Solovay models and forcing extensions

We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly- absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions.

0# and inner models

In this paper we examine the cardinal structure of inner models that satisfy GCH but do not contain 0#. We show, assuming that 0# exists, that such models necessarily contain Mahlo cardinals of high order, but without further assumptions need not contain a cardinal κ which is κ-Mahlo. The principal tools are the Covering Theorem for L and the technique of reverse Easton iteration.Let I denote the class of Silver indiscernibles for L and 〈iα ∣ α ϵ ORD〉 its increasing enumeration. Also fix an inner model M of GCH not containing 0# and let ωα denote the ωα of the model M[0#], the least inner model containing M as a submodel and 0# as an element.

◇ at Mahlo cardinals

Given a Mahlo cardinal k and a regular ε such that ω1 < ε < k we show that ◇k(cf = ε) holds in V provided that there are only non-stationarily many β < k with o(β) ≥ ε in K.

Stretchings

A structure is locally finite if every finitely generated substructure is finite; local sentences are universal sentences all models of which are locally finite. The stretching theorem for local sentences expresses a remarkable reflection phenomenon between the finite and the infinite models of local sentences. This result in part requires strong axioms to be proved; it was studied by the second named author, in a paper of this Journal, volume 53. Here we correct and extend this paper; in particular we show that the stretching theorem implies the existence of inaccessible cardinals, and has precisely the consistency strength of Mahlo cardinals of finite order. And we present a sequel due to the first named author:(i) decidability of the spectrum Sp(φ) of a local sentence φ, below ωω; where Sp(φ) is the set of ordinals α such that φ has a model of order type α(ii) proof that bethω = sup{Sp(φ): φ local sentence with a bounded spectrum}(iii) existence of a local sentence φ such that Sp(φ) contains all infinite ordinals except the inaccessible cardinals.

Regressive partitions and Borel diagonalization

Several rather concrete propositions about Borel measurable functions of several variables on the Hilbert cube (countable sequences of reals in the unit interval) were formulated by Harvey Friedman [F1] and correlated with strong set-theoretic hypotheses. Most notably, he established that a “Borel diagonalization” proposition P is equivalent to: for any a ⊆ co and n ⊆ ω there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. In later work (see the expository Stanley [St] and Friedman [F2]), Friedman was to carry his investigations further into propositions about spaces of groups and the like, and finite propositions. He discovered and analyzed mathematical propositions which turned out to have remarkably strong consistency strength in terms of large cardinal hypotheses in set theory.In this paper, we refine and extend Friedman's work on the Borel diagonalization proposition P. First, we provide more combinatorics about regressive partitions and n-Mahlo cardinals and extend the approach to the context of the Erdös cardinals In passing, a combinatorial proof of a well-known result of Silver about these cardinals is given. Incorporating this work and sharpening Friedman's proof, we then show that there is a level-by-level analysis of P which provides for each n ⊆ ω a proposition almost equivalent to: for any a ⊆ co there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. Finally, we use the combinatorics to bracket a natural generalization Sω of P between two large cardinal hypotheses.