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). 



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



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> **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 $V$. 



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> **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).
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Questions about Prikry forcing and Cohen forcing
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**](http://www.ams.org/mathscinet-getitem?mr=347606), *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.


