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Location: Chapter 24, solutions to Problems 6 and 7

Problem 6

It is (formula given in the solution)

... such that for every \(n\) and \(m\), \(A\overline{n}\,\overline{m}=(Z\,\overline{m})\overline{0}((\oplus)(A(\overline{n}(P\,\overline{m}))\overline{n}))\). Again, such a bird \(A\) can be ...

It should become

... such that for every \(n\) and \(m\), \(A\overline{n}\,\overline{m}=Z\,\overline{m}\overline{0}(\oplus(A\overline{n}(P\,\overline{m}))\overline{n})\). Again, such a bird \(A\) can be ...

Problem 7

It is (two expressions to correct)

Note: the exponentiating bird is represented in the following with the symbol \(\varepsilon\).

... for any positive number \(m\), \(\varepsilon\,\overline{n}\,\overline{m}=\otimes(\varepsilon\,\overline{n}\,\overline{m})\overline{n}\). Equivalently we want a bird \(\varepsilon\) such that for all \(n\) and \(m\), \(\varepsilon\,\overline{n}\,\overline{m}=Z\,\overline{m}\overline{1}(\otimes(\varepsilon\,\overline{n}\,\overline{m})\overline{n})\). Again, such a bird \(\varepsilon\) can be ...

It should become

... for any positive number \(m\), \(\varepsilon\,\overline{n}\,\overline{m}=\otimes(\varepsilon\,\overline{n}(P\,\overline{m}))\overline{n}\). Equivalently we want a bird \(\varepsilon\) such that for all \(n\) and \(m\), \(\varepsilon\,\overline{n}\,\overline{m}=Z\,\overline{m}\overline{1}(\otimes(\varepsilon\,\overline{n}(P\,\overline{m}))\overline{n})\). Again, such a bird \(\varepsilon\) can be ...

Short explanation

In the correction to Problem 6, besides a few redundant parentheses that can go away just for ease of reading, the crucial point is the term \(A\overline{n}(P\,\overline{m})\): with the formula given in the text, \(A\) would be applied, incorrectly, to the two terms \((\overline{n}(P\,\overline{m}))\) and \(\overline{n}\), leaving the sum bird to act on a single term. The correction restores the intended interplay between \(\oplus\) and \(A\).

As for Problem 7, the expressions simply forgot to decrease \(m\) by one when "unfolding" one step in the recursive definition of the exponentiation.