Sy Friedman and I recently submitted the paper 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 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 of as a parameter. The argument uses Justin Moore‘s technique of the Mapping Reflection Principle, and provides us with a well-ordering. In this sense, the result is best possible.

In earlier work, I had shown that is consistent with a projective well-ordering of the reals. The result with Velickovic dramatically improves this; for example, if and holds, then there is a projective well-ordering of Note that this is an implication rather than just a consistency result, and does not require that the universe is a forcing extension of The point here is that the parameter can be chosen in so that it is “projective in the codes,” and then provides us with enough coding machinery to transform the definition of the well-ordering into a projective one.

In the new paper, Friedman and I show that in fact under the projective definition is best possible, For this we need to combine the coding technique giving the 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 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 well-ordering in that case. Also, since turns out, perhaps unexpectedly, to provides us with definable well-orderings, it would be interesting to see that does not suffice for this.

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This is a very interesting question (and I really want to see what other answers you receive). I do not know of any general metatheorems ensuring that what you ask (in particular, about consistency strength) is the case, at least under reasonable conditions. However, arguments establishing the proof theoretic ordinal of a theory $T$ usually entail this. You […]

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You may be interested in the following paper: Lorenz Halbeisen, and Norbert Hungerbühler. The cardinality of Hamel bases of Banach spaces, East-West Journal of Mathematics, 2, (2000) 153-159. There, Lorenz and Norbert prove a few results about the size of Hamel bases of arbitrary infinite dimensional Banach spaces. In particular, they show: Lemma 3.4. If $K\ […]

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For brevity's sake, say that a theory $T$ is nice if $T$ is a consistent theory that can interpret Peano Arithmetic and admits a recursively enumerable set of axioms. For any such $T$, the statement "$T$ is consistent" can be coded as an arithmetic statement (saying that no number codes a proof of a contradiction from the axioms of $T$). What […]