1
THE UPPER SHIFT KERNEL THEOREMS
by
Harvey M. Friedman*
Ohio State University
October 9, 2010
DRAFT
1. INFINITE UPPER SHIFT KERNEL THEOREM.
2. FINITE UPPER SHIFT KERNEL THEOREM.
3. TEMPLATES.
1. INFINITE UPPER SHIFT KERNEL THEOREM.
Here we sketch a proof of the Infinite Upper Shift Kernel
Theorem from a suitable large cardinal assumption. The
reversal will be available later.
A digraph is a pair G = (V,E), where V is a nonempty set of
vertices and E
⊆
V
2
is a set of edges. We say that G is on
V.
A kernel in (V,E) is commonly defined in graph theory as a
set S
⊆
V such that
i. No element of S connects to any element of S.
ii. Every element of V\S connects to some element of S.
We now fix A
⊆
Q. We study a fundamental class of digraphs
associated with A, which we call the Adigraphs. An A,k
digraph is a digraph (A
k
,E), where E is an order invariant
subset of A
2k
in the following sense. For all x,y
∈
A
2k
, if
x,y have the same order type then x
∈
E
↔
y
∈
E.
An Adigraph is an A,kdigraph for some k
≥
1.
Note that for each A
⊆
Q, there are finitely many A,k
digraphs.
A downward Adigraph (A,kdigraph) is an Adigraph (A,k
digraph) where for every edge (x,y), max(x) > max(y).
The upper shift ush:Q
→
Q is defined by ush(q) = q+1 if q
≥
0; q otherwise. This lifts to ush:Q
k
→
Q
k
by acting
coordinatewise. We can now use ush to give forward images
of subsets of Q
k
.
SRP stands for "stationary Ramsey property". We say that
λ
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has the kSRP if and only if
λ
is an infinite cardinal, and
every f:[
λ
]
k
→
2 is constant on some [S]
k
, where S is a
stationary subset of
λ
. Here [S]
k
is the set of all
unordered k tuples from S.
INFINITE UPPER SHIFT KERNEL THEOREM
. There exists 0
∈
A
⊆
Q
such that every downward Adigraph has a kernel containing
its upper shift.
Note that we are talking about Con(SRP), rather than 1
Con(SRP).
Also, A
⊆
Q can be taken to be recursive in the jump, as
well as the sequence of kernels for the Adigraphs.
INFINITE UPPER SHIFT KERNEL THEOREM(k)
. There exists 0
∈
A
⊆
Q such that every downward A,kdigraph has a kernel
containing its upper shift.
There is a small k such that the Infinite Upper Shift
Kernel Theorem(k) also has the same metamathematical
properties. I.e., is also provably equivalent, over ACA
0
, to
Con(SRP). Thus we do NOT get a hierarchy on the dimension
k.
It remains to give a small k. I have been postponing this
kind of investigation for some time, waiting for the
independent statements to stabilize.
We will now prove the Infinite Upper Shift Kernel Theorem
in ACA
0
+ Con(SRP). Here SRP = ZFC + {there exists
λ
with
the kSRP}
k
.
We expect to use the ideas and techniques of the BRT book
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 Fall '08
 JOSHUA
 Math, Set Theory, Countable set, Qk, upper shift kernel, shift kernel theorem

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