1
P
0
1
INCOMPLETENESS: finite set equations
by
Harvey M. Friedman
December 4, 2005
‘Beautiful’ is a word used by mathematicians with a semi
rigorous meaning.
We give an ‘arguably beautiful’ explicitly
P
0
1
sentence
independent of ZFC. See Proposition A from section 1.
1.
P
0
1
INDEPENDENT STATEMENT.
We use [1,n] for the discrete interval {1,...,n}.
Let A [1,n]
k
. We write A’ = [1,n]
k
\A. This treats [1,n]
k
as the ambient space.
Let R [1,n]
3k
[1,n]
k
. We define
R<A> = {y [1,n]
k
: (
$
x A
3
)(R(x,y))}.
We say that R is strictly dominating if and only if for all
x,y [1,n]
k
, if R(x,y) then max(x) < max(y).
We start with a basic finite fixed point theorem.
THEOREM 1.1. For all k,n
≥
1 and strictly dominating R
[1,n]
3k
[1,n]
k
, there exists A [1,n]
k
such that R<A’> =
A. Furthermore, A [1,n]
k
is unique.
We can obviously take complements, obtaining what we call
the ‘complementation theorem’ for R<A>.
THEOREM 1.2. For all k,n
≥
1 and strictly dominating R
[1,n]
3k
[1,n]
k
, there exists A [1,n]
k
such that R<A> =
A’. Furthermore, A [1,n]
k
is unique.
Here is a trivial modification of Theorem 1.1 with the same
proof. We call this the ‘complementation theorem’ for
R<A\{t}
k
>.
THEOREM 1.3. For all k,n,t
≥
1 and strictly dominating R
[1,n]
3k
[1,n]
k
, there exists A [1,n]
k
such that R<A\{t}
k
>
= A’. Furthermore, A [1,n]
k
is unique.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
We now incorporate the free set concept.
Let R [1,n]
3k
[1,n]
k
. We say that A is R free if and only
if A and R<A> are disjoint.
The reader will recall the following familiar ‘maximal free
set’ theorem.
THEOREM 1.4. For all k,n
≥
1, every R [1,n]
3k
[1,n]
k
has
a maximal free set.
Note that the equation in Theorems 1.2, R<A> = A’, asserts
that A is a very strong kind of free set.
THEOREM 1.5. For all k,n
≥
1, every strictly dominating R
[1,n]
3k
[1,n]
k
has a free set A such that R<A> = A’.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 JOSHUA
 Math, Set Theory, Equations, Russell's paradox, Axiom of choice, ZFC

Click to edit the document details