# Regarding Lemma 21.9 of Jech

Mathematics Asked by Hanul Jeon on December 22, 2020

Jech states the following lemma in his book:

Lemma 21.9 Let $$j:Vto M=operatorname{Ult}_U(V)$$ be an elementary embedding with a critical point $$kappa$$. If $$mathbb{P}in M$$ is a forcing poset such that $$|mathbb{P}|lekappa$$ and $$G$$ is a $$V$$-generic filter over $$mathbb{P}$$, then $$M[G]$$ is closed under $$kappa$$-sequences. (i.e., $$({^kappa}M[G])^{V[G]}subseteq M[G]$$.)

(The original statement involves with $$lambda$$-supercompactness, and states $$M[G]$$ is closed under $$lambda$$-sequences under the hypothesis $$|mathbb{P}|lelambda$$. I will only consider the measurable case for simplicity.)

He starts with the proof as follows:

It suffices to show that if $$fin V[G]$$ is a function from $$kappa$$ into ordinals, then $$fin M[G]$$. (…)

I do not understand this point. I can see that if $$M[G]$$ is closed under functions $$kappato mathrm{Ord}$$, then the standard Mostowski argument with some coding shows $$M[G]$$ is closed under functions from $$kappa$$ to $$H_{kappa^+}^{V[G]}$$. Hence I tried to prove it in another way:

My attempt. Let $$fin V[G]$$, $$f:kappato M[G]$$. Take $$p_0in G$$ such that $$p_0Vdash dot{f}text{ is a function from kappa to M^mathbb{P}}.$$
For each $$alpha, take
$$A_alpha := {ple p_0mid exists sigmain M^mathbb{P}[pVdash dot{f}(alpha)=sigma]}.$$
For each $$alpha and $$pin A_alpha$$, choose $$sigma_{alpha,p}in M^mathbb{P}$$ that witnesses $$pin A_alpha$$. Since the choice is made over $$V$$, $$langle sigma_{alpha,p}midalpha,pranglein V$$.
Furthermore, we can choose $$g_{alpha,p}:kappato V$$ such that $$sigma_{alpha,p}=[g_{alpha,p}]_U$$.

Now define a function $$g_p$$ as $$g_p(alpha)(xi) := g_{alpha,p}(xi)$$
if it is defined. Then $$g_p$$ sends $$xi$$ to a partial function over $$kappa$$.
Take $$h_p=[g_p]_Uin M$$. Then $$M$$ thinks $$h_p$$ is a partial function from $$j(kappa)$$ to $$M^mathbb{P}$$. Moreover, we have

• $$p’le pimplies operatorname{dom} h_psubseteq operatorname{dom} h_{p’}$$, and
• $$p’le p implies p’Vdash h_p(alpha)=h_{p’}(alpha)$$ (Here $$h_p(alpha)$$ and $$h_{p’}(alpha)$$ themselves are treated as a single $$M^mathbb{P}$$-name.)

Let $$h(alpha):= h_p(alpha)$$ for some $$p$$. Then $$h$$ is a partial function from $$j(kappa)$$. Since $$A_alphacap G$$ is nonempty for each $$alpha$$, $$h_p(alpha)$$ is defined for all $$alpha. Moreover, by definition of $$A_alpha$$, we have $$f(alpha)=h(alpha)$$ for $$alpha. Hence $$f=hupharpoonright kappain M[G]$$.

My questions are as follows:

1. Why just showing $$({^kappa}mathrm{Ord})^{V[G]}subseteq M[G]$$ suffices to prove Lemma 21.9?
2. Is my argument correct?

Thank you for any help in advance.

Let me give a brief answer to (1).

If $$A$$ is any set, fix in $$M[G]$$ a well-ordering of $$A$$, and let $$alpha$$ be the order type of this well-ordering. If $$fcolonkappato A$$ is any sequence, then $$f^*colonkappatoalpha$$ given by the composition of $$f$$ with the isomorphism is a function in $$mathrm{Ord}^kappa$$, but then by composing again with the inverse isomorphism, which is in $$M[G]$$, we get $$f$$.

Correct answer by Asaf Karagila on December 22, 2020

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