Double Recursion Theorems

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II. Pseudo-uniform Reducibility 133Theorem d (Th. 13, Ch. 2). // all recursive sets are definablein S and S is consistent, then not only is S undecidable, but the pair(P,R) is recursively inseparable.The above theorems combine notions of recursive function theorywith those of first-order systems. We shall now obtain generalizationsof them of a purely recursive function theoretic nature and wewill also show that the conclusions of <strong>Theorems</strong> C\ and Ci hold underweaker hypotheses. The results that follow are those of Smullyan[1963 A].§3. Pseudo-uniform Reducibility. We know that if everyr.e. set is reducible to a set a, then a must be non-recursive (in facteven generative). Suppose that every recursive set is reducible to a.Does it necessarily follow that a must be non-recursive? Certainlynot, for if a is any non-empty set whose complement is non-empty,then every recursive set A is reducible to a as follows: Take anynumber a\ in a and any number a? not in a and define g(x) = ai ifx 6 A, and g(x) = a% if x £ A. Since A is recursive, so is the functiong(x) (why?) and for all a;, x € A «-»• g(x) € a, so g(x) reduces A toa. Thus, it is not true that if all recursive sets are reducible to a,then a must be non-recursive.On the other hand, as we saw in Part I of this chapter, if thecollection of all recursive sets is uniformly reducible to a, then ais not only non-recursive, but even generative. Thus, to establishthe non-recursivity of a set a, the hypothesis that all recursive setsare reducible to a is too weak, whereas the hypothesis of uniformreducibility is stronger than necessary (as we will see). And so weturn to a notion of intermediate strength.We will say that a collection C of number sets is pseudo-uniformlyreducible to a if there is a recursive function g(x,y)—which we calla pseudo-uniform reduction of C to a—such that for any set A in C,there is some number i such that g(i, x) (as a function of x) reducesA to a (i.e. for all x, x € A «-» g(i,x) 6 a). If C is a collection of r.e.sets, our definition does not require that for any index i of a memberA of C, g(i,x) reduces A to a, but only that there is some numberi (not necessarily even an index of A) such that g(i,x) reduces A toa. Thus, the notion of pseudo-uniform reducibility is much weakerthan uniform reducibility. We shall now show that if the collectionof all recursive sets is pseudo-uniformly reducible to a, then a mustbe non-recursive. We first show the following stronger fact.
134 Chapter XI. Three Special TopicsTheorem 3. A sufficient condition for a set a to be non-recursiveis that there is a recursive function f(x) such that for every recursiveset A, there is at least one number i such that i
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Recursion Theory forMetamathematics
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To Blanche
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xiiContentsII Undecidability and Re
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xivContents§2. Weak Double Product
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2 Chapter 0. Prerequisitesinfinite
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4 Chapter 0. PrerequisitesMore spec
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6 Chapter 0. PrerequisitesA, then t
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8 Chapter 0. PrerequisitesOne also
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In G.I.T. we gave a proof of the fo
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12 Chapter 0. PrerequisitesTheorem
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14 Chapter 0. Prerequisitesk ^ k +
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16 Chapter 0. Prerequisitesthe prov
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18 Chapter 0. PrerequisitesTheorem
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20 Chapter 0. Prerequisitesshowed t
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22 Chapter 0. PrerequisitesEh(n,h}
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Chapter IRecursive Enumerability an
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26 Chapter I. Recursive Enumerabili
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Chapter IIUndecidability and Recurs
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40 Chapter II. Undecidability and R
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44 Chapter II. UndecidabiHty and Re
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Chapter IIIIndexingFor the remainin
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50 Chapter III. Indexingsynonymous
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52 Chapter III. IndexingTheorem 2
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54 Chapter III. Indexingsuch that f
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56 Chapter III. Indexingsive functi
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Chapter IVGenerative Sets and Creat
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60 Chapter IV. Generative Sets and
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68 Chapter V. Double Generativity a
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Page 177 and 178: 160 References[14] Raymond M. Smull
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Double Recursion Theorems

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