MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In the paper "On the consistency strength of projective uniformization" Woodin proves a lemma "Assume $M$ is a model of ZFC that is $\Sigma^{1}_{3}$-absolute. Then $M\vDash\forall x\in\mathbb{R}\,[x^{\sharp} \mathrm{\ exists}]$." He then goes on to say after the proof "It in fact now follows by a theorem of Martin-Solovay [6] that $\Sigma^{1}_{3}$-absoluteness is equivalent to the existence of $S^{\sharp}$ for every set $S$."

[6] Martin, D. A. and Solovay, R. M., A basis theorem for $\Sigma^{1}_{3}$ sets of reals, Ann. of Math. 89 (1969), 138-160.

When I read this article of Martin and Solovay I have trouble seeing the connection with the assertion that $S^{\sharp}$ exists for all sets $S$. I was wondering if anyone could clarify this for me.

share|cite|improve this question
up vote 6 down vote accepted

Rupert, the Martin-Solovay paper shows the absoluteness result follows form the existence of measurable cardinals. The measurables are used in the construction of certain trees (now called Martin-Solovay trees), where some ordinals are chosen by means of the measure to serve as witnesses (of ill-foundedness of some branches, say). There are standard arguments to tighten up this approach so indiscernibles (coming from sharps) suffice for this. (This is closely related to how $\Pi^1_1$-determinacy can be established form sharps rather than measurables.)

I suspect Martin's draft of a book on determinacy has the details, but I do not know whether you have access to it (and I do not have my copy handy at the moment to double-check). In any case, Ralf Schindler and I have a joint paper where we explain in detail how $\Sigma^1_3$-absoluteness is equivalent to the existence of sharps. The paper is "Projective well-orderings of the reals", Arch. Math. Logic (2006) 45:783–793. It is available at my webpage, see Theorem 3.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.