## 数学代写|数理逻辑代写Mathematical logic代考|MATH4810

2023年1月6日

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## 数学代写|数理逻辑代写Mathematical logic代考|Substitution

In this section we define how to substitute a term $t$ for a variable $x$ in a formula $\varphi$ at the places where $x$ occurs free, thus obtaining a formula $\psi$. We wish to define the substitution so that $\psi$ expresses the same about $t$ as $\varphi$ does about $x$. We start with an example to illustrate our objective and to show why a certain care is necessary. Let
$$\varphi:=\exists z z+z \equiv x .$$
In $\mathfrak{N}$ the formula $\varphi$ says that $x$ is even; more precisely:
$$(\mathfrak{N}, \beta) \models \varphi \quad \text { iff } \quad \beta(x) \text { is even. }$$
If we replace the variable $x$ by $y$ in $\varphi$, we obtain the formula $\exists z z+z \equiv y$, which states that $y$ is even. But if we replace the variable $x$ by $z$, we obtain the formula $\exists z z+z \equiv z$, which no longer says that $z$ is even; in fact, this formula is valid in $\mathfrak{N}$ regardless of the assignment for $z$ (because $0+0=0$ ). In this case the meaning is altered because at the place where $x$ occurred free, the variable $z$ gets bound. On the other hand, we obtain a formula which expresses the same about $z$ as $\varphi$ does about $x$ if we proceed as follows: First, we introduce a new bound variable $u$ in $\varphi$, and then in the formula $\exists u u+u \equiv x$ thus obtained we replace $x$ by $z$. It is immaterial which variable $u$ (distinct from $x$ and $z$ ) we choose. However, for certain technical purposes it is useful to make a fixed choice.

In the preceding example we replaced only one variable, but in our exact definition we specify the procedure for simultaneously replacing several variables: With a given formula $\varphi$, pairwise distinct variables $x_0, \ldots, x_r$ and arbitrary terms $t_0, \ldots, t_r$, we associate a formula $\varphi \frac{t_0 \ldots t_r}{x_0 \ldots x_r}$, which is said to be obtained from $\varphi$ by simultaneously substituting $t_0, \ldots, t_r$ for $x_0, \ldots, x_r$. The reader should note that $x_i$ has to be replaced by $t_i$ only if
$$x_i \in \operatorname{free}(\varphi) \quad \text { and } \quad x_i \neq t_{i \cdot}$$
In the following inductive definition this is explicitly taken into account in the quantifier step; in the other steps it follows immediately.

## 数学代写|数理逻辑代写Mathematical logic代考|Sequent Rules

A mathematical proof proceeds from one statement to the next until it finally arrives at the assertion of the theorem in question. The individual statements depend on certain hypotheses. These can either be hypotheses of the theorem or additional hypotheses temporarily assumed in the course of the proof. For example, if one wants to prove an intermediate claim $\varphi$ by contradiction, one adds $\neg \varphi$ to the hypotheses; if a contradiction results, then $\varphi$ has been proved, and the additional assumption $\neg \varphi$ is dropped.

This observation leads us to describe a stage in a proof by listing the corresponding assumptions and the respective claim. If we call a nonempty list (sequence) of formulas a sequent, then we can use sequents to describe “stages in a proof”. For instance, the “stage” with assumptions $\varphi_1, \ldots, \varphi_n$ and claim $\varphi$ is rendered by the sequent $\varphi_1 \ldots \varphi_n \varphi$. The sequence $\varphi_1 \ldots \varphi_n$ is called the antecedent and $\varphi$ the succedent of the sequent $\varphi_1 \ldots \varphi_n \varphi$. From Lemma II.4.3 it follows that the formulas which constitute a sequent are uniquely determined. In particular, the antecedent and the succedent are well-defined.

In terms of sequents, the indirect proof sketched above can be represented schematically as follows:
$$\begin{array}{lrr} \varphi_1 \ldots \varphi_n & \neg \varphi & \psi \ \varphi_1 \ldots \varphi_n & \neg \varphi & \neg \psi \ \hline \varphi_1 \ldots \varphi_n & \varphi \end{array}$$
Thus (+) describes the following argument: If under the assumptions $\varphi_1, \ldots, \varphi_n$ and (the additional assumption) $\neg \varphi$ one can obtain both the formula $\psi$ and its negation $\neg \psi$ (that is, a contradiction), then from the assumptions $\varphi_1, \ldots, \varphi_n$ one can infer $\varphi$.

In the following we shall use the letters $\Gamma, \Delta, \ldots$ to denote (possibly empty) sequences of formulas. Then we can write sequents as $\Gamma \varphi \psi, \Delta \psi, \ldots$ and the scheme (+) in the form

As in ( $+)$, we use spaces between elements in a sequent merely for easier reading.

# 数理逻辑代写

## 数学代写|数理逻辑代写Mathematical logic代考|Substitution

$$\varphi:=\exists z z+z \equiv x$$

$(\mathfrak{N}, \beta) \models \varphi \quad$ iff $\quad \beta(x)$ is even.

$$x_i \in \operatorname{free}(\varphi) \text { and } x_i \neq t_i .$$

## 数学代写|数理逻辑代写Mathematical logic代考|Sequent Rules

Ibegin{array}{Irr} Ivarphi_1 \ddots Ivarphi_n \& Ineg Ivar

## 有限元方法代写

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## MATLAB代写

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