In set theory, definitely the notion of a Woodin cardinal.

First, it is not an entirely straightforward notion to guess. Significant large cardinals were up to that point defined as critical points of certain elementary embeddings. This is not the case here: Woodin cardinals need not be measurable. If $\kappa$ is Woodin, then $V_\kappa$ is a model of set theory where there is a proper class of strong cardinals. Woodinness requires more, namely, that these strong embeddings move some predicates correctly.

Second, the definition turned out to identify a pivotal point in inner model theory: for notions weaker than Woodinness, the corresponding canonical inner models carry $\mathbf\Delta^1_3$ well-orderings of the reals. This is no longer true once we have Woodin cardinals. This is closely tied up to the complexity of the comparison process: given two set models that look like initial segments of canonical inner models, how hard is to compare them to tell which one carries more information? For notions weaker than Woodinness, this process is carried out in an essentially linear fashion: disagreements between the models are witnessed by some measures, and repeatedly using these least measures to form ultrapowers eventually lines the models up: One of the iterates ends up as an initial segment of the other, and whichever one is longer comes from the model which originally has more information.With Woodin cardinals and beyond this process is no longer enough. Instead, comparisons sometimes need to retrace steps, and rather than a linear iteration, at the end we have tree-like structures. Identifying these crucial differences allowed us to develop inner model theory beyond this point. This in turn has led to many results and to the identification of deep connections between large cardinals and descriptive set theory. Literally, thanks to the presence of Woodin cardinals, the set-theoretic landscape grew and transformed significantly.

Third, Woodinness also turns out to be the notion needed to carry out certain forcing constructions. Some consistency results that were not expected now could be established, thanks to the identification of genuinely new forcing notions that use the Woodin cardinals in an essential way. Other constructions that were known from significant large cardinals were improved to their optimal form.

The story of how the notion of Woodinness was identified is actually quite nice. Kunen had used huge cardinals to prove the consistency of the existence of saturated ideals on $\omega_1$. The embeddings witnessing hugeness end up lifting to the generic embeddings witnessing saturation in the appropriate forcing extension. In 1984, Foreman, Magidor and Shelah identified an entirely new way of finding models with saturated ideals. Their construction improved the large cardinal notion needed from hugeness to supercompactness. What is significant is that the generic embedding is no longer a lifting of an old genuine embedding. Indeed, their construction preserves $\omega_1$. As a consequence of their results, in May of the same year, Woodin showed that supercompactness implies that all projective sets of reals are Lebesgue measurable (and more). Conversations between Shelah and Woodin quickly led to the realization that much weaker cardinals than supercompact sufficed for this result. Through a series of refinements, the notions of what we now call Shelah and Woodin cardinals were identified, with the latter being of precisely the right strength: all this happened while developments in determinacy and inner model theory showed the deep connections between these fields, and the pivotal role that Woodin cardinals played in this connection. This all happened quite quickly: by the time the Shelah-Woodin paper appeared in print, the importance of Woodin cardinals was already recognized.

MR1074499 (92m:03087). Shelah, Saharon; Woodin, Hugh. *Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable*. Israel J. Math. **70** (1990), no. 3, 381–394.

weren'tcrucial to further understanding. $\endgroup$ – Nik Weaver Dec 3 '18 at 16:256more comments