This is a question from Jech's Set Theory (Ex. 17.12) which I'm reading at the moment and pretty much stuck on.



> 
> If $D$ is a normal measure on $\kappa$
>  and $\{ \aleph\_\alpha \colon
> > 2^{\aleph\_\alpha} \le
> > \aleph\_{\alpha+\beta}\} \in D$ (for
>  some constant $\beta < \kappa$), then $2^\kappa
> > \le \aleph\_{\kappa + \beta}$
> 
> 
> 


He gives the following hint: If $f$ is such that $f(\aleph\_\alpha) = \aleph\_{\alpha+\beta}$ for all $\alpha < \kappa$, then $[f]\_D = (\aleph \_{ \kappa+j(\beta)})^M$


I think that I am just confused about the whole representation in $M$ and how to use it to solve this problem. Hints, partial or complete solutions are most welcomed.



Normal measures and Elementary Embeddings
The question you've stated isn't the question in Jech, you've made a minor typo. Here's the actual problem:



> 
> If $\beta < \kappa$ and {$\aleph \_{\alpha} : 2^{\aleph \_{\alpha}} \leq \aleph \_{\alpha + \beta}$} $\in D$ and $D$ is a normal measure on $\kappa$, then $2^{\aleph \_{\kappa}} \leq \aleph \_{\kappa + \beta}$
> 
> 
> 


Note that since $\kappa$ is measurable, $\aleph \_{\kappa} = \kappa$.


Okay, now we know that a normal measure extends the club filter, and the set of cardinals below $\kappa$ is club in $\kappa$, hence it makes sense in the hint to define $f(\aleph \_{\alpha}) = \aleph \_{\alpha + \beta}$ without specifying how $f$ acts on non-cardinals. Following my comment, let $g(\aleph \_{\alpha}) = 2^{\aleph \_{\alpha}}$. Then $g \leq f$ almost everywhere, and so:



> 
> $M \vDash [g] \leq [f]$
> 
> 
> 


i.e.



> 
> $M \vDash j(g)(\kappa) \leq j(f)(\kappa)$
> 
> 
> 


i.e.



> 
> $M \vDash 2^{\kappa} \leq \aleph \_{\kappa + j(\beta)}$
> 
> 
> 


Since $\beta < \kappa$, $j(\beta) = \beta$. Thus there is an injection from $(2^{\kappa})^M$ to $\aleph \_{\kappa + \beta} ^M$. Since $P(\kappa) = P^M(\kappa)$, it means there's an injection from $2^{\kappa}$ to $\aleph \_{\kappa + \beta}^M$. Finally, $\aleph \_{\kappa + \beta} ^M \leq \aleph \_{\kappa + \beta}$ since $M \subseteq V$.


