Results of mine I
sketched in

H. Friedman and S. G SImpson, Issues and Problems in Reverse Mathematics,
to appear in the Proceedings of the Boulder Conference of Summer, 1999

indicate that this extremely innocent looking statement cannot be proved
without infinite Ramsey theory!!

It is not hard to determine for all sixteen of the single inequations, whether

5) For all f in V_0 there exists A in K_0 such that the given Boolean
inequation holds of A,fA.

holds. The strongest statement that comes up is:

6) For all f in V_0 there exists A in K_0 such that A union fA not= N.

The equational classification problem becomes much more interesting when we
restrict the class of multivariate functions V_0 to the subclass V_1
obeying the inequality

f(x) > |x|

where we take |x| to be the sup norm of x (i.e., the largest coordinate of
x). In the case of arity 1, we may simply write f(x) > x.

Here we come accross the extremely intriguing Boolean equation

A = (fA)'.

The statement

7) For all f in V_1 there exists A in K_0 such that A = (fA)'.

holds. The proof is by an elegant recursion, and furthermore such a set A
is always infinite and unique! And things are not any simpler if the arity
of f is assumed to be 1.

It is not hard to determine the truth values of 7) for all of the other 15
equations. There are no other nontrivial ones.

It seems possible to spend a lifetime examining the unique solution to A =
(fA)' for very elementary functions f, even of arity 1, trying to
characterize and compare them as the function f is adjusted, and also
running computer investigations on the unique solutions.

What about the inequations with one function from V_1 and one set from K?
Here all sixteen instances are fairly trivial.

4. MAIN TEST QUESTIONS.

In mathematics generally, every substantial classification problem usually
comes with test questions. These are individual statements about the
classification that follow from the classification itself. They generally
cannot be solved without doing the classification. Test questions are a way
of communicating information about the classification in digestible form,
since substantial classifications generally involve rather long and
detailed statements.

Recall the main classification problem for Boolean equations:

1) For all f_1,...,f_n in V there exists A_1,...,A_m in K such that E holds
of the A's and their images under the f's.

The first test question is called dimensionality, and it asserts the following:

8) If an instance of 1) holds using only functions in V of arity 2, then it
holds as stated.

The second test question is called finiteness, and it asserts the following:

9) If an instance of 1) holds with "A_1,...,A_m in K" replaced by
"arbitrarily large finite A_1,...,A_m", then it holds as stated.

Obviously, we can also consider these test questions for the main
classification problems for inequations, and beyond.

We do not assert that these are reasonable test questions for any choice of
(V,K). But they do make sense and are quite plausible for the cases that we
have discussed up to now in section 3 and the cases taken up later in this
abstract.

In particular, we have verified that both dimensionality and finiteness
hold for 1) where

i) n = m = 1;
ii) V is V_0 or V_1
iii) K is K_0.

In the case of Boolean inequations, we have also obtained precisely the
same result.

5. TWO FUNCTIONS AND THREE SETS.

Let f be a multivariate function from N into N. We say that f is of linear
growth if and only if there are rational constants c,d > 0 such that the
inequality

c|x| <= f(x) <= d|x|

holds with finitely many exceptions.

We say that f is of expansive linear growth if and only if there are
rational constants c,d > 1 such that inequality

c|x| <= f(x) <= d|x|

holds with finitely many exceptions.

Here | | is taken to be the sup norm, but the results will hold for any l_p
norm, 0 < p <= infinity.

There are many other natural classes of multivariate functions from N into
N for which the results claimed still hold, and for which the conjectures

We focus on the following principal classification problem, which is
a special case of our main classification problem for Boolean
equations:

10) For all multivariate functions f,g from N into N of expansive linear
growth, there exist infinite A,B,C containedin N such that the given
Boolean equation holds of A,B,C,fA,fB,fC,gA,gB,gC.

THEOREM 11. There is an instance of 10) that is provably equivalent to
the 1-consistency of Mahlo cardinals of finite order, over ACA. Conjectures
13 and 14 below each imply the 1-consistency of Mahlo cardinals
of finite order, over ACA.

CONJECTURE 12. Informally speaking, it is necessary and sufficient to use
Mahlo cardinals of all finite orders in order to determine the truth values
of all instances of 10). In particular, every instance of 10) is either
refutable in RCA_0, provable in ACA, or provably equivalent to the
1-consistency of Mahlo cardinals of finite order over ACA, with all three
possibilities present. Furthermore, in the first of the three cases, the
refutation proof can be given with fewer than 100,000 symbols in RCA_0 with
abbreviations.

We also focus the corresponding principal test questions -
dimensionality and finiteness, respectively:

CONJECTURE 13. (Dimensionality). If an instance of 10) holds with
"multivariate" replaced by "binary", then it holds as stated.

CONJECTURE 14. (Finiteness). If an instance of 10) holds with "infinite"
replaced by
"arbitrarily large finite", then it holds as stated.

We conjecture that 13 and 14 are each provably equivalent, over ACA, to the
1-consistency of Mahlo cardinals of finite order.

6. FIRST PARTIAL RESULTS (TWO FUNCTIONS, THREE SETS).

It is convenient to modify the classification problem 10) in the following
way.

10') For all multivariate functions f,g from N into N of expansive linear
growth, there exist infinite A properlycontainedin B properlycontainedin C
containedin N such that the given Boolean equation holds of
A,B,C,fA,fB,fC,gA,gB,gC.

Observe that 10') is likely to be significantly easier than 10), and is
also nearly as natural as 10). Unfortunately, we have been unable to solve
10') at this very early stage in the development of Boolean relation theory.

We need to restrict the Boolean equations considered. For this purpose, we
consider sets of Boolean equations. Recall the discussion from section 1
about Boolean equations and sets of elementary inclusions.

There are some particularly simple elementary inclusions in set variables
E_1,...,E_n. There are the disjointness conditions

E_i intersect E_j = emptyset.

And there are the binary covering conditions

E_i containedin E_j union E_k.

We now cut down 10') as follows.

10'') For all multivariate functions f,g from N into N of expansive linear
growth, there exist infinite A properlycontainedin B properlycontainedin C
containedin N such that the given finite set of disjointness and binary
covering conditions hold of A,B,C,fA,fB,fC,gA,gB,gC.

We are hopeful to be able to solve this classfication problem and prove the
associated dimensionality and finiteness conjectures. However, at this time
we only claim to be able to handle the following:

10''') For all multivariate functions f,g from N into N of expansive linear
growth, there exist infinite A properlycontainedin B properlycontainedin C
containedin N such that the given finite set of disjointness and binary
covering conditions that include "C,gC are disjoint" hold of
A,B,C,fA,fB,fC,gA,gB,gC.

THEOREM 15. Informally speaking, it is necessary and sufficient to use
Mahlo cardinals of all finite orders in order to determine the truth values
of all instances of 10'''). In particular, every instance of 10''') is either
refutable in RCA_0, provable in ACA, or provably equivalent to the
1-consistency of Mahlo cardinals of finite order over ACA, with all three
possibilities present. Furthermore, in the first of the three cases, the
refutation proof can be given with fewer than 100,000 symbols in RCA_0 with
abbreviations.

THEOREM 16. (Dimensionality). The following is provably equivalent to the
1-consistency of Mahlo cardinals of finite order over ACA. If an instance
of 10''') holds with "multivariate" replaced by "binary", then it holds as
stated.

THEOREM 17. (Finiteness). The following is provably equivalent to the
1-consistency of Mahlo cardinals of finite order over ACA. If an instance
of 10''') holds with "finite" replaced by "arbitrarily large finite", then
it holds as stated.

7. COMPLAINTS (DEFENSES).

Normal mathematicians will quickly figure out how to complain. I.e., to
defend normal mathematics against the totally unwelcome intrusion of
foundational issues.

Here are some of the most obvious complaints (defenses):

1. The main conjectures have not been established.

This is a rather weak complaint (defense) for several reasons:

a) the main conjectures - e.g., that it is necessary and sufficient to use
large cardinals to classify 10) or 10') - are likely to get proved sooner
or later (the necessity has already been established for a quite simple
instance);
b) the partial results towards the main conjectures are likely to
strengthen considerably in the near future, so much so that they should
have similar reactions to the main conjectures, especially in light of the
main conjectures;
c) especially in light of partial results, the main conjectures -
especially the necessity and sufficiency of large cardinals to classify 10)
or 10') - will be believed to be true by all set theorists and practically
all logicians. This boils down to the sufficiency of large cardinals to
resolve all questions of a certain simple and very constrained kind. This
is nearly certain.

2. An arbitrary multivariate function from N into N is not part of normal
mathematics.

It appears that the results should still hold if we restrict attention to
semilinear multivariate functions from N into N. In particular, we know
that the same simple instance still corresponds to Mahlo cardinals of
finite order - in fact, we get provable equivalence to the consistency of
Mahlo cardinals of finite order (rather than 1-consistency).

This means that the graph is a semilinear subset of the appropriate
Cartesian power of N. I.e., a Boolean combination of halfplanes defined
with integer coefficients.

3. An arbitrary semilinear function from N into N is not part of normal
mathematics. If it is going to be a function, it had better be linear, or
at least a polynomial.

We are going for linear functions with integer coefficients defined on a
finite intersection of halfplanes defined with integer coefficients. We are
also going for polynomials with integer coefficients defined on halfplanes
defined with integer coefficients.

4. An arbitrary subset of N is not part of normal mathematics.

Here the idea is that only finite subsets of N are to be fully trusted. We
plan to meet this crucial objection by considering only functions f,g from
[0,t]^k into N, where t >> k,n,c,d and f,g obey the expansive linear
inequality everywhere on [0,t]^k, and look for A,B,C containedin [0,t] with
at least n elements so that A,B,C,fA,fB,fC,gA,gB,gC obeys the given Boolean
equality. However, this does not seem to be quite enough to get outside of
ZFC.

We use the crucial concept of exponentially related sets. Let E_1,...,E_r
be subsets of N. We say that they are exponentially related if and only if
for all n >= 1, the number of elements of E_1,...,E_r in [0,n],
respectively, are within the base 2 exponential of each other.

If we always demand exponentially related A,B,C, then the purely finite
treatment in the first paragraph runs right into the consistency of Mahlo
cardinals of finite order (not 1-consistency), with explicit iterated
exponential bounds on t. In fact, we expect theorems to the effect that for
certain classes of Boolean equations, if there are solutions for
arbitrarily large t, then there are exponentially related solutions for
arbitrarily large t, even with bounds on t, and where this assertion is
equivalent to the consistency of Mahlo cardinals of finite order.

5. An arbitrary Boolean equation in 9 set variables is not part of normal
mathematics.

We never seem to need the terms fC,gA, and so we can just use
A,B,C,fA,fB,gB,gC.

6. An arbitrary Boolean equation in 7 set variables is not part of normal
mathematics.

We can just use disjointness conditions and binary covering conditions. I.e.,

A_i intersect A_j = emptyset.
A_i containedin A_j union A_k.

7. These problems were not worked on by people before Friedman made them up.

This is true of many important new theories in mathematics that are set up
to analyze clearly recognizable mathematical or physical phenomena. Boolean
relation theory is in that category. It is obvious that it could have been
a mainline investigation coming out of many branches of mathematics,
especially combinatorics.

The real question is this. Is Boolean relation theory going to be viewed as
a natural, compelling, accessible, informative, deep, exciting, well
motivated new mathematical theory cutting across virtually the whole range
of mathematics?

8. These problems were not famous celebrated open questions in mathematics
like the Riemann hypothesis.

??????????

**********

This is the 80th 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
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
46:Tamism  7/14/99  11:25AM
47:Subalgebras/Reverse Math  7/14/99  11:36AM
48:Continuous Embeddings/Reverse Mathematics  7/15/99  12:24PM
49:Ulm Theory/Reverse Mathematics  7/17/99  3:21PM
50:Enormous Integers/Number Theory  7/17/99  11:39PN
51:Enormous Integers/Plane Geometry  7/18/99  3:16PM
52:Cardinals and Cones  7/18/99  3:33PM
53:Free Sets/Reverse Math  7/19/99  2:11PM
54:Recursion Theory/Dynamics 7/22/99 9:28PM
55:Term Rewriting/Proof Theory 8/27/99 3:00PM
56:Consistency of Algebra/Geometry  8/27/99  3:01PM
57:Fixpoints/Summation/Large Cardinals  9/10/99  3:47AM
57':Restatement  9/11/99  7:06AM
58:Program A/Conjectures  9/12/99  1:03AM
59:Restricted summation:Pi-0-1 sentences  9/17/99  10:41AM
60:Program A/Results  9/17/99  1:32PM
61:Finitist proofs of conservation  9/29/99  11:52AM
62:Approximate fixed points revisited  10/11/99  1:35AM
63:Disjoint Covers/Large Cardinals  10/11/99  1:36AM
64:Finite Posets/Large Cardinals  10/11/99  1:37AM
65:Simplicity of Axioms/Conjectures  10/19/99  9:54AM
66:PA/an approach  10/21/99  8:02PM
67:Nested Min Recursion/Large Cardinals  10/25/99  8:00AM
69:Baby Real Analysis  11/1/99  6:59AM
70:Efficient Formulas and Schemes  11/1/99  1:46PM
71:Ackerman/Algebraic Geometry/1  12/10/99  1:52PM
72:New finite forms/large cardinals  12/12/99  6:11AM
73:Hilbert's program wide open?  12/20/99  8:28PM
74:Reverse arithmetic beginnings  12/22/99  8:33AM
75:Finite Reverse Mathematics  12/28/99  1:21PM
76: Finite set theories  12/28/99  1:28PM
77:Missing axiom/atonement  1/4/00  3:51PM