Narrow Results

We identify a particular mouse, $M^{\mathrm{\text{ld}}}$, the minimal ladder mouse, that sits in the mouse order just past $M_n^{♯}$ for all $n$, and we show that $ℝ\cap M^{\mathrm{\text{ld}}}=Q_{\omega +1}$, the set of reals that are ${\mathrm{\Delta }}_{\omega +1}^1$ in a countable ordinal. Thus $Q_{\omega +1}$ is a mouse set.

This is analogous to the fact that $ℝ\cap M_1^{♯}=Q_3$ where $M_1^{♯}$ is the sharp for the minimal inner model with a Woodin cardinal, and $Q_3$ is the set of reals that are ${\mathrm{\Delta }}_3^1$ in a countable ordinal.

More generally $ℝ\cap M_{2n+1}^{♯}=Q_{2n+3}$. The mouse $M^{\mathrm{\text{ld}}}$ and the set $Q_{\omega +1}$ compose the next natural pair to consider in this series of results. Thus we are proving the mouse set theorem just past projective.

Some of this is not new. $ℝ\cap M^{\mathrm{\text{ld}}}\subseteq Q_{\omega +1}$ was known in the 1990s. But $Q_{\omega +1}\subseteq M^{\mathrm{\text{ld}}}$ was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin’s proof.

Asperó and Schindler have completely solved the Axiom $(\ast )$ vs. ${\mathrm{\text{MM}}}^{++}$ problem. They have proved that if ${\mathrm{\text{MM}}}^{++}$ holds then Axiom $(\ast )$ holds, with no additional assumptions. The key question now concerns the relationship between ${\mathrm{\text{MM}}}^{++}$ and Axiom ${(\ast )}^{+}$. This is because the foundational issues raised by the problem of Axiom $(\ast )$ vs. ${\mathrm{\text{MM}}}^{++}$ arguably persist in the problem of Axiom ${(\ast )}^{+}$ vs. ${\mathrm{\text{MM}}}^{++}$. The first of our two main theorems is that Axiom ${(\ast )}^{+}$ is equivalent to Axiom ${(\ast )}^{++}$, and as a corollary we show that Axiom ${(\ast )}^{+}$ fails in all the known models of ${\mathrm{\text{MM}}}^{++}$. This suggests that ${\mathrm{\text{MM}}}^{++}$ actually refutes Axiom ${(\ast )}^{+}$. Our second main theorem is that the ${\mathrm{\text{AD}}}^{+}$ Conjecture holds assuming ${\mathrm{\text{MM}}}^{++}(c)$. This is the strongest partial result known on this conjecture which is one of the central open problems of ${\mathrm{\text{AD}}}^{+}$-theory and $\mathrm{\Omega }$-logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both ${\mathrm{\Sigma }}_2$-expressible and which implies ${\mathrm{\text{MM}}}^{++}(c)$, on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the ${\mathrm{\Sigma }}_1^2$-sentences with no parameters. Such sentences are those which simply assert the existence of a set $A\subseteq ℝ$ for which some property involving only quantification over $ℝ$ holds.

An elementary embedding $j:M\to N$ between two inner models of $\text{ZFC}$ is cardinal preserving if $M$ and $N$ correctly compute the class of cardinals. We look at the case $N=V$ and show that there is no nontrivial cardinal preserving elementary embedding from $M$ into $V$, answering a question of Caicedo.

We prove that one can construct various kinds of automata over finite words for which some elementary properties are actually independent from strong set theories like $T_n=:ZFC$$+$ “There exist (at least) $n$ inaccessible cardinals”, for integers $n\ge 0$. In particular, we prove independence results for languages of finite words generated by context-free grammars, or accepted by 2-tape or 1-counter automata. Moreover we get some independence results for weighted automata and for some related finitely generated subsemigroups of the set ${\mathbb{Z}}^{3\times 3}$ of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA.

This is the story of my relationship with Patrick Dehornoy and his family, which began in 1996 and ended when Patrick passed away.

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics.

These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC.

In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.

We prove that if λ is a strong limit singular cardinal and κ a regular uncountable cardinal < λ, then NS_κλ, the non-stationary ideal over , is nowhere precipitous. We also show that under the same hypothesis every stationary subset of can be partitioned into λ^κ disjoint stationary sets.

We present a general lemma which allows proving determinacy from Woodin cardinals. The lemma can be used in many different settings. As a particular application we prove the determinacy of sets in , n ≥ 1. The assumption we use to prove determinacy is optimal in the base theory of determinacy.

We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality (see Secs. 4 and 5). They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC (see Sec. 6). It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without power set (see [4]).

We lift Jensen's coding method into the context of Woodin cardinals. By a theorem of Woodin, any real which preserves a "strong witness" to Woodinness is set-generic. We show however that there are class-generic reals which are not set-generic but preserve Woodinness, using "weak witnesses".

We prove determinacy for open length ω₁ games. Going further we introduce, and prove determinacy for, a stronger class of games of length ω₁, with payoff conditions involving the entire run, the club filter on ω₁, and a sequence of ω₁ disjoint stationary subsets of ω₁. The determinacy proofs use an iterable model with a class of indiscernible Woodin cardinals, and we show that the games precisely capture the theory of the minimal model for this assumption.

We extend the results of Laver on using inverse limits to reflect large cardinals of the form, there exists an elementary embedding L_α(V_λ+1) → L_α(V_λ+1). Using these inverse limit reflection embeddings directly and by broadening the collection of U(j)-representable sets, we prove structural results of L(V_λ+1) under the assumption that there exists an elementary embedding j : L(V_λ+1) → L(V_λ+1). As a consequence we show the impossibility of a generalized inverse limit X-reflection result for X ⊆ V_λ+1, thus focusing the study of L(ℝ) generalizations on L(V_λ+1).

In this paper, we show that the failure of the unique branch hypothesis ($UBH$) for tame iteration trees implies that in some homogenous generic extension of $V$ there is a transitive model $M$ containing $\mathrm{\text{Ord}}\cup ℝ$ such that $M\models {AD}_{ℝ}+\mathrm{\Theta }$ is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math.66(4) (2014) 903–923; Core models with more Woodin cardinals, J. Symbolic Logic67(3) (2002) 1197–1226] for tame trees.

We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in $V$ can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal $𝜃$ in $V$ is ${𝜃}^{+}$-supercompact in HOD.

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class $C$ there is a $C$-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing $C$-strongness of some cardinal.

We introduce bounded category forcing axioms for well-behaved classes $\mathrm{\Gamma }$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{{\lambda }_{\mathrm{\Gamma }}^{+}}$ modulo forcing in $\mathrm{\Gamma }$, for some cardinal ${\lambda }_{\mathrm{\Gamma }}$ naturally associated to $\mathrm{\Gamma }$. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation ${\lambda }_{\mathrm{\Gamma }}=\omega$ — to classes $\mathrm{\Gamma }$ with ${\lambda }_{\mathrm{\Gamma }}>\omega$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $\mathrm{\Gamma }$ with ${\lambda }_{\mathrm{\Gamma }}={\omega }_1$ giving rise to pairwise incompatible theories for $H_{{\omega }_2}$.

Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle ${\mathrm{\text{SR}}}^{-}$ introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above by subtleness, and contains all large cardinals that can be characterized through the validity of the principle ${\mathrm{\text{SR}}}^{-}$ for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak ${\mathrm{\Pi }}_n^1$-indescribability, can be canonically characterized through localized versions of the principle ${\mathrm{\text{SR}}}^{-}$. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak ${\mathrm{\Pi }}_1^1$-indescribability.

Assume that there is no transitive class model of $\mathrm{\text{ZFC}}$ with a Woodin cardinal. Let $\nu$ be a singular ordinal such that $\nu >{\omega }_2$ and $\mathrm{\text{cf}}(\nu )<|\nu |$. Suppose $\nu$ is a regular cardinal in K. Then $\nu$ is a measurable cardinal in K. Moreover, if $\mathrm{\text{cf}}(\nu )>\omega$, then $o^K(\nu )\ge \mathrm{\text{cf}}(\nu )$.