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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|MATH301

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|The Collection of Axioms Is Recursive

In this section we will exhibit two $\Delta$-formulas that are designed to pick out the axioms of our deductive system.

Proposition 4.11.1. The collection of Gödel numbers of the axioms of $N$ is recursive.

Proof. The formula AxiomOfN is easy to describe. As there are only a finite number of $\mathrm{N}$-axioms, a natural number $a$ is in the set AxiomOFN if and only if it is one of a finite number of Gödel numbers. Thus
$\operatorname{AxiomOfN}(a)$ is:
$$\begin{gathered} a=\overline{\Gamma(\forall x) \neg S x=0\urcorner} \vee \ a=\overline{\Gamma(\forall x)(\forall y)[S x=S y \rightarrow x=y]\urcorner} \vee \ \vdots \ \vee a=\overline{\Gamma(\forall x)(\forall y)[(x<y) \vee(x=y) \vee(y<x)]\urcorner} . \end{gathered}$$
(To be more-than-usually picky, we need to change the $x$ ‘s and $y$ ‘s to $v_1$ ‘s and $v_2$ ‘s, but you can do that.)

Proposition 4.11.2. The collection of Gödel numbers of the logical axioms is recursive.

Proof. The formula that recognizes the logical axioms is more complicated than the formula AxiomOfN for two reasons. The first is that there are infinitely many logical axioms, so we cannot just list them all. The second reason that this group of axioms is more complicated is that the quantifier axioms depend on the notion of substitutability, so we will have to use our results from Section 4.10.

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

It is probably difficult to remember at this point of our journey, but our goal is to prove the Incompleteness Theorem, and to do that we need to write down an $\mathcal{L}{N T}$-sentence that is true in $\mathfrak{N}$, the standard structure, but not provable from the axioms of $N$. Our sentence, $\theta$, will “say” that $\theta$ is not provable from $N$, and in order to “say” that, we will need a formula that will identify the (Gödel numbers of the) formulas that are provable from $N$. To do that we will need to be able to code up deductions from $N$, which makes it necessary to code up sequences of formulas. Thus, our next goal will be to settle on a coding scheme for sequences of $\mathcal{L}{N T}$-formulas.

We have been pretty careful with our coding up to this point. If you check, every Gödel number that we have used has been even, with the exception of 3 , which is the garbage case in Definition 4.7.1. We will now use numbers with smallest prime factor 5 to code sequences of formulas.

Suppose that we have the sequence of formulas
$$D=\left\langle\phi_1, \phi_2, \ldots, \phi_k\right\rangle .$$
We will define the sequence code of $D$ to be the number
$$\left.r D\urcorner=5^{\left.r \phi_1\right\urcorner} 7^{\left.r \phi_2\right\urcorner} \cdots p_{k+2} \phi_k\right\urcorner .$$
So the exponent on the $(i+2)$ nd prime is the Gödel number of the $i$ th element of the sequence. You are asked in the Exercises to produce several useful $\mathcal{L}_{N T}$-formulas relating to sequence codes.

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|The Collection of Axioms Is Recursive

$\operatorname{AxiomOfN}(a)$是:
$$\begin{gathered} a=\overline{\Gamma(\forall x) \neg S x=0\urcorner} \vee \ a=\overline{\Gamma(\forall x)(\forall y)[S x=S y \rightarrow x=y]\urcorner} \vee \ \vdots \ \vee a=\overline{\Gamma(\forall x)(\forall y)[(x<y) \vee(x=y) \vee(y<x)]\urcorner} . \end{gathered}$$
(为了比通常更挑剔，我们需要将$x$和$y$更改为$v_1$和$v_2$，但您可以这样做。)

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

$$D=\left\langle\phi_1, \phi_2, \ldots, \phi_k\right\rangle .$$

$$\left.r D\urcorner=5^{\left.r \phi_1\right\urcorner} 7^{\left.r \phi_2\right\urcorner} \cdots p_{k+2} \phi_k\right\urcorner .$$

## MATLAB代写

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