Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert space $\cal H$ by taking the unit ball of $M$ to be the intersection of that of $\cal H$ with the halfspaces defined by $(x, \pm a) \le 1 - \delta_a$ for certain unit vectors $a$ and small $\delta_a >0$. They then need to show that the $\cal H$-unit ball is definable in $M$ using a first order language with symbols for vector addition and membership of the $M$-unit ball. With this in view, in their Lemma 2.9, they claim certain of the vectors $a$ are definable, but they don't have what they need to apply the lemma they appeal to for this (they have an inequality $\|b\| \le (1 - \delta)^{-1}$ but their Lemma 2.3 needs the opposite inequality).

So this looks like a bug. I suspect it can be fixed, e.g., by taking the unit ball to be the convex hull of the proposed one and the vectors $\pm a$. But this would be quite disruptive to the rest of the argument, I suspect.

My questions are (1) have I missed something so that Shelah and Stern's proof does actually go through more or less as its stands, or (2) is there another reference that gives a correct proof of these results.

share|improve this question
add comment

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.