E 1

If \(\mathbb{P}\) is a poset, an automorphism of \(\mathbb{P}\) is a \(1-1\) and onto \(i: \mathbb{P}\to\mathbb{P}\) which preserves \(\leq\) and satisfies \(i(\unicode{x1D7D9}) = \unicode{x1D7D9}\); thus also \(i_*(\check{x}) = \check{x}\) for each \(x\). \(\mathbb{P}\) is called almost homogeneous iff for all \(p,q\in\mathbb{P}\), there is an automorphism \(i\) of \(\mathbb{P}\) such that \(i(p)\) and \(q\) are compatible. Suppose that \(\mathbb{P}\in M\) and \(\mathbb{P}\) is almost homogeneous in \(M\). Show that if \(p\Vdash\phi(\check{x}_1,\dots,\check{x}_n)\), then \(\unicode{x1D7D9}\Vdash\phi(\check{x}_1,\dots,\check{x}_n)\); thus, either \(\unicode{x1D7D9}\Vdash\phi(\check{x}_1,\dots,\check{x}_n)\) or \(\unicode{x1D7D9}\Vdash\lnot\phi(\check{x}_1,\dots,\check{x}_n)\).

By 7.13 (c) if \(p\Vdash\phi(\tau_1,\dots,\tau_n)\), then \(i(p)\Vdash\phi(i_*(\tau_1),\dots,i_*(\tau_n))\) for any automorphism \(i\).

Assume \(p\Vdash\phi(\check{x}_1,\dots,\check{x}_n)\), want \(\unicode{x1D7D9}\Vdash\phi(\check{x}_1,\dots,\check{x}_n)\). Let \(G\) be any generic filter, suppose \(M[G]\) thinks \(\lnot\phi(x_1,\dots,x_n)\), and let it be forced by \(q\in G\), then there is an automorphism such that \(i(p)\) and \(q\) are compatible, but \(i(p)\) forces \(\phi(x_1,\dots,x_n)\).

E 2

Show that any \(\operatorname{Fn}(I,J,\kappa)\) is almost homogeneous.

Let \(p,q\in \mathbb{P}=\operatorname{Fn}(I,J,\kappa)\), just define \(i\) such that \(\operatorname{dom}(i(p))\cap \operatorname{dom}(q) = 0\).

E 3

In \(M\), let \(I\) and \(J\) be uncountable, \(\mathbb{P}= \operatorname{Fn}(I,2)\) and \(\mathbb{Q}= \operatorname{Fn}(J,2)\). Let \(\phi(x)\) be a formula. Show that \[ \unicode{x1D7D9}\Vdash_\mathbb{P}\phi(\check{\alpha})^{\mathbf{L}(\mathcal{P}(\omega))} \text{ iff } \unicode{x1D7D9}\Vdash_\mathbb{Q}\phi(\check{\alpha})^{\mathbf{L}(\mathcal{P}(\omega))}. \]

Follow hint, if \(M\) thinks \(I\) and \(J\) are equinumerous, directly apply Lemma 7.13 (c) since \(M\) will contain the isomorphism \(i: \mathbb{P}\to\mathbb{Q}\).

More generally say \(M\models \left\lvert I\right\rvert\leq \left\lvert J\right\rvert\) and let \(H\) be \(\operatorname{Fn}(I,J,\omega_1)^M\)-generic over \(M\); then \(M[H]\models \left\lvert I\right\rvert=\left\lvert J\right\rvert\). Witnessed by \(\bigcup H\) by Lemma 6.2.

If \(G\) if \(\mathbb{P}\)-generic over \(M[H]\), then \(G\) is \(\mathbb{P}\)-generic over \(M\) because being generic relativises down.

\(\mathcal{P}(\omega)\cap M[H][G] = \mathcal{P}(\omega) \cap M[G]\) (why no new subsets of \(\omega\) because \(\operatorname{Fn}(I,J,\omega_1)^M\) is countably closed)

So \(M[H][G]\) and \(M[H]\) see the same \(\mathbf{L}(\mathcal{P}(\omega))\) (need to confirm with definition of \(L(X)\)).

Since \(G\) was arbitrary this shows that if \(\mathbb{P},M\) forces, \(\mathbb{P},M[H]\) also forces.

Use E1 and repeat the argument with negation to show the reverse.

Likewise for \(\mathbb{Q}\).

This allows us to reduce to the easy case that directly applied Lemma 7.13 (c).

E 4

Follow hint. Let \(M\models \left\lvert I\right\rvert\geq \omega_1\), let \(G\) be \(\mathbb{P}=\operatorname{Fn}(I,2)\)-generic over \(M\), and let \(N = (L(\mathcal{P}\left(\omega\right)))^{M[G]}\). If choice holds in \(N\), let \(N\models \kappa = \left\lvert\mathcal{P}\left(\omega\right)\right\rvert\), by E1 \[ \unicode{x1D7D9}\Vdash_\mathbb{P}\left( \check{\kappa} = \left\lvert\mathcal{P}\left(\omega\right)\right\rvert \right)^{\mathbf{L}(\mathcal{P}\left(\omega\right))}. \]

Take \(J\) such that \(M\models \left\lvert J\right\rvert>\kappa\), and let \(\mathbb{Q}=\operatorname{Fn}(J\times\omega,2)\) and by E3, \[ \unicode{x1D7D9}\Vdash_\mathbb{Q}\left( \check{\kappa} = \left\lvert\mathcal{P}\left(\omega\right)\right\rvert \right)^{\mathbf{L}(\mathcal{P}\left(\omega\right))}. \] so let \(H\) be \(\mathbb{Q}\)-generic over \(M\), then \(M[H] \models \left(\kappa = \left\lvert\mathcal{P}\left(\omega\right)\right\rvert\right)^{L(\mathcal{P}\left(\omega\right))}\). The problem here is this forcing notion adds at least \(\left\lvert J\right\rvert\) many reals to \(M[H]\), that is \(M[H]\models \left\lvert\mathcal{P}\left(\omega\right)\right\rvert \geq \left\lvert J\right\rvert\). As \(\mathbb{Q}\) is ccc, the forcing preserves cardinals which gives the contradiction.