## BPFA and projective well-orderings of the reals

Sy Friedman and I recently submitted the paper ${\sf BPFA}$ and projective well-orderings of the reals to The Journal of Symbolic Logic. The preprint is available at my papers page.

In a previous paper, Boban Velickovic and I showed that if ${\sf BPFA},$ the bounded version of the proper forcing axiom, holds, then one can define a well-ordering of the reals using what is in essence a subset $C$ of $\omega_1$ as a parameter. The argument uses Justin Moore‘s technique of the Mapping Reflection Principle, and provides us with a $\Delta_1(C)$ well-ordering. In this sense, the result is best possible.

In earlier work, I had shown that ${\sf BPFA}$ is consistent with a projective well-ordering of the reals. The result with Velickovic dramatically improves this; for example, if $\omega_1=\omega_1^L$ and ${\sf BPFA}$ holds, then there is a projective well-ordering of ${\mathbb R}.$ Note that this is an implication rather than just a consistency result, and does not require that the universe is a forcing extension of $L.$ The point here is that the parameter $C$ can be chosen in $L$ so that it is “projective in the codes,” and ${\sf MA}_{\omega_1}$ then provides us with enough coding machinery to transform the $\Delta_1(C)$ definition of the well-ordering into a projective one.

In the new paper, Friedman and I show that in fact under ${\sf BPFA}+\omega_1=\omega_1^L,$ the projective definition is best possible, $\Sigma^1_3.$ For this we need to combine the coding technique giving the $\Delta_1(C)$ well-ordering with a powerful coding device in the absence of sharps, what Friedman calls David’s trick. The point now is that the forcing required to add the witnesses that make the coding work is proper, and ${\sf BPFA}$ suffices to grant in the universe the existence of these objects.

As a technical problem, it would be interesting to see whether the appeal to the mapping reflection principle can be eliminated here. We only obtain a $\Sigma^1_4$ well-ordering in that case. Also, since ${\sf BPFA}$ turns out, perhaps unexpectedly, to provides us with definable well-orderings, it would be interesting to see that ${\sf MA}_{\omega_1}$ does not suffice for this.