1
EQUATIONAL BOOLEAN RELATION THEORY
by
Harvey M. Friedman
Ohio State University
[email protected]
http://www.math.ohiostate.edu/~friedman/
9/03/02
NOTE: THIS IS ONLY THE FIRST THREE SECTIONS, WHICH CONTAINS
THE PROOF FROM LARGE CARDINALS.
Abstract. Equational Boolean Relation Theory concerns the
Boolean equations between sets and their forward images under
multivariate functions. We study a particular instance of
equational BRT involving two multivariate functions on the
natural numbers and three infinite sets of natural numbers.
We prove this instance from certain large cardinal axioms
going far beyond the usual axioms of mathematics as
formalized by ZFC. We show that this particular instance
cannot be proved in ZFC, even with the addition of slightly
weaker large cardinal axioms, assuming the latter are
consistent.
1. EQUATIONAL BOOLEAN RELATION THEORY.
Equational Boolean Relation Theory (equational BRT) concerns
the Boolean equations between sets and their images under
multivariate functions. We formally present equational BRT in
this section.
To be fully rigorous, we need to distinguish, say, a function
of two variables from A into A and a function of one variable
from A
2
into A. For this reason, we use the following
definition of multivariate function.
A multivariate function is a pair f = (g,k), where k
≥
1 (the
arity of f) and dom(g) is a set of ordered ktuples. We put
no other restrictions on g.
We define dom(f) to be dom(g), and write f(x
1
,...,x
k
) =
g(<x
1
,...,x
k
>), where <x
1
,...,x
k
> is the ordered ktuple with
coordinates shown. In practice, we need not be so careful
about multivariate functions.
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Equational BRT is based on the following crucial notion of
forward image. Let f be a multivariate function and A be a
set. We define
fA = {f(x
1
,...,x
k
): k is the arity of f and x
1
,...,x
k
A}.
We could write f[A
k
] for this forward image construction, but
it is particularly convenient to suppress the arity and write
fA. In this way, f defines a special kind of operator from
sets to sets.
A BRT setting is a pair (V,K), where V is a set of
multivariate functions and K is a set of sets. Typically, V
and K are naturally related so that one is interested in the
forward images of elements of V on elements of K, although in
equational BRT, no restrictions are placed on the choice of
V,K.
We use N for the set of all nonnegative integers. We say that
f is a multivariate function from A into B if and only if f
is a multivariate function with dom(f) = A
k
and rng(f) B,
where the arity of f is k.
We use MF(A,B) for the set of all multivariate functions from
A into B, and MF(A) for the set of all multivariate functions
from A into A.
We use S(A) for the set of all subsets of A, and INF(A) for
the set of all infinite subsets of A.
We say that f MF(N) is strictly dominating if and only if
for all x dom(f), f(x) > max(x).
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 Fall '08
 JOSHUA
 Math, Equivalence relation, Lemma, Order type, equational BRT

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