We prove a reflection property, with respect to forcibility of \(\Sigma_2\) sentences, for \(L(V_\delta)\), where \(\delta\) is the least ordinal \(\gamma\) which is a Woodin cardinal in \(L(V_\gamma)\).
We prove that the forcing axiom \(MA^{1.5}_{\aleph_2}(\mbox{stratified})\) implies \(\Box_{\omega_1, \omega_1}\). Using this implication, we show that the forcing axiom \(MM_{\aleph_2}(\aleph_2\mbox{-c.c.})\) is inconsistent. We also derive weak Chang's Conjecture from \(MA^{1.5}_{\aleph_2}(\mbox{stratified})\) and use this second implication to give another proof of the inconsistency of \(MM_{\aleph_2}(\aleph_2\mbox{-c.c.})\).
We show that the Proper Forcing Axiom for forcing notions of size \(\aleph_1\) is consistent with the continuum being arbitrarily large. In fact, assuming GCH holds and \(\kappa\geq\omega_\)$ is a regular cardinal, we prove that there is a proper and \(\aleph_2\)-c.c. forcing P giving rise to a model of this forcing axiom together with \(2^{\aleph_0}=\kappa\) and which, in addition, satisfies all statements of the form \(H(\aleph_2)\models \forall x\exists y\varphi(x, y)\), where \(\varphi(x, y)\) is a restricted formula with the property that for every \(a\in H(\aleph_2)\) and every inner model M of CH with \(a\in M\) there is, in M, a suitably nice proper poset adding some b such that \(\varphi(a, b)\). In particular, P forces Moore's Measuring principle, Baumgartner's Axiom for \(\aleph_1\)-dense sets of reals, and Todorcevic's P-ideal Dichotomy for \(\aleph_1\)-generated ideals on \(\omega_1\), among other statements. In particular, all these statements are simultaneously compatible with a large continuum.
We introduce a new method for building models of CH, together with \(\Pi_2\) statements over \(H(\omega_2)\), by forcing. Unlike other forcing constructions in the literature, our construction adds new reals, although only \(\aleph_1\)-many of them. Using this approach, we build a model in which a very strong form of the negation of Club Guessing at \(\omega_1\) known as Measuring holds together with CH, thereby answering a well-known question of Moore. This construction can be described as a finite-support weak forcing iteration with side conditions consisting of suitable graphs of sets of models with markers. The CH-preservation is accomplished through the imposition of copying constraints on the information carried by the condition, as dictated by the edges in the graph.
We show that Martin's Maximum\({}^{++}\) implies Woodin's \({\mathbb P}_{\rm max}\) axiom \((*)\). This answers a question from the 1990's and amalgamates two prominent axioms of set theory which were both known to imply that there are \(\aleph_2\) many real numbers.
Starting from the existence of a weakly compact cardinal, we build a generic extension of the universe in which GCH holds and all \(\aleph_2\)-Aronszajn trees are special and hence there are no \(\aleph_2\)-Souslin trees. This result answers a longstanding open question from the 1970's.
We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of \(\omega_1\) is measured by some club subset of \(\omega_1\). The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from D. Aspero and M.A. Mota's "Few new reals" about parameterized measuring principles. Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for \(\sigma\)-centered forcings, as well as from BPFA. We also consider strong versions of Measuring in the absence of the Axiom of Choice.
We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to DC-preserving symmetric submodels of forcing extensions. Hence, ZF+DC not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in ZF+DC. Our results confirm ZF+DC as a natural foundation for a significant portion of "classical mathematics" and provide support to the idea of this theory being also a natural foundation for a large part of set theory.
We introduce bounded category forcing axioms for well-behaved classes \(\Gamma\). These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe \(H_{\lambda^+_\Gamma}\) modulo forcing in \(\Gamma\), for some cardinal \(\lambda_\Gamma\) naturally associated to \(\Gamma\). These axioms naturally extend projective absoluteness for arbitrary set-forcing—in this situation \(\lambda_\Gamma = \omega\)—to classes \(\Gamma\) with \(\lambda_\Gamma > \omega\). 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 \(\Gamma\) with \(\lambda_\Gamma = \omega_1\) giving rise to pairwise incompatible theories for \(H_{\omega_2}\).
Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal \(\kappa\). We show the consistency of \(E^{\lambda^{++},\lambda^{++}}_{\lambda\text{-club}}\), the relation of equivalence modulo the non-stationary ideal restricted to \(S^{\lambda^{++}}_\lambda\) in the space \((\lambda^{++})^{\lambda^{++}}\), being continuously reducible to \(E^{2,\lambda^{++}}_{\lambda^+\text{-club}}\), the relation of equivalence modulo the non-stationary ideal restricted to \(S^{\lambda^{++}}_{\lambda^+}\) in the space \(2^{\lambda^{++}}\). Then we that for \(\kappa\) ineffable \(E^{2, \kappa}_{\text{reg}}\), the relation of equivalence modulo the non-stationary ideal restricted to regular cardinals in the space \(2^{\kappa}\), is \(\Sigma^1_1\)-complete. We finish by showing, for \(\Pi_2^1\)-indescribable \(\kappa\), that the isomorphism relation between dense linear orders of cardinality \(\kappa\) is \(\Sigma^1_1\)-complete.
I construct, in ZFC, a forcing notion that collapses \(\aleph_3\) and preserves all other cardinals. The existence of such a forcing answers a question of Uri Abraham from 1983.
The familiar continuum \(\mathbb{R}\) of real numbers is obtained by a well-known procedure which, starting with the set of natural numbers \(\mathbb{N}=\omega\), produces in a canonical fashion the field of rationals \(\mathbb{Q}\) and, then, the field \(\mathbb{R}\) as the completion of \(\mathbb{Q}\) under Cauchy sequences (or, equivalently, using Dedekind cuts). In this article, we replace \(\omega\) by any infinite suitably closed ordinal \(\kappa\) in the above construction and, using the natural (Hessenberg) ordinal operations, we obtain the corresponding field \(\kappa\)-\(\mathbb{R}\), which we call the field of the \(\kappa\)-reals. Subsequently, we study the properties of the various fields \(\kappa\)-\(\mathbb{R}\) and develop their general theory, mainly from the set-theoretic perspective. For example, we investigate their connection with standard themes such as forcing and descriptive set theory.
I define a homogeneous product forcing for adding many clubs of \(\omega_1\) with finite conditions. I use this forcing to build models of \(b(\omega_1)=\aleph_2\), together with \(2^{\aleph_0}=2^{\aleph_1}\) large and very strong failures of Club Guessing at \(\omega_1\).
This is a (non-exhaustive) collection of results addressing the question "If \(A\) is such that \(P(A)\), does there exists a \(B\) such that \(Q(B)\) and \(B\) is definable from \(A\)?'', for various properties \(P(x)\), \(Q(x)\), as well as closely related questions. The focus is on classical combinatorial properties at the level of \(H(\omega_2)\).
I answer a question of Shelah by showing that if \(\kappa\) is a regular cardinal such that \(2^{{<}\kappa}=\kappa\), then there is a \({<}\kappa\)-closed partial order preserving cofinalities and forcing that for every club-sequence \(\langle C_\delta\mid \delta\in \kappa^+\cap cf(\kappa)\rangle\) with \(ot(C_\delta)=\kappa\) for all \(\delta\) there is a club \(D\subseteq\kappa^+\) such that \(\{\alpha<\kappa\mid \{C_\delta(\alpha+1), C_\delta(\alpha+2)\}\subseteq D\}\) is bounded for every \(\delta\). This forcing is built as an iteration with \({<}\kappa\)-supports and with symmetric systems of submodels as side conditions.
We separate various weak forms of Club Guessing at \(\omega_1\) in the presence of \(2^{\aleph_0}\) large, Martin's Axiom, and related forcing axioms. We also answer a question of Abraham and Cummings concerning the consistency of the failure of a certain polychromatic Ramsey statement together with the continuum large. All these models are generic extensions via finite support iterations with symmetric systems of structures as side conditions, possibly enhanced with \(\omega\)-sequences of predicates, and in which the iterands are taken from a relatively small class of forcing notions. We also prove that the natural forcing for adding a large symmetric system of structures (the first member in all our iterations) adds \(\aleph_1\)-many reals but preserves CH.
For any given uncountable cardinal \(\kappa\) with \(\kappa^{{<}\kappa}=\kappa\), we present a forcing that is \(<\kappa\)-directed closed, has the \(\kappa^+\)-c.c. and introduces a lightface definable well-order of \(H(\kappa^+)\). We use this to define a global iteration that does this for all such \(\kappa\) simultaneously and is capable of preserving the existence of many large cardinals in the universe.
We define the \(\aleph_{1.5}\)-chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom; in fact, \(MA^{1.5}_{<\kappa}\) implies \(MA_{<\kappa}\). Also, \(MA^{1.5}_{<\kappa}\) implies certain uniform failures of club-guessing on \(\omega_1\) that do not seem to have been considered in the literature before. We show, assuming CH and given any regular cardinal \(\kappa\geq\omega_2\) such that \(\mu^{\aleph_0}< \kappa\) for all \(\mu < \kappa\) and such that \(\diamondsuit(\{\alpha<\kappa\,:\, cf(\alpha)\geq\omega_1\})\) holds, that there is a proper \(\aleph_2\)-c.c. partial order of size \(\kappa\) forcing \(2^{\aleph_0}=\kappa\) together with \(MA^{1.5}_{<\kappa}\).
We develop a new method for building forcing iterations with symmetric systems of structures as side conditions. Using this method we prove that the forcing axiom for the class of all finitely proper posets of size \(\aleph_1\) is compatible with \(2^{\aleph_0}> \aleph_2\). In particular, this answers a question of Moore by showing that mho does not follow from this arithmetical assumption.
We isolate natural strengthenings of Bounded Martin's Maximum which we call \(BMM^*\) and \(A\)-\(BMM^{*,++}\) (where \(A\) is a universally Baire set of reals), and we investigate their consequences. We also show that if \(A\)-\(BMM^{*,++}\) holds true for every set of reals \(A\) in \(L({\mathbb R})\), then Woodin's axiom \((*)\) holds true. We conjecture that \(MM^{++}\) implies \(A\)-\(BMM^{*,++}\) for every \(A\) which is universally Baire.
We study the spectrum of forcing notions between the iterations of \(\sigma\)-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of \(\alpha\)-proper forcings for indecomposable countable ordinals \(\alpha\) as well as the Axiom A forcings. We focus on the bounded forcing axioms for the hierarchy of \(\alpha\)-proper forcings and connect them to a hierarchy of weak club guessing principles. We show that they are, in a sense, dual to each other. In particular, these weak club guessing principles separate the bounded forcing axioms for distinct countable indecomposable ordinals. In the study of forcings completely embeddable into an iteration of \(\sigma\)-closed followed by ccc forcing, we present an equivalent characterization of this class in terms of Baumgartner's Axiom A. This resolves a well-known conjecture of Baumgartner from the 1980's.
There exist sentences \(\psi_1\) and \(\psi_2\) which are \(\Pi_2\) over the
structure \(\langle H(\omega_2), \in, \omega_1\rangle\) such that
(1) \(\psi_{2}\) can be forced by a proper forcing not adding \(\omega\)-sequences of ordinals;
(2) If there exists a strongly inaccessible limit of measurable cardinals, then \(\psi_1\)
can be forced by a proper forcing which does not add \(\omega\)-sequences of ordinals;
(3) The conjunction of \(\psi_1\) and \(\psi_2\) implies that \(2^{\aleph_0} = 2^{\aleph_1}\).
Assuming \(2^{\aleph_0}=\aleph_1\) and \(2^{\aleph_1}=\aleph_2\), we build a partial order that forces the existence of a well-order of \(H(\omega_2)\) lightface definable over \(\langle H(\omega_2), \in\rangle\) and that preserves cardinal exponentiation and cofinalities.
Several situations are presented in which there is an ordinal \(\gamma\) such that \(\{ X \in [\gamma]^{\aleph_0} \,:\, X \cap \omega_1 \in S \textrm{ and } ot(X) \in T \}\) is a stationary subset of \([\gamma]^{\aleph_0}\) for all stationary \(S\), \(T \subseteq \omega_1\). A natural strengthening of the existence of an ordinal \(\gamma\) for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of \(H_{\omega_2}\) and the existence of sharps for all reals. Also, an optimal model separating BSPFA and BMM is produced and it is shown that a strong form of BMM involving only parameters from \(H_{\omega_2}\) implies that every function from \(\omega_1\) into \(\omega_1\) is bounded on a club by a canonical function.
There is a partial order \(\mathbb P\) preserving stationary subsets of \(\omega_1\) and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to \(\omega_1\) over V also collapses \(\omega_1\) over \(V^{\mathbb P}\). The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real \(c\), there are, for every real \(a\) in \(V[c]\), sets \(A\) and \(B\) such that \(c\) is Cohen generic over both \(L[A]\) and \(L[B]\) but \(a\) is constructible from \(A\) together with \(B\).
We present several forcing posets for adding a non-reflecting stationary subset of \(\mathcal P_{\omega_1}(\lambda)\), where \(\lambda \geq \omega_2\). We prove that PFA is consistent with dense non-reflection in \(\mathcal P_{\omega_1}(\lambda)\), which means that every stationary subset of \(\mathcal P_{\omega_1}(\lambda)\) contains a stationary subset which does not reflect to any set of size \(\aleph_1\). If \(\lambda\) is singular with countable cofinality, then dense non-reflection in \(\mathcal P_{\omega_1}(\lambda)\) follows from the existence of squares.
By forcing over a model of ZFC + GCH (above \(\aleph_0\)) with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all leves \(H(\kappa^+)\) (\(\kappa\geq \omega_2\) a regular cardinal) is a well-order of \(H(\kappa^+)\) definable over the structure \(\langle H(\kappa^+), \in\rangle\) by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular supercompactness. It is also possible to define varaiants of this construction which, in addition to forcing a locally defined well-order of the universe, preserve many of the \(n\)-huge cardinals from the ground model (for all \(n\)).
It is possible to control to a large extent, via semiproper
forcing, the parameters \((\beta_0,\,\beta_1)\) measuring the
guessing density of the members of any given antichain of
stationary subsets of \(\omega_1\) (assuming the existence of an
inaccessible limit of measurable cardinals). Here, given a pair
\((\beta_0,\,\beta_1)\) of ordinals, we will say that a stationary set
\(S\subseteq\omega_1\) has guessing density \((\beta_0,\,\beta_1)\) if \(\beta_0 = \gamma(S)\)
and \(\beta_1 = sup\{\gamma(S^\ast)\,:\,S^\ast\subseteq S,\,S^\ast\textrm{
stationary}\}\), where \(\gamma(S^\ast)\) is, for every stationary
\(S^\ast\subseteq\omega_1\), the infimum of the set of ordinals
\(\tau\leq\omega_1+1\) for which there a function \(F:S^\ast\longrightarrow \mathcal
P(\omega_1)\) with \(ot(F(\nu))<\tau\) for all \(\nu\in S^\ast\) and with
\(\{\nu\in S^\ast\,:\,g(\nu)\in F(\nu)\}\) stationary for every
\(\alpha<\omega_2\) and every canonical function \(g\) for \(\alpha\). This work
involves an analysis of iterations of models of set theory
relative to sequences of measures on possibly distinct measurable
cardinals.
As an application of these techniques I show how to force, from
the existence of a supercompact cardinal, a model of \(PFA^{++}\) in
which there is a well-order of \(H(\omega_2)\) definable, over \(\langle
H(\omega_2),\in\rangle\), by a formula without parameters.
This paper is mainly a survey of recent results concerning the possibility of building forcing extensions in which there is a simple definition, over the structure \(\langle H(\omega_2), \in\rangle\) and without parameters, of a prescribed member of \(H(\omega_2)\) or of a well-order of \(H(\omega_2)\). Some of these results are in conjunction with strong forcing axioms like \(PFA^{++}\) or \(MM\), some are not. I also observe (Corollary 4.4) that the existence of certain objects of size \(\aleph_1\) follows outright from the existence of large cardinals. This observation is motivated by an attempt to extend the \(PFA^{++}\) result to a result mentioning \(MM^{++}\).
Given any subset \(A\) of \(\omega_1\) there is a proper partial order which forces that the predicate \(x\in A\) and the predicate \(x \in \omega_1 \setminus A\) can be expressed by ZFC-provably incompatible \(\Sigma_3\) formulas over the structure \(\langle H_{\omega_2} , \in, NS_{\omega_1}\rangle\). Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of \(H_{\omega_2}\) definable over \(\langle H_{\omega_2}, \in, NS_{\omega_1}\rangle\) by a provably antisymmetric \(\Sigma_3\) formula with two free variables. The proofs of these results involve a technique for manipulating the guessing properties of club-sequences defined on stationary subsets of \(\omega_1\) at will in such a way that the \(\Sigma_3\) theory of \(\langle H_{\omega_2} , \in, NS_{\omega_1}\rangle\) with countable ordinals as parameters is forced to code a prescribed subset of \(\omega_1\). On the other hand, using theorems due to Woodin it can be shown that, in the presence of sufficiently strong large cardinals, the above results are close to optimal from the point of view of the Levy hierarchy.
Several results are presented concerning the existence or nonexistence, for a subset \(S\) of \(\omega_1\), of a real \(r\) which works as a robust code for \(S\) with respect to a given sequence \((S_\alpha : \alpha < \omega_1)\) of pairwise disjoint stationary subsets of \(\omega_1\), where "robustness" of \(r\) as a code may either mean that \(S \in L[r, (S^*_\alpha \,:\, \alpha < \omega_1)]\) whenever each \(S^*_\alpha\) is equal to \(S_\alpha\) modulo nonstationary changes, or may have the weaker meaning that \(S \in L[r, (S_\alpha \cap C : \alpha < \omega_1)]\) for every club \(C \subseteq \omega_1\). Variants of the above theme are also considered which result when the requirement that \(S\) gets exactly coded is replaced by the weaker requirement that some set is coded which is equal to \(S\) up to a club, and when sequences of stationary sets are replaced by decoding devices possibly carrying more information (like functions from \(\omega_1\) into \(\omega_1\)).
For every uncountable regular cardinal \(\kappa\), every \(\kappa\)-Borel partition of the space of all members of \([\kappa]^{\kappa}\) whose enumerating function does not have fixed points has a homogeneous club.
We prove that a form of the Erd\H{o}s property (consistent with \(V = L[H_{\omega_2}]\) and strictly weaker than the Weak Chang's Conjecture at \(\omega_1\)), together with Bounded Martin's Maximum implies that Woodin's principle \(\psi_{AC}\) holds, and therefore \(2^{\aleph_0} = \aleph_2\). We also prove that \(\psi_{AC}\) implies that every function \(f:\omega_1 \longrightarrow \omega_1\) is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum fails.
After presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we consider the bounded forcing axiom for a class of partially ordered sets \(\Gamma_1\) such that, letting \(\Gamma_0\) be the class of all stationary-set-preserving partially ordered sets, one can prove the following:
(1) \(\Gamma_0\subseteq \Gamma_1\).
(2) \(\Gamma_0 = \Gamma_1\) if and only if \(NS_{\omega_1}\) is \(\aleph_1\)-dense.
(3) If \(P\notin \Gamma_1\), then \(BFA(\{P\})\) fails.
We call the bounded forcing axiom for \(\Gamma_1\) Maximal Bounded Forcing Axiom (MBFA). Finally we
prove MBFA consistent relative to the consistency of an inaccessible \(\Sigma_2\)-correct cardinal which is a limit of strongly compact cardinals.
We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of \((\omega_2, \omega_2)\)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for \(L\) at the level \(\omega_2\), and therefore with the existence of an \(\omega_2\)-Suslin tree. We also show that the axiom we call \(BMM_{\aleph_3}\) implies \(\aleph_2^{\aleph_1} = \aleph_2\), as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size \(\aleph_2\). Finally, we give an example of a so-called boldface bounded forcing axiom implying \(2^{\aleph_0}=\aleph_2\).