# How does one go to prove two forcing extensions are equivalent?

I want to know how does someone usually go, that is, what is the canonical way to go proving that having two notions of forcing $P$ and $Q$, the respective generic extensions $P[G]$ and $Q[H]$ are equivalent.

More specific, how would the proof of: forcing a Cohen real $\kappa$ many times and forcing $\kappa$ Cohen reals give the same forcing extension go?

Is it by showing that the complete Boolean algebras associated are the same by proving either $P$ is dense on $B(Q)$ or $Q$ is dense in $B(P)$? But doesn't that require too much information about these completions?

Or is it by finding an automorphism between the generics? Any automorphism will do?

Thank you for your time, it is my first question and I am not sure if it is appropriate I hope it's not too basic, but I am seriously having problems understanding this.

Kind regards,

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## 1 Answer

Usually the two forcing notions will be given as partial orders $P$ and $Q$. I would say most of the time, in order to show that two forcing extensions are equivalent, you show that $P$ can be densely embedded into $Q$ or vice versa. You can also show that both $P$ and $Q$ can be densely embedded into the same partial order.

A natural candidate for this "master p.o." is of course the completion of $P$ or $Q$, but sometimes something else is combinatorially simpler. Finally you can show that from a generic of $P$ you can construct a generic for $Q$ and the other way around.

I don't really understand what you mean by finding an automorphism between the generics. Automorphism of what? Of the p.o.? In this case you force with the same p.o., getting two different generics, and you now want to show that both give the same forcing extension. But this doesn't seem to be the original question.

Finally, it is not clear to me what you mean by "forcing $\kappa$ Cohen reals" vs "forcing Cohen real $\kappa$ times".
Do you mean forcing with a finite support product of Cohen forcing, thus adding $\kappa$ Cohens side by side, vs. iterating Cohen forcing of length $\kappa$, say with finite supports?

To show that the last two forcings are equivalent (iteration vs product) I would show that the iteration has a dense subset that is isomorphic to the finite support product.

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Thank you for your answer!! And sorry if I was too vague. I do mean iterations vs product, and to be fair I wasn't sure what I meant by automorphism between the generics, I suppose I was trying to say get one generic from the other as you pointed out. I'll try this. – maria Jul 22 '12 at 16:56