KernStruThm100910

# KernStruThm100910 - 1 THE UPPER SHIFT KERNEL THEOREMS by...

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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 A-digraphs. 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 A-digraph is an A,k-digraph for some k 1. Note that for each A Q, there are finitely many A,k- digraphs. A downward A-digraph (A,k-digraph) is an A-digraph (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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2 has the k-SRP 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 A-digraph 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 A-digraphs. INFINITE UPPER SHIFT KERNEL THEOREM(k) . There exists 0 A Q such that every downward A,k-digraph 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 k-SRP} k . We expect to use the ideas and techniques of the BRT book
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KernStruThm100910 - 1 THE UPPER SHIFT KERNEL THEOREMS by...

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