Résume | This talk discusses the power of "almost" order preserving collapsing functions, which map large ordinals (uncountable resp. non-recursive) to smaller ones (countable resp. recursive). More precisely, I will consider collapsing functions in the context of dilators (J.-Y. Girard): Let $D$ be a dilator, i.e. a particularly uniform function from ordinals to ordinals. It can happen that we have $D(\alpha)>\alpha$ for every ordinal $\alpha$, so that $D$ has no fixed-point. The best we can expect is a collapsing function $D(\alpha)\rightarrow\alpha$ that is almost order preserving, in a sense that will be made precise in the talk. If such a function exists, then $\alpha$ is called a Bachmann-Howard fixed-point of $D$. I will show that the following holds over a weak base theory: The statement that "every dilator has a Bachmann-Howard fixed-point" is equivalent to the existence of admissible sets, and hence to $\Pi^1_1$-comprehension (full details can be found in arXiv:1809.06759). |