# General theory of the reals in Solovay-like models

Solovay's model is a famous model of $$\sf ZF$$ where we start in $$L$$ with $$\kappa$$ inaccessible, and we collapse all the ordinals below $$\kappa$$ to be countable, without collapsing $$\kappa$$ itself, and then considering $$L(\Bbb R)$$.

Solovay proved that in that model every set of reals has nice regularity properties such as Lebesgue measurability, the Baire property, Ramsey property, etc.

Truss proved that we can mimic the same construction from any limit ordinal, and we simply pay the price that $$\omega_1$$ is singular, and the majority of properties no longer make "inherent sense" with their simplified definitions (as many rely on $$\sf DC$$) and while we can switch to "code-based definitions", this is generally hard to do, so we don't really do it very often.

Other, similar, models are often used in the study of Determinacy axioms. Here we start with a sequence of Woodin cardinals, $$\delta_n$$ with $$\delta$$ being their supremum, and we collapse all the Woodin cardinals to be countable without collapsing $$\delta$$ itself. Again we look at $$L(\Bbb R)$$, and here things work out like in Solovay's model, mainly because the large cardinals imply that $$\delta$$ is in fact regular in $$L(x)$$ for any $$x\in V_\delta$$. In particular, the $$\omega$$-sequence of the Woodin cardinals is not computable from any of the reals.

But now we can wonder. What happens when we take any limit of large enough cardinals?

Definition. We say that $$\delta$$ is a sharp cardinal if for every $$x\in V_\delta$$, $$x^\#$$ exists.

Remark. Note that if $$\delta$$ is a limit of Ramsey cardinals, then it is a sharp cardinal.

Suppose now that $$\delta$$ is a sharp cardinal, then we can look at $$L(\Bbb R)$$ when we collapse every $$\alpha<\delta$$ to be countable. Then this is a Solovay-style model satisfying $$\sf DC$$ and nice regularity properties.

We know have some understanding of the theory of this model when $$\delta$$ is a limit of Woodin cardinals.

What can we say about the theory of the reals in Solovay models obtained from sharp cardinals well below Woodin cardinals, e.g. limits of measurable, Ramsey, or even just the least sharp cardinal?

• A natural problem would be to look at consequences of AD that don't hold in the inaccessible Solovay model, but might hold in a Solovay model starting from a stronger large cardinal hypothesis. One such example that might be relevant is the conjunction of projective uniformization, projective Lebesgue measurability and projective Baire property, which was conjectured to be equivalent to PD, but was shown by Steel to hold in an appropriate $V^{Col(\omega, \lambda)}$ under a weaker large cardinal hypothesis. This is discussed in "Projective uniformization revisited" by Hauser and Schindler.
– Haim
Feb 5, 2020 at 3:38
• Thanks, Haim. This is the sort of question I'm asking here, yes. I'm kind of interested in some commonalities, though. Which I expect we can find in the least sharp cardinal. (The dullest cardinal? I don't know, there's a joke about blades here, figure it out, you're a smart guy.) Feb 5, 2020 at 10:33