Assume V = L ⊨ ZF + DC + Θ > ω2 + μ is a normal fine measure on.

We show that it is relatively consistent with ZFC that there is a projective wellfounded relation with rank higher than all projective prewellorderings.

We describe the inner models with representatives in all equivalence classes of thin equivalence relations in a given projective pointclass of even level assuming projective determinacy. The main result shows that these models are characterized by their correctness and the property that they correctly compute the tree from the appropriate scale. The main step towards this characterization shows that the tree from a scale can be reconstructed in a generic extension of an iterate of a mouse. We then construct models (...) with this property as generic extensions of iterates of mice under the assumption that the corresponding projective ordinal is below ω2ω2. On the way, we consider several related problems, including the question when forcing does not add equivalence classes to thin projective equivalence relations. For instance, we show that if every set has a sharp, then reasonable forcing does not add equivalence classes to thin provably View the MathML source Δ1/3 equivalence relations, and generalize this to all projective levels. (shrink)

Fix a cardinal κ. We can ask the question: what kind of a logic L is needed to characterize all models of cardinality κ up to isomorphism by their L-theories? In other words: for which logics L it is true that if any models A and B of cardinality κ satisfy the same L-theory then they are isomorphic?It is always possible to characterize models of cardinality κ by their Lκ+,κ+-theories, but we are interested in finding a “small” logic L, i.e., (...) the sentences of L are hereditarily of smaller cardinality than κ. For any cardinal κ it is independent of ZFC whether any such small definable logic L exists. If it exists it can be second order logic for κ=ω and fourth order logic or certain infinitary second order logic Lκ,ω source for uncountable κ. All models of cardinality κ can always be characterized by their theories in a small logic with generalized quantifiers, but the logic may be not definable in the language of set theory. Our work continues and extends the work of Ajtai [Miklos Ajtai, Isomorphism and higher order equivalence, Ann. Math. Logic 16 181–203]. (shrink)

We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal$M_1$, there is a model of$\mathsf {ZFC}$in which$\mbox {NS}_{\omega _{1}}$is$\aleph _2$-saturated and${\Delta }_{1}$-definable with$\omega _1$as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with$\mbox {NS}_{\omega _{1}}$saturated and${\Delta }_{1}$-definable with a ladder system$\vec {C}$and a full Suslin treeTas parameters. Both results rely on (...) a new coding technique whose presentation is the main goal of this article. (shrink)

We show that under $\mathsf {BMM}$ and “there exists a Woodin cardinal, $"$ the nonstationary ideal on $\omega _1$ cannot be defined by a $\Pi _1$ formula with parameter $A \subset \omega _1$. We show that the same conclusion holds under the assumption of Woodin’s $(\ast )$ -axiom. We further show that there are universes where $\mathsf {BPFA}$ holds and $\text {NS}_{\omega _1}$ is $\Pi _1(\{\omega _1\})$ -definable. Lastly we show that if the canonical inner model with one Woodin cardinal (...) $M_1$ exists, there is a generic extension of $M_1$ in which $\text {NS}_{\omega _1}$ is saturated and $\Pi _1(\{ \omega _1\} )$ -definable, and $\mathsf {MA_{\omega _1}}$ holds. (shrink)

The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. $\underset{\sim}{\Pi}$ determinacy implies that for every thin Σ 1 2 equivalence relation there is a Δ 1 3 real, N, over which every equivalence class is generic--and hence there is a good Δ 1 2 (N ♯ ) wellordering of the equivalence classes. Analogous results are obtained for Π 1 2 and Δ 1 2 quasilinear orderings and $\underset{\sim}{\Pi}^1_2$ determinacy is shown to imply that (...) every Π 1 2 prewellorder has rank less than $\underset{\sim}{\delta}^1_2$. (shrink)

The Axiom of Projective Determinacy implies the existence of a universal $\utilde{\Pi}^{1}_{n}\setminus\utilde{\Delta}^{1}_{n}$ set for every $n \geq 1$. Assuming $\text{\upshape MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\utilde{\Pi}^{1}_{1}\setminus\utilde{\Delta}^{1}_{1}$ set. In ZFC there is a universal $\utilde{\Pi}^{0}_{\alpha}\setminus\utilde{\Delta}^{0}_{\alpha}$ set for every $\alpha$.

In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo–Fraenkel set theory $\mathrm {ZFC}^{-}$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\mathrm {ZFC}^{-}$ that are bi-interpretable, but not isomorphic—even $\langle H_{\omega _1},\in (...) \rangle $ and $ \langle H_{\omega _2},\in \rangle $ can be bi-interpretable—and there are distinct bi-interpretable theories extending $\mathrm {ZFC}^{-}$. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. (shrink)

In this paper we consider whether L(R) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.

We show that is consistent with the existence of a -definable wellorder of the reals and a -definable ω-mad subfamily of [ω]ω.

We show that the existence of a ${\Pi^1_1}$ -definable mad family is consistent with the existence of a ${\Delta^{1}_{3}}$ -definable well-order of the reals and ${\mathfrak{b}=\mathfrak{c}=\aleph_3}$.

Using countable support iterations of S-proper posets, we show that the existence of a definable wellorder of the reals is consistent with each of the following: , and.

We extend the work of Fischer et al. [6] by presenting a method for controlling cardinal characteristics in the presence of a projective wellorder and 2ℵ0>ℵ2. This also answers a question of Harrington [9] by showing that the existence of a Δ31 wellorder of the reals is consistent with Martinʼs axiom and 2ℵ0=ℵ3.

If there is no inner model with ω many strong cardinals, then there is a set forcing extension of the universe with a projective well-ordering of the reals.

If A ⊆ ω1, then there exists a cardinal preserving generic extension [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ][x ] of [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ] by a real x such that1) A ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ] and A is Δ1HC in [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ];2) x is minimal over [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ], that is, if a set Y belongs to [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][x ], then either x ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A, Y ] or Y (...) ∈ [MATHEMATICAL DOUBLE-STRUCK CAPITAL L][A ].The forcing we use implicitly provides reshaping of the given set A. (shrink)

In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

Any model of ZFC + GCH has a generic extension (made with a poset of size ℵ 2 ) in which the following hold: MA + 2 ℵ 0 = ℵ 2 +there exists a Δ 2 1 -well ordering of the reals. The proof consists in iterating posets designed to change at will the guessing properties of ladder systems on ω 1 . Therefore, the study of such ladders is a main concern of this article.

In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.