KernStrrctThy082910

KernStrrctThy082910 - 1 Kernel Structure Theory by Harvey...

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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 [Fr09-10]. 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 G|A.
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2 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. What else can be used in Proposition 2.1 in addition to the
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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KernStrrctThy082910 - 1 Kernel Structure Theory by Harvey...

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