What does it mean to say that a forcing "collapses cardinals"?

Question

I hear the following terminology a lot: "So-and-so forcing collapses cardinals." Does this just mean that certain cardinals in the ground model are no longer cardinals in the forcing extension?

If not, what does it mean?

Answer 1

A forcing collapses cardinals iff (by definition) some cardinal of the ground model is no longer a cardinal in the forcing extension.

Naturally, this means that there is some $\kappa$ in the ground model whose cardinality in the extension is strictly smaller than $\kappa$ (e.g., let $\kappa$ be the first cardinal that witnesses the definition above). Note that there may still be ordinals $\beta$ such that the $\aleph_\beta$ of the ground model is the $\aleph_\beta$ of the extension, even if some cardinals strictly smaller than $\aleph_\beta$ have been collapsed. For example, suppose that the $\aleph_1$ of the ground model is no longer a cardinal, while cardinals larger than $\aleph_1$ are preserved (this is achieved via the forcing usually denoted $\mathrm{Col}(\omega,\aleph_1)$). This means that the $\aleph_1$ of the ground model is a countable ordinal in the extension. Since the $\aleph_2$ of the ground model is still a cardinal, it must now be the $\aleph_1$ of the extension. Similarly, $\aleph_3$ becomes $\aleph_2$, etc, but $\aleph_\omega$ stays $\aleph_\omega$ and, for any $\beta\ge\omega$, $\aleph_\beta$ stays $\aleph_\beta$. Note that this applies to $\aleph_{\omega_1^V}$, even though $\omega_1^V$ is no longer a cardinal. In this case, $\aleph_{\omega_1^V}$ is now a cardinal of cofinality $\omega$, and $\aleph_{\omega_1}$ is a larger cardinal, simply because $\omega_1=\omega_2^V>\omega_1^V$.

That said, just because a forcing is a collapse forcing, it does not mean that cardinals are collapsed when it is applied. For example, $\mathrm{Col}(\omega,\omega)$ is just Cohen forcing, which does not collapse any cardinals. Another example is $\mathrm{Col}(\omega,<\omega_1)$. In choiceless contexts, the notation is used sometimes even if some non-well-ordered cardinals are involved. A word of caution is also in order: To say that a forcing collapses a cardinal $\kappa$ does not quite mean that $\kappa$ is no longer a cardinal in the extension. For instance, it may be that $\kappa$ is now a smaller cardinal than it was originally.