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Unformatted text preview: 1 THE UPPER SHIFT KERNEL THEOREMS by Harvey M. Friedman* Ohio State University October 5, 2010 DRAFT 1. INFINITE UPPER SHIFT KERNEL THEOREM. 2. FINITE UPPER SHIFT KERNEL THEOREM. 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. The induced subdigraph GA, A ⊆ V, is the digraph (A,E ∩ A 2 ). A kernel in (V,E) is commonly defined 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 the family G k (A) of all digraphs (Ak,E), k ≥ 1, such that i. ( ∀ x,y ∈ A k )(x E y → max(x) > max(y)). ii. If x,y are elements of A2k with the same order type, then x ∈ E ↔ y ∈ E. Note that for each A ⊆ Q, G k (A) is finite. The upper shift ush:Q → Q is defined by ush(q) = q+1 if q ≥ 0; q otherwise. SRP stands for "stationary Ramsey property". We say that λ 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 digraph in G(A) has a kernel containing its 2 upper shift. We will prove the Infinite Upper Shift Kernel Theorem in ACA + Con(SRP). Here SRP = ZFC + {there exists a cardinal λ with the kSRP} k . LEMMA 1. Let λ be least with the 2k+1SRP. Let f:[ λ ] k → λ obey min(A) > 0 → f(A) < min(A). There exists stationary S ⊆ λ such that f is constant on [S] k . (This can be improved with k+1 instead of 2k+1). Proof: Let λ ,f be as given. Then λ is an uncountable regular cardinal. Let g:[ λ ] k+1 → 2 be defined as follows. Let α 1 < ... < α 2k+1 . Set g( α 1 ,..., α k+1 ) = 0 if f( α 1 ,..., α k ) = f( α 1 , α k+1 ,..., α 2k ); 1 otherwise. Let E ⊆ λ \{0} be stationary, where g is constant on [E] 2k+1 ....
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 Fall '08
 JOSHUA
 Math, Set Theory, Countable set, upper shift kernel, shift kernel theorem

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