Andrés Eduardo Caicedo

Projective Well-orderings of the Reals

If there is no inner model with ω many strong cardinals, then there is a set forcing extension of the universe with a projective well-ordering of the reals.

Cardinal preserving elementary embeddings

Say that an elementary embedding j : N → M is cardinal preserving if CAR^M = CAR^N = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V. We also show that no ultrapower embedding j : V → M induced by a set extender is cardinal preserving, and present some results on the large cardinal strength of the assumption that there is a cardinal preserving j : V → M.

Real-valued Measurable Cardinals and Well-orderings of the Reals

We show that the existence of atomlessly measurable cardinals is incompatible with the existence of well-orderings of the reals in L(ℝ), but consistent with the existence of well-orderings of the reals that are third-order definable in the language of arithmetic. Specifically, we provide a general argument that, starting from a measurable cardinal, produces a forcing extension where c is real-valued measurable and there is a Δ 2 2 -well-ordering of ℝ. A variation of this idea, due to Woodin, gives Σ 1 2 -well-orderings when applied to L[μ] or, more generally, Σ 1 2 (Hom∞) if applied to nice inner models, provided enough large cardinals exist in V. We announce a recent result of Woodin indicating how to transform this variation into a proof from large cardinals of the Ω-consistency of real-valued measurability of c together with the existence of Σ 1 2 -definable well-orderings of ℝ. It follows that if the Ω-conjecture is true, and large cardinals are granted, then this statement can always be forced.

Downward transference of mice and universality of local core models

If M is a proper class inner model of ZFC and omega_2^M=omega_2, then every sound mouse projecting to omega and not past 0-pistol belongs to M. In fact, under the assumption that 0-pistol does not belong to M, K^M \| omega_2 is universal for all countable mice in V. Similarly, if M is a proper class inner model of ZFC, delta>omega_1 is regular, (delta^+)^M = delta^+, and in V there is no proper class inner model with a Woodin cardinal, then K^M \| delta is universal for all mice in V of cardinality less than delta.

Regressive functions on pairs

We compute an explicit upper bound for the regressive Ramsey numbers by a combinatorial argument, the corresponding function being of Ackermannian growth. For this, we look at the more general problem of bounding g(n,m), the least l such that any regressive function f:[m,l]^[^2^]->N admits a min-homogeneous set of size n. An analysis of this function also leads to the simplest known proof that the regressive Ramsey numbers have a rate of growth at least Ackermannian. Together, these results give a purely combinatorial proof that, for each m,g(@?,m) has a rate of growth precisely Ackermannian, considerably improve the previously known bounds on the size of regressive Ramsey numbers, and provide the right rate of growth of the levels of g. For small numbers we also find bounds on their values under g improving those provided by our general argument.