Questions about Prikry forcing and Cohen forcing

Question

I have some questions.

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The first one is about the product of Prikry's forcing. Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, \mathbb{P}_{U_2}$ be the corresponding Prikry forcings. Let $G\times H$ be $\mathbb{P}_{U_1}\times \mathbb{P}_{U_2}-$generic over $V$. It is easily seen that in the extension there are new subsets of $\omega$ (for example if $(x_n: n<\omega), (y_n: n<\omega)$ are the Prikry sequences added by $G, H$ respectively, then $\{ n<\omega: x_n < y_n\}$ is such a set).

Question 1.1 Is $\kappa$ preserved in the extension $V[G\times H]$? Do $V$ and $V[G\times H]$ have the same cardinals?

Remark. Though the question remained unanswered in general, but by Yair Hayut's very nice answer, given a normal measure $U$ on $\kappa, \mathbb{P}_U^2$ does not collapse cardinals. His proof extends easily to show that for any natural number $n>1, \mathbb{P}_U^n$ is forcing isomorphism to $\mathbb{P}_U\times \mathbb{C},$ hence it preserves cardinals.

Question 1.2. What is the least cardinal $\lambda$ such that $\mathbb{P}_U^\lambda$ collapses some cardinals? What is the least cardinal $\delta$ such that $\mathbb{P}_U^\delta$ collapses $\kappa$? Are $\lambda$ and $\delta$ equal?

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My second question is about Cohen forcing. Let $\kappa$ be a Mahlo cardinal, let $\mathbb{P}$ be the reverse Easton iteration of $Add(\alpha,1)$ for all inaccessible cardinals $\alpha\leq \kappa,$ and let $G$ be $\mathbb{P}-$generic over $V$.

Question 2. Suppose $\alpha$ is an inaccessible cardinal $\leq \kappa.$ Is there an $H\in V[G]$ which is $Add(\alpha,1)^V-$generic over $V$? (of course the answer is yest for the least inaccessible).

Answer 1

This should be a comment - but it is too long:

Assume that $U = U_1 = U_2$. I want to show that $\mathbb{P}_U ^2 \cong \mathbb{P}_U\times \mathbb{C}$ where $\mathbb{C}$ is the Cohen forcing.

Let $\{ {\alpha^0}_i\}_{i <\omega}, \{ {\alpha^1}_i \}_{i<\omega}$ be the two Prikry sequences.

Set $\{ \gamma_n \}_{n<\omega} = \{ {\alpha^0}_i\}_{i <\omega} \cup \{ {\alpha^1}_i \}_{i<\omega}$ (the Prikry sequence) and $f:\omega \rightarrow P(2)\setminus \{\emptyset\}$, $f(n) = \{ i < 2 | \gamma_n \in \{ {\alpha^i}_m \}_{m < \omega} \}$ (the Cohen real).

This gives us an isomorphism: send conditions from the dense set $( \langle s_0, A\rangle , \langle s_1, A\rangle )$ (the same $A$, $\min A > \max s_0 \cup s_1$) in $\mathbb{P}_U^2$ to $(\langle s_0\cup s_1, A \rangle, f\restriction |s_0 \cup s_1|) \in \mathbb{P}_U \times \mathbb{C}$ (we can calculate $f\restriction |s_0 \cup s_1|$ since we can only add elements in the Prikry sequence above the $\max s_0 \cup s_1$). This is an order preserving bijection between those two posets.

Edit: The answer for Question 1.2 depends on the exact support that you're using:

For finite support - $\mathbb{P}^\omega$ trivially collapses $\kappa$ to $\omega$ - the function that assign to each $n$ the first element in the $n$-th Prikry sequence is onto $\kappa$.

For full support - $\mathbb{P}^\omega$ also collapses $\kappa$: in the generic extension there is a surjection from $(2^{\aleph_0})^V$ to $\kappa$.

Choose a $\omega$-Jonsson function on $\kappa$, i.e. function $f:[\kappa]^\omega \rightarrow \kappa$, such that for every $x\subset \kappa, |x|=\kappa$, $f^{\prime\prime}([x]^\omega) = \kappa$. Let $\{\alpha^j_i \}_{i<\omega}$ be the Prikry sequence added by the $j$ component of $\mathbb{P}^\omega$.

We define a function $g: (\omega^\omega)^V \rightarrow \kappa$ by $g(z) = f(\{\alpha_{z(n)}^n | n < \omega\})$ .

I claim that $g$ is surjective: Let $p = \langle g_i, A_i | i < \omega \rangle \in \mathbb{P}^\omega$ and $\alpha < \kappa$. WLOG, $A=A_i$ for every $i$. Choose $y\in A^\omega$ such that $f(y) = \alpha$ and extends each $g_i$ by the corresponding element of $y$. Since the sequence of lengths of $g_i$ is real from $V$ - the new condition forces $\alpha \in \text{im }g$.

Since we're dealing with Prikry forcing there is at least one more support that we should consider - the Magidor support, namely the conditions are all elements of the form $\langle g_i, A_i |i < \delta\rangle$ such that $\{i<\delta | g_i \neq \emptyset \}$ is finite. In this case (as long as $\delta < \kappa$), we can apply the same idea as above and get that $\mathbb{P}^\delta \cong \mathbb{P}\times \mathbb{D}$ where $\mathbb{D}$ is a forcing that adds a generic function from $\omega$ to the set of all finite, non empty subsets of $\delta$, so as long as $\delta < \kappa$ - $\kappa$ is not collapsed.

When $\delta = \kappa$ this argument doesn't work, so I don't know if $\kappa$ is collapsed by the Magidor power $\mathbb{P}^\kappa$ or not.

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Remark 1. The answer to question 1.1 is yes, even if $U_1\neq U_2.$

Theorem. Let $U,V$ be normal measures on $\kappa.$ Then forcing with $\mathbb{P}_U\times \mathbb{P}_V$ preserves all cardinals.

Proof. It suffices to consider the case where $U$ is not equal to $V$. So let $A^*\in U$ such that $\kappa-A^*\in V.$ Let $W=\{ X\subset\kappa: X\cap A^*\in U, X\cap (\kappa-A^*)\in V \}.$ It is easily seen that $W$ is $\kappa-$complete filter on $\kappa$ which is Rowbottom (for any $f: [D]^{<\omega}\to \lambda<\kappa, D\in W$, there is $E\in W, E\subset D$ such that $card(f'' [E]^{<\omega}) \leq \omega$ ). So we can define $\mathbb{P}_W$ and by Devlin's paper "Some Remarks on Changing Cofinalities"¹, forcing with $\mathbb{P}_W$ preserves cardinals. As above argument, we have a forcing isomorphism from the dense subset $\{((s, A)(t, B))\in \mathbb{P}_U\times \mathbb{P}_V: A\subset A^*, B\subset \kappa-A^*, max(s\cup t)< min(A), min(B) \}$ of $\mathbb{P}_U\times \mathbb{P}_V$ to $\mathbb{P}_W\times \mathbb{C}$ given by $((s, A)(t, B))\to ((s\cup t, A\cup B), f\restriction |s\cup t|),$ where $f$ is defined as above argument. The result follows.

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1. Keith J. Devlin, Some remarks on changing cofinalities, J. Symbolic Logic 39 (1974), 27--30.

Remark 2. $\kappa$ is collapsed by the Magidor power $\mathbb{P}^\kappa$, by the following argument:

For any $\delta<\kappa,$ we may factor $\mathbb{P}^\kappa$ as $\mathbb{P}^\delta \times \mathbb{P}^{\kappa -\delta}$, so by the above argumets we can conclude that all cardinals $<\kappa$ are collapsed, so $\kappa$ is collapsed since it is singular in the extension.

Answer 2

Concerning question 2, Yair Hayut's comment is exactly right, and this kind of situation is just the kind of situation that I had aimed to analyze with those results.

The basic fact is that if one performs forcing $\mathbb{P}_0$ of size $\delta$ followed by nontrivial strategically $\leq\delta$-closed forcing $\mathbb{Q}$, then the extension $V\subset V[g][H]$ has the $\delta^+$-approximation property, which means that any set $A\in V[g][H]$ with $A\subset V$ and $A\cap a\in V$ whenever $a$ has size at most $\delta$ in $V$, then $A$ is already in $V$. Thus, there can be in the extension no fresh subsets of a larger regular cardinal $\alpha$, where $A\subset\alpha$ is fresh over $V$ if $A\notin V$ but all initial segments of $A$ below $\alpha$ are in $V$.

Your iteration admits a closure point at the least inaccessible, and so it can add no fresh subsets above the least inaccessible. In particular, it adds no $V$-generic Cohen sets using $\text{Add}(\alpha,1)^V$, since this forcing would add a fresh subset of $\alpha$.

You can find details in my paper Joel David Hamkins, Extensions with the approximation and cover properties have no new large cardinals, Fund. Math. 180 (2003), no. 3, 257--277. see also my blog.

One can use this idea to show that if you add a Cohen subset to $\kappa$ and then to $\lambda>\kappa$, then you kill all supercompact cardinals between $\kappa$ and $\lambda$.