# $\mathfrak{p}=\mathfrak{b}=\mathfrak{a}=\aleph_1$ and $\mathfrak{d}=\mathfrak{c}=\kappa$

Asumme tha in $M$, $CH$ holds and $\kappa>\aleph_0$ and $\kappa^{\aleph_0}=\kappa$. Let $K$ be $Fn(\kappa,2)$-generic over $M$.

Question:

Then we can say in $M[K]$ that:

$(i)$ $\mathfrak{p}=\mathfrak{b}=\mathfrak{a}=\aleph_1$ and $\mathfrak{d}=\mathfrak{c}=\kappa$ ?

Where

$\mathfrak{c}=2^{\aleph_0}$ the size of the continuum.

$\mathfrak{d}$, is the least size of a $\mathfrak{d}$ominating family.

$\mathfrak{b}$, is the least size of an un$\mathfrak{b}$ounded family.

$\mathfrak{p}$, is the least size of family $\mathcal{E}\subseteq [\omega]^\omega$ such that $\mathcal{E}$ has the SFIP and there does not exist any $\mathfrak{p}$seuod-intersection of $\mathcal{E}$.

$\mathfrak{a}$, is the least size of an infinite m$\mathfrak{a}$d family.

• Why the vote to close? Commented Mar 14, 2015 at 1:42

## 1 Answer

Yes, these values are correct for the Cohen model. (Self-promotion: See the table in Section 11 of my chapter in the Handbook of Set Theory. The pre-publication version is on my web site at http://www.math.lsa.umich.edu/~ablass/hbk.pdf .)

• Hello Andreas Blass, I would like to see proof of this question, where I can see ?. thanks Commented Mar 14, 2015 at 1:29
• @Angel All these facts are in Section 11.3 of the handbook chapter cited in my answer. Unfortunately, you may need to look at some of the earlier sections of the chapter for some of the terminology used there. Although I don't have my copy of the Bartoszynski-Judah book "Set theory: On the structure of the real line" handy, I'm pretty sure all this information is there also. Commented Mar 14, 2015 at 14:10