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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|MATH160 SoUndNESS OF N

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

We show the soundness and completeness theorems for the proof system $\mathcal{N}$ for FOL. As in the case of PL, soundness is an easy task.

Theorem 11.1 For every $\Gamma \subseteq \mathrm{WFF}{\mathrm{FOL}}, A \in \mathrm{WFF}{\mathrm{FOL}}: \Gamma \vdash_{\mathcal{N}} A \Rightarrow \Gamma \models A$.
Proof. Axioms A0-A3 and MP are the same as for PL and their validity follows from the proof of the soundness Theorem $6.15$ for PL. Validity of A4 was shown in Exercise 9.10.
It suffices to show that $\exists$ I preserves truth, i.e. that $M \models B \rightarrow C$ implies $M \models \exists x B \rightarrow C$ for arbitrary structure $M$ (in particular, one for which $M=\Gamma$ ), provided $x$ is not free in $C$. In fact, it is easier to show the contrapositive implication from $M \not \exists \exists x \rightarrow C$ to $M \not \models$ $B \rightarrow C$. So suppose $M \not \exists x B \rightarrow C$, i.e., $M \not \models_v \exists x B \rightarrow C$ for some
$v$. Then $M=v \exists x B$ and $M \forall_v C$. Hence $M={v[x \mapsto a]} B$ for some
a. Since $M \not \models_v C$ and $x \notin \mathcal{V}(C)$, it follows from Lemma $9.7$ that also $M \not \models_{v[x \mapsto \underline{a}]} C$, hence $M \not \bigoplus_{v[x \mapsto \underline{a}]} B \rightarrow C$, i.e., $M \not \models B \rightarrow C$. QED (11.1)
By the same argument as in Corollary 6.16, every satisfiable FOL theory is consistent or, equivalently, inconsistent FOL theory is unsatisfiable:
Corollary 11.2 $\Gamma \vdash_{\mathcal{N}} \perp \Rightarrow \operatorname{Mod}(\Gamma)=\varnothing$.
Proof. If $\Gamma \vdash_{\mathcal{N}} \perp$ then, by the theorem, $\Gamma \models \perp$, i.e., for any $M: M \models$ $\Gamma \Rightarrow M \models \perp$. But there is no $M$ such that $M \models \perp$, so $\operatorname{Mod}(\Gamma)=\varnothing$.
QED (11.2)

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Completeness of $\mathcal{N}$

As in the case of $\mathrm{PL}$, we prove the opposite of Corollary $11.2$, namely, that every consistent FOL-theory is satisfiable. Starting with a consistent theory $\Gamma$, we have to show that there is a model satisfying $\Gamma$. The procedure is thus very similar to the one applied for PL (which you might repeat before reading this section). Its main point was expressed in Lemma 6.18, which has the following counterpart:

Lemma 11.3 The following formulations of completeness are equivalent:
(1) For any $\Gamma \subseteq \mathrm{WFF}{\mathrm{FOL}}: \Gamma H_N \perp \Rightarrow \operatorname{Mod}(\Gamma) \neq \varnothing$ (2) For any $\Gamma \subseteq \mathrm{WFF}{\mathrm{FOL}}: \Gamma \models A \Rightarrow \Gamma \vdash_{\mathcal{N}} A$.
Proof. (1) $\Rightarrow$ (2). Assuming (1) and $\Gamma \models A$, we consider first the special case of a closed $A$. Then $\operatorname{Mod}(\Gamma, \neg A)=\varnothing$ and so $\Gamma, \neg A \vdash_{\mathcal{N}} \perp$ by (1). By Deduction Theorem $\Gamma \vdash_N \neg A \rightarrow \perp$, so $\Gamma \vdash_N A$ by PL.
This result for closed $A$ yields the general version: If $\Gamma \models A$ then $\Gamma \models \forall(A)$ by Fact $9.27$. By the argument above $\Gamma \vdash_N \forall(A)$, and so $\Gamma \vdash_{\mathcal{N}} A$ by Lemma 8.24.(1) and MP.
(2) $\Rightarrow$ (1). This is shown by exactly the same argument as for PL in the proof of Lemma 6.18.

QED (11.3)

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代 考|SoUndNESS OF N

$v$. 然后 $M=v \exists x B$ 和 $M \forall_v C$. 因此 $M=v[x \mapsto a] B$ 对于一些
a. 自从 $M F_v C$ 和 $x \notin \mathcal{V}(C)$ ，它逪循引理 $9.7$ 那也 $M \nvdash_{v[x \mapsto a]} C$ ，因此 $M \bigoplus_{v[x \mapsto a]} B \rightarrow C$ ，那是，
$M \not \models B \rightarrow C$. QED $(11.1)$

QED (11.2)

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代 考|Completeness of $\mathcal{N}$

(1) 对于任何 $\Gamma \subseteq \mathrm{WFFFOL}: \Gamma H_N \perp \Rightarrow \operatorname{Mod}(\Gamma) \neq \varnothing(2)$ 对于任何
$\Gamma \subseteq \mathrm{WFFFOL}: \Gamma \models A \Rightarrow \Gamma \vdash_{\mathcal{N}} A$.

$(2) \Rightarrow(1)$. 这在引理 $6.18$ 的证明中由与 $P L$ 完全相同的论证表明。
QED (11.3)

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