,
,
,
By clause 1 of Definition 15.3.6, d = ( E, S, D) (?x)?(G, x) hence d ?(G).-partition tree S, every function f : parts(S) ? parts(T ) and E smaller than the code of S such that properties i-ii) hold, there exists a non-empty part ? of S ,
} with c s = ( F s , T s , C s ), the set of the acceptable parts ? of T s such that (?, T s ) ? C s forms an infinite, directed acyclic graph G (F ) ,
, Suppose that F is sufficiently generic and let P and G be the generic path and the generic real, respectively. For any ? 0 1 (? 0 1 ) formula ?(G), ?(G) holds iff c s P(s) ?(G) for
, F such that c P(s) ?(G), and let ? = P(s). ? If ? ? ? 0 1 then ?(G) can be expressed as (?x)?(G, x) where ? ? ? 0
+?), so ?(G, w) holds by continuity, hence ?(G) holds. ? If ? ? ? 0 1 then ?(G) can be expressed as (?x)?(G, x) ,
G, w) holds, so ?(G) holds. path and the generic real, respectively. For any ? 0 n+2 (? 0 n+2 ) formula ?(G), ?(G) holds iff c s ?(G) for ,
, By Lemma 15.3.11, ?(G, w) holds, hence ?(G) holds. ? If ? ? ? 0 2 then ?(G) can be expressed as (?x)?(G, x) where ? ? ? 0 for every infinite k ?-partition tree S, every function f : parts(S) ? parts(T ), every w and E smaller than the code of S such that the followings hold i) (E ? , dom(S)) EM extends (F f (?) , dom(T )) for each ? < parts(S) ii) S f-refines ?<parts(S) T, ? If ? ? ? 0 2 then ?(G) can be expressed as (?x)?(G, x) where ? ? ? 0 1. By clause 1 of Definition 15.3.6, for every part ? of T such that
, ? If ? ? ? 0 n+3 then ?(G) can be expressed as (?x)?(G, x) where ? ? ? 0
The set D is / 0 ?-p.r. By Lemma 15.3.6 and since #S ? #T whenever S ? f T , D is upward-closed under the refinement relation, hence it is a promise for T. By clause 2. of Definition 15.3.6, d = ( F, T, D) (?x)¬?(G, w, x), hence d ¬?(G, w) for some w ? A. ? In case n > 0, let U = {w ? ? : (?d ? Ext(c))d ?(G, w)}. By Lemma 15.3.13 and Lemma 15.3.14, U ? ? 0 n+2 , thus U = A. Fix some w ? U?A. If w ? U A then by definition of U, there exists a condition d extending c such that d ?(G, w) ,
, By definition of a condition, the set G is R-transitive. By Lemma 15.3.5, G is infinite. By Lemma 15.3.15 and Lemma 15.3.14, G preserves non-? 0 1 definitions relative to C. By Lemma 15.3.16 and Lemma 15.3.14, G preserves non-? 0 n+2 definitions relative to C, Let P and G be the corresponding generic path and generic real, respectively
, = (?x)?(G, x) where ? ? ? 0
, Does SRT 2 2 imply IPT 2 2 over RCA 0 ?
By [Pat15e], IPT 2 2 and 2-RWKL are equivalent over RCA 0 ,
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,
, , p.117
,
,
see pseudo Ramsey's theorem R rainbow Ramsey theorem ,
,
, , p.139
, , p.138
, , p.135
, type weak weak König's lemma
,
,
,
see Ramsey's theorem RWKL see Ramsey-type weak König's lemma RWWKLsee Ramsey-type weak weak König's lemma S s-n-m theorem ,
see free set theorem Shoenfield's limit lemma ,
, G k?1 satisfying c and such that ? e i (G i ,U,V ) is essential for each i < k. Fix some x ? ?. By definition of being essential, there are some finite sets R 0 ,. .. , R k?1 > x such that for every y ? ?, there are finite sets S 0 ,. .. , S k?1 > y such that ? e i (G i , R i , S i ) holds for each i < k. Let R = R i and fix some y ? ?. There are finite sets S 0
F k?1 ? E k?1 ,Y ) is a valid extension of c for some infinite set Y ? X. Let z ? Y. In particular, by the definition of being a condition extending c, z ? X, z > max(E 0 ,. .. , E k?1 ) and F i ? E i ? {z} is pseudo-homogeneous for color i for each i < k. Therefore ?(R, S) holds, as witnessed by E 0 ,