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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|MATH591 The Relationship Between Enumerability and Decidability

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|The Relationship Between Enumerability and Decidability

We have just seen that the set of “logically true” sentences can be listed by means of an enumeration procedure. Is it possible to go farther than this and decide whether an arbitrary given sentence is “logically true”? The enumeration procedure given above does not help to solve this problem. For example, if we want to test a sentence $\varphi$ for validity we might start the enumeration procedure in 1.6 and wait to see whether $\varphi$ appears; we obtain a positive decision as soon as $\varphi$ is added to the list. But as long as $\varphi$ has not appeared, we cannot say anything about $\varphi$, since we do not know whether $\varphi$ will never appear (because it is not valid) or whether it will appear at a later time. In fact, we shall show (cf. Theorem 4.1) that the set of valid $S_{\infty}$-sentences is not decidable.
On the other hand, if a set is decidable, we can conclude that it is enumerable:
Theorem. Every decidable set is enumerable.
Proof. Suppose $W \subseteq \mathbb{A}^$ is decidable and $\mathfrak{P}$ is a decision procedure for $W$. To list $W$, generate the strings of $\mathbb{A}^$ in lexicographic order, use $\mathfrak{P}$ to check for each string $\zeta$ thus obtained whether it belongs to $W$ or not, and, if the answer is positive, add $\zeta$ to the list.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Computable Functions

Let $\mathbb{A}$ and $\mathbb{B}$ be alphabets. A procedure that for each input from $\mathbb{A}^$ yields a word in $\mathbb{B}^$ determines a function from $\mathbb{A}^$ to $\mathbb{B}^$. A function whose values can be computed in this way by a procedure is said to be computable. An example of a computable function is the length function $l$, which assigns to every $\zeta \in \mathbb{A}^*$ the length of $\zeta$ (in decimal notation as a word over ${0, \ldots, 9}$ ).

Whereas our discussion of procedures deals mainly with the notions of enumerability and decidability, many presentations of the theory of computability start with the computability of functions as the key concept. Both approaches are equivalent in the sense that the above notions are definable from each other. The following exercise shows that the notion of computable function can be reduced to both the notion of enumerability and the notion of decidability.

1.12 Exercise. Let $\mathbb{A}$ and $\mathbb{B}$ be alphabets, $# \notin \mathbb{A} \cup \mathbb{B}$ and $f: \mathbb{A}^* \rightarrow \mathbb{B}^$. Show that the following are equivalent: (i) $f$ is computable. (ii) $\left{\zeta # f(\zeta) \mid \zeta \in \mathbb{A}^\right}$ is enumerable.
(iii) $\left{\zeta # f(\zeta) \mid \zeta \in \mathbb{A}^\right}$ is decidable. The set $\left{\zeta # f(\zeta) \mid \zeta \in \mathbb{A}^\right}$ can be considered as the graph of $f$, and hence the equivalences in 1.12 can be formulated as follows: A function $f: \mathbb{A}^* \rightarrow \mathbb{B}^*$ is computable if and only if its graph is enumerable (decidable).

## MATLAB代写

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