We know that if $\kappa>2^{\aleph_0}$ and $\kappa^{<\kappa}=\kappa$, then there is a c.c.c. forcing which forces $\sf MA+2^{\aleph_0}=\kappa$. Traditionally, we even start with $\sf GCH$, which simplifies even further the cardinal arithmetic, although going through the proof it seems excessive.

Consider the following scenario.

You wake up in a universe; to your left there is a regular cardinal $\kappa$, not smaller than the continuum, which itself is greater than $\aleph_1$. Do you want to try and force $\sf MA+2^{\aleph_0}=\kappa$.

Can we always do it? Can we always do it with a c.c.c. forcing? The latter clearly has a negative answer. If, say, $2^{\aleph_1}=\kappa^+$, then we must either add $\kappa^+$ new reals (which we don't want) or collapse $\kappa^+$ (which we cannot do), since $\sf MA$ implies that $2^{\aleph_0}=2^{\aleph_1}$.

So, what and when can we do things, and how can we do them?

Suppose that $\kappa\geq2^{\aleph_0}$ is a regular cardinal.

1. Can we always force $\sf MA+2^{\aleph_0}=\kappa$ with a proper forcing? (E.g. $\operatorname{Add}(\kappa,1)*\Bbb P_\kappa$, where $\Bbb P_\kappa$ is the standard iteration. What if $\kappa=2^{\aleph_0}$ already in the ground model?)

2. Under which assumptions can we just use a c.c.c.\ forcing? In particular, does $\sf ZFC$ prove that there is always a c.c.c. forcing that forces $\sf MA$, or do things like "For all $\kappa\geq\omega$, $2^\kappa=\kappa^{++}$" get in the way?

We know that if $\kappa>2^{\aleph_0}$ and $\kappa^{<\kappa}=\kappa$, then there is a c.c.c. forcing which forces $\sf MA+2^{\aleph_0}=\kappa$. Traditionally, we even start with $\sf GCH$, which simplifies even further the cardinal arithmetic, although going through the proof it seems excessive.

Consider the following scenario.

You wake up in a universe; to your left there is a regular cardinal $\kappa$, not smaller than the continuum, which itself is greater than $\aleph_1$. Do you want to try and force $\sf MA+2^{\aleph_0}=\kappa$.

Can we always do it? Can we always do it with a c.c.c. forcing? The latter clearly has a negative answer. If, say, $2^{\aleph_1}=\kappa^+$, then we must either add $\kappa^+$ new reals (which we don't want) or collapse $\kappa^+$ (which we cannot do), since $\sf MA$ implies that $2^{\aleph_0}=2^{\aleph_1}$.

So, what and when can we do things, and how can we do them?

Suppose that $\kappa\geq2^{\aleph_0}$ is a regular cardinal.

1. Can we always force $\sf MA+2^{\aleph_0}=\kappa$ with a proper forcing? (E.g. $\operatorname{Add}(\kappa,1)*\Bbb P_\kappa$, where $\Bbb P_\kappa$ is the standard iteration. What if $\kappa=2^{\aleph_0}$ already in the ground model?)

2. Under which assumptions can we just use a c.c.c. forcing? In particular, does $\sf ZFC$ prove that there is always a c.c.c. forcing that forces $\sf MA$, or do things like "For all $\kappa\geq\omega$, $2^\kappa=\kappa^{++}$" get in the way?