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An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Computer Science & Applied Mathematics)

Peter B. Andrews

Chapter 6

Formalized Number Theory - all with Video Answers

Educators

Section 1

Cardinal Numbers and the Axiom of Infinity

Problem 1

Prove the following theorems:
$\forall p_{o \beta}=E_{o(\alpha \beta)(\alpha \beta)} p_{o \beta} p_{o \beta}$

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Problem 2

Prove the following theorems:
$\forall p_{o \beta} \forall q_{\mathrm{ox}}=E_{\mathrm{o}(\mathrm{o} \alpha)(\mathrm{o} \beta)} p_{\mathrm{o} \beta} q_{\mathrm{o} \alpha} \supset E_{\mathrm{o}(\mathrm{o} \beta)(\mathrm{o} \alpha)} q_{\mathrm{o \alpha}} p_{\mathrm{o} \beta}$

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Problem 3

Prove the following theorems:
$\forall p_{\mathrm{o} \beta} \forall q_{\mathrm{o} \alpha} \forall r_{\mathrm{o} \gamma}=E_{\mathrm{o}(\mathrm{o} \gamma)(\mathrm{o} \beta)} p_{\mathrm{o} \beta} q_{\mathrm{o \alpha}} \wedge E_{\mathrm{o}(\mathrm{o} \gamma)(\mathrm{o} \alpha))} q_{\mathrm{o} \alpha} r_{\mathrm{o} \gamma} \supset E_{\mathrm{o}(\mathrm{o} \gamma)(\mathrm{o} \beta)} p_{\mathrm{o} \beta} r_{\mathrm{o} \gamma}$

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Problem 4

Prove the following theorems:
$\forall p_{o \beta} \forall q_{\mathrm{o} \beta}=E_{\mathrm{o}(\mathrm{o} \beta)(\mathrm{o} \beta)} p_{\mathrm{o} \beta} q_{\mathrm{o} \beta}=\cdot \overline{\overline{p_{\mathrm{o} \beta}}}=\overline{\overline{q_{\mathrm{o} \beta}}}$

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Problem 5

Prove the following theorems:
$E_{\mathrm{o}(\alpha x)(\mathrm{o} \beta)}\left[\mathrm{Q}_{\mathrm{o} \beta \rho} x_\beta\right]\left[\mathrm{Q}_{\mathrm{ox \alpha}} y_\alpha\right]$

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Problem 6

Prove the following theorems:
$[N C]_{\circ(\circ(\rho \beta))} \overline{\overline{p_{o \beta}}}$

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Problem 7

Prove the following theorems:
$I n f^1=\operatorname{Inf} f^{\mathrm{II}}$

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Problem 8

Prove the following theorems:
$\mathbb{N}_{\mathrm{o} \sigma}=\bigcap_{\mathrm{o}(\mathrm{o} \sigma)}\left[\lambda p_{\mathrm{o} \sigma} \cdot p_{\mathrm{o} \sigma} 0_\sigma \wedge \forall x_\sigma \cdot p_{\mathrm{o} \sigma} x_\sigma \supset p_{\mathrm{o} \sigma} \cdot S_{\sigma \sigma} x_\sigma\right]$

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02:35

Problem 9

Prove the following theorems:
$\bar{n} p_{o t} \supset \exists x_1^1 \ldots \exists x_1^n \cdot \bigwedge_{i=1}^n p_{0 t} x_1^i \wedge \bigwedge_{1 \leq i<j \leq n} x_1^i \neq x_1^j$ for each positive integer $n$. (Hint: use induction.)

Kumar Vaibhav
Kumar Vaibhav
Numerade Educator

Problem 10

Prove the following theorems:
Show that if $\mathscr{M}$ is a model of $Q_0$ with exactly $n$ individuals (members of $\mathscr{D}), \mathscr{M} \vDash \bar{n}=Q_{\text {o(o)(or) }}\left[\lambda x_1 T_{\mathrm{o}}\right]$ and $\mathscr{M} \vDash \bar{m}=\left[\lambda p_{o t} F_{\mathrm{o}}\right]$ for each integer $m>n .(\bar{m}$ is defined on p. 209.)

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01:32

Problem 11

Prove the following theorems:
Show that if $n$ is any positive integer and $\mathscr{M}$ is any model of $Q_0$, then $\mathscr{M} \vDash \exists p_{o t} \bar{n}_a p_{o t}$ if and only if $\mathscr{M}$ has at least $n$ individuals.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 12

Prove the following theorems:
Is there a sentence $S$ such that for every general model $\mathscr{M}$, it is true that $\mathscr{M} \mathrm{FS}$ if and only if $\mathscr{M}$ has an infinite domain of individuals?

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