1
LECTURE NOTES ON BABY BOOLEAN RELATION THEORY*
Harvey M. Friedman
Department of Mathematics
Ohio State University
[email protected]
December 4, 2000
October 3, 2001
Abstract. This is an introduction to the most primitive form
of the new Boolean relation theory, where we work with only
one function and one set. We give eight complete
classifications. The thin set theorem (along with a slight
variant), and the complementation theorem are the only
substantial cases that arise in these classifications.
INTRODUCTION.
Let f be a multivariate function of arity k. Let A be a set.
We write fA for f[A
k
].
Z = set of all integers, N = set of all nonnegative integers.
THEOREM. Let f be a multivariate function from Z into Z.
There exists an infinite A Z such that ????.
QUIZ: Find four common mathematical symbols so that this is
true and highly nontrivial.
Here is the answer to the quiz.
THIN SET THEOREM. Let f be a multivariate function from Z
into Z. There exists an infinite A Z such that fA ≠ Z.
DIGRESSION: TST is provable in ACA
0
’ = RCA
0
+ "for all x,n,
the nth jump of x exists". We have shown that TST is not
provable in ACA
0
. See [FS00] or the proof below, which is
better. It is open whether RCA
0
proves TST
ACA
0
’, or even
whether RCA
0
proves TST
Æ
ACA
0
. RCA
0
cannot prove TST
ACA
0
since ACA
0
is not finitely axiomatizable. We have also shown
that TST for k = 2 is not provable in WKL
0
. See [CGH00].
Proof of TST: Use RT = Ramsey’s theorem. Wlog, replace Z by
N. Let p be the number of order types of ktuples. Set H(x) =
f(x) if f(x)
£
p; p+1 otherwise. Let A N be infinite and H
homogeneous in the sense that the value of H at tuples from A
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depend only on their order types. Then the values of f at
tuples of any given order type from A are either all the same
number
£
p, or all > p. In any event, at least one of the
numbers {0,.
..,p} is omitted by f on A. QED
We can immediately ask what else can be put there other than
the Boolean inequation fA ≠ Z. Think of Z as the universal
set.
This is what BRT = Boolean relation theory is all about. In
these lecture notes, we only consider “baby” BRT, where we
have just one function and one set.
1. SOME BOOLEAN ALGEBRA.
The notion of Boolean algebra is a very robust concept with
several different definitions. We give a particularly elegant
definition.
By way of motivation, the clearest examples of Boolean
algebras are the Boolean fields of sets. These are structures
(W,
∅
,S, ,
«
,’), where W is a family of subsets of the set S
closed under pairwise union, pairwise intersection, and
complement relative to S. Here ’ is complementation relative
to S.
We let
2
be the structure ({0,1},0,1,+,•,), where x+y =
min(1,x+y), x•y is multiplication, and x = 1x.
A Boolean algebra is a structure B = (B,0,1,+,
•
,), where 0,1
B, +,•:B2
Æ
B, :B
Æ
B, such that every equation that
holds universally in
2
also holds universally in B.
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 Fall '08
 JOSHUA
 Math, Set Theory, Boolean Algebra, Propositional calculus, Laws of Form

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