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There are infinitely many binary strings in $\{0, 1\}^*$, while there are -finite number of strings in $\{ 0, 1\}^n$. And that's precisely what makes it impossible to compute some functions since -NAND-TM programs are finite objects. +It is important to get the intuition here. Part of the difficulty of expanding the domain of functions from $\{0,1\}^n$ to $\{0,1\}^*$ is that there are only finitely many strings of $n$ bits but infinitely many of arbitary length. This difference explains why we cannot apply the same approach to computing every function that we used with functions on $\{0,1\}^n$, which relied on enumerating the finite set of outputs. + +However, there is another distinction which suggests why there are some functions we cannot compute: NAND-TM consists of objects with finite descriptions, while functions $\{0,1\}^* \to \{0,1\}$ have infinitely long description lengths. We can imagine a NAND-TM program as a compressed representation of the functions with ``nice patterns'', but the number of possible functions is so large that it overwhelms the ability of a finitely-described NAND-TM program to capture all of them. \subsection{Proof:} @@ -259,6 +259,9 @@ \subsection{Exercise} Consider the set $P(\mathbb{N})$ of all subsets of $\mathbb{N}$. Show that there is no one-to-one and onto function between $\mathbb{N}$ and $P(\mathbb{N})$. +\subsection{Exercise} +Is the set of NAND-TM programs countable? What about the set of functions $\{0,1\}^* \to \{0,1\}$? How can we get an alternate proof of Theorem 1 using these facts? + \section{Reduction} In the preceding section, we showed that there is some function from $\{0, 1\}^*$ to $\{0, 1\}$ that can't be simulated by any NAND-TM program, i.e. an uncomputable function. 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