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		It is easy to check that . Using the n-ary recursion theorem (Theorem 2.5), show that . Then generalize this argument to show that .

		10-10 The purpose of this exercise is to extend Aex and Uaex paradigms from Chapter 7 to incorporate both Ap and Uap types of additional information. We illustrate this extension by defining paradigm below. First recall from Chapter 7 that the asymptotic agreement between two partial functions  h  and  q  (denoted: aa( h , q )) is and the asymptotic disagreement between  h  and  q  (denoted: ad( h ,  q )) is 1 - aa( h ,  q ).

		10.53 Definition Let dl, .

		(a) A scientist just in case, for each p with  j p that is d1-conforming with f, we have that and .

		(b)

		Similarly to the above, define , , and paradigms. Give simple arguments for the following.

		(a) For all , .

		(b) For all , .

		10-11 Show that, for all d1 > 0 and all d2, d3 with d2 + d3 < 1, .

		Hint: Without loss of generality, assume that dl = 2/m, d2 = l/m, d3 = (m - l - 1)/m for integers and . Let n0 = 0. For each , let n2i+1 = mi+1 + n2i, and n2i+2 = n2i+1 + (i + 1). · m. Let . Then, use C, the class of functions , each of which satisfy:

		1. For all , f(x) = 0.

		2. For all j and all , f(x) = f(j).

		10-12 Suppose d1, d2, and d3 are real numbers such that and d2+d3 < 1. Use a technique similar to the one of Exercise 10-11 to show that . Also, verify that this result implies .