1
Kernel Structure Theory
by
Harvey M. Friedman
Ohio State University
August 29, 2010
ABSTRACT
One of the major directions of research not touched on in
the BRT book is
to find an explicitly
Π
0
1
sentence which can only be proved
using large cardinals, and which arguably represents clear
and compelling information in the finite mathematical
realm.
We have been recently engaged in this search, and have
announced a long series of successively simpler and more
convincing examples. See [Fr0910].
We present a recent version of this work in progress.
1. Digraphs, Kernels, Downward, Order Invariance.
2. Kernel Closure Theorem.
3. Kernel Closure Templates.
4. Kernel Tower Theorems.
5. Kernel Tower Templates.
6. Towards Unification with BRT.
1. Digraphs, Kernels, Downward, Order Invariance.
A digraph is a pair G = (V,R), where R
⊆
V
×
V. The
vertices of G are the elements of V, and the edges of G are
the
elements of R. We say that G is a digraph on V.
We say that E is independent in G if and only if E is a
subset of V, where there is no edge from any element of E
to any element of E.
We say that E is a kernel in G if and only if E is
independent in G, and for all x in V\E, there is an edge
from x to some element of E.
A kernel of A in G is a kernel of the induced subdigraph
GA.
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Let Q be the set of all rational numbers. We say that a
digraph on Q
k
is downward if and only if for all edges
(x,y), max(x) > max(y).
We say that x,y
∈
Q
k
are order equivalent if and only if for
all 1
≤
i,j
≤
k, x
i
< x
j
↔
y
i
< y
j
.
We say that A
⊆
Q
k
is order invariant if and only if for
all order equivalent x,y
∈
Q
k
, x
∈
A
↔
y
∈
A.
A digraph on Q
k
is order invariant if and only if its edge
set is order invariant as a subset of Q
2k
.
2. Kernel Closure Theorem.
Let A
⊆
Q
k
. The upper shift of A is the set of all y
∈
Q
k
obtained by starting with some x
∈
A, and adding 1 to all
nonnegative coordinates of x.
PROPOSITION 2.1. Kernel Closure Theorem. In every downward
order invariant digraph on Q
k
, the kernel of some (E
∪
{0})
k
contains its upper shift.
THEOREM 2.2. Proposition 2.1 is provable in SRP
+
but not
from any consequence of SRP that is consistent with RCA
0
.
Proposition 2.1 is provably equivalent, over WKL
0
, to
Con(SRP). These results hold if we fix k to be any
particular sufficiently large integer in Proposition 2.1.
3. Kernel Closure Templates.
Note how the Kernel Closure Theorem is based on the
functions ush:Q
k
→
Q
k
, where ush(x) is the result of adding
1 to all nonnegative coordinates of x.
What else can be used in Proposition 2.1 in addition to
ush?
We can of course view the one dimensional upper shift ush:Q
→
Q as fundamental here, where ush:Q
k
→
Q
k
is obtained by
ush acting on the coordinates.
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 Fall '08
 JOSHUA
 Math, Empty set, Qk, Kernel Closure Theorem, order invariant digraph

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