# forcing and set theory [closed]

Let $\kappa , \lambda , \theta$ be infinte cardinals de $M$ represents a c.t.m, and $P=Fn( \kappa\ \times \omega,2)$. Show that $(\lambda^{\theta})^{M[G]}\leq((max( \kappa, \lambda)^{\theta})^{M}$. as building function names of $\theta$ in $\lambda$

## closed as off-topic by Andrés E. Caicedo, Stefan Kohl, Monroe Eskew, Igor Belegradek, S. Carnahan♦Jun 22 '14 at 3:56

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Andrés E. Caicedo, Stefan Kohl, Monroe Eskew, Igor Belegradek, S. Carnahan
If this question can be reworded to fit the rules in the help center, please edit the question.

• It might help to check out the FAQ and spend a little more time writing your question. Explain why you're interested, what's the motivation? Describe previous work on the topic, etc. – Jonathan Beardsley Jun 20 '14 at 22:58
• Is this homework? If not, what is the motivation? – Noah Schweber Jun 21 '14 at 0:16
• Questions tend to end with question marks. – Michael Greinecker Jun 21 '14 at 1:03

Any function $f:\theta\to\lambda$ in the extension can be completely described by giving, in the ground model, for each $i\in\theta$, a "nice name" for $f(i)$. A nice name amounts to a maximal antichain of conditions that decide particular values (in $\lambda$) for $f(i)$. So what you need to do is count (1) maximal antichains in your forcing (remember that it has the countable antichain condition), (2) count how many ways there are to label a maximal antichain with members of $\lambda$, and (3) count how many ways there are to assign such a labeled antichain (= nice name) to each element of $\theta$.