This is a survey lecture about elementary embeddings from $V$ into transitive classes $M$ that are definable in a generic extension $V[G]$. Such embeddings are often the result of assumptions about ideals on sets $X$ and the quotient algebra $\mathcal P(X)$. Why are they interesting?

• They often settle questions like the CH or GCH.
• They can imply properties such as ``$\aleph_\omega$ is Jonsson."
• Non-regular ultrafilters on small cardinals can be constructed using them—even ultrafilters giving very small ultrapowers.
• If the associated ideals are `"determined" then they imply the existence of inner models of cardinals such as $n$-huge cardinals. (The inner models are NOT fine structural.)
• This gives equiconsistency results....

Speculation: They MAY be relevant to questions like ``Kunen's Question."