One formulation of the Axiom of Choice says that for each...

One formulation of the Axiom of Choice says that for each collection $u$ of nonempty pairwise disjoint sets (whose elements have type $\tau$ ), there is a set $r$ containing exactly one element from each set in $u$. This can be expressed in $\mathscr{F}^\omega$ by $$ \begin{aligned} & A C_1^\tau: \forall u_{(\mathrm{r}))} \cdot \forall p_{(\mathrm{r} \mathrm{t}}\left[u_{(\mathrm{t}(\mathrm{r})} p_{(\mathrm{t})} \supset \exists x_{\mathrm{r}} p_{(\mathrm{r})} x_{\mathrm{r}}\right] \\ & \wedge \forall p_{(\mathrm{t})} \forall q_{(\mathrm{r})}\left[u_{(\mathrm{t})} p_{(\mathrm{r})} \wedge u_{(\mathrm{t})} q_{(\mathrm{t})} \wedge p_{(\mathrm{t})} \neq q_{(\mathrm{t})}\right. \\ & \left.\supset \forall x_{\mathrm{t}} \sim p_{(\mathrm{t})} x_{\mathrm{r}} \wedge q_{(\mathrm{t})} x_{\mathrm{t}}\right] \\ & \supset \exists r_{(\mathrm{t})} \forall p_{(\mathrm{t})} \cdot u_{(\mathrm{t})} p_{(\mathrm{r})} \supset \exists_1 x_{\mathrm{r}} \cdot p_{(\mathrm{t})} x_{\mathrm{t}} \wedge r_{(\mathrm{t})} x_{\mathrm{t}} . \\ & \end{aligned} $$ A second formulation of the Axiom of Choice says that there is a function (a special sort of binary relation) $f$ which chooses from each nonempty set $p$ (whose elements have type $\tau$ ) a member of that set. This can be expressed in $\mathscr{F}^\omega$ by $$ \begin{aligned} & A C_2^{\mathrm{\tau}}: \exists f_{(\mathrm{t} \mathrm{r} \tau)} \forall p_{(\mathrm{t})} \cdot\left[\exists x_{\mathrm{t}} p_{(\mathrm{t})} x_{\mathrm{t}} \supset \exists_1 x_{\mathrm{r}} f_{(\mathrm{t}(\mathrm{r}) \mathrm{r})} p_{(\mathrm{r})} x_{\mathrm{r}}\right] \\ & \wedge \forall x_\tau \cdot f_{(t) t) p} p_{(t)} x_t \supset p_{(t)} \mathbf{x}_t \\ & \end{aligned} $$ $\vdash A C_1^{((\mathrm{t}) \mathrm{q})} \supset A C_2^{\varepsilon}$

One formulation of the Axiom of Choice says that for each collection $u$ of nonempty pairwise disjoint sets (whose elements have type $\tau$ ), there is a set $r$ containing exactly one element from each set in $u$. This can be expressed in $\mathscr{F}^\omega$ by
$$
\begin{aligned}
& A C_1^\tau: \forall u_{(\mathrm{r}))} \cdot \forall p_{(\mathrm{r} \mathrm{t}}\left[u_{(\mathrm{t}(\mathrm{r})} p_{(\mathrm{t})} \supset \exists x_{\mathrm{r}} p_{(\mathrm{r})} x_{\mathrm{r}}\right] \\
& \wedge \forall p_{(\mathrm{t})} \forall q_{(\mathrm{r})}\left[u_{(\mathrm{t})} p_{(\mathrm{r})} \wedge u_{(\mathrm{t})} q_{(\mathrm{t})} \wedge p_{(\mathrm{t})} \neq q_{(\mathrm{t})}\right. \\
& \left.\supset \forall x_{\mathrm{t}} \sim p_{(\mathrm{t})} x_{\mathrm{r}} \wedge q_{(\mathrm{t})} x_{\mathrm{t}}\right] \\
& \supset \exists r_{(\mathrm{t})} \forall p_{(\mathrm{t})} \cdot u_{(\mathrm{t})} p_{(\mathrm{r})} \supset \exists_1 x_{\mathrm{r}} \cdot p_{(\mathrm{t})} x_{\mathrm{t}} \wedge r_{(\mathrm{t})} x_{\mathrm{t}} . \\
&
\end{aligned}
$$
A second formulation of the Axiom of Choice says that there is a function (a special sort of binary relation) $f$ which chooses from each nonempty set $p$ (whose elements have type $\tau$ ) a member of that set. This can be expressed in $\mathscr{F}^\omega$ by
$$
\begin{aligned}
& A C_2^{\mathrm{\tau}}: \exists f_{(\mathrm{t} \mathrm{r} \tau)} \forall p_{(\mathrm{t})} \cdot\left[\exists x_{\mathrm{t}} p_{(\mathrm{t})} x_{\mathrm{t}} \supset \exists_1 x_{\mathrm{r}} f_{(\mathrm{t}(\mathrm{r}) \mathrm{r})} p_{(\mathrm{r})} x_{\mathrm{r}}\right] \\
& \wedge \forall x_\tau \cdot f_{(t) t) p} p_{(t)} x_t \supset p_{(t)} \mathbf{x}_t \\
&
\end{aligned}
$$
$\vdash A C_1^{((\mathrm{t}) \mathrm{q})} \supset A C_2^{\varepsilon}$

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