In this paper we prove the equiconsistency of ``Every omega_1 tree which is first order definable over H_{omega_1} has a cofinal branch'' with the existence of a Pi^1_1 reflecting cardinal. The proof uses a definable version of Ramsey theorem on aleph_1 which is again equiconsistent with a Pi^1_1 reflecting cardinal. We also prove that the addition of $MA$ to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.
We prove that in the Cohen extension adding $\aleph_3$ generic reals to a model of $ZFC+CH$ containing a simplified $(ω_1,2)$-morass, gap-2 morass-definable $η_1$-orderings with cardinality $\aleph_3$ are order-isomorphic. Hence it is consistent that the $2^{\aleph_0}=\aleph_3$ and that morass-definable $η_1$-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of $\mathbb{R}$ over $ω$ that are gap-2 morass-definable. The constructions use a simplified gap-2 morass, and commutativity with morass-maps and morass-embeddings, to extend a transfinit…
We prove that in a continuous $\aleph_0$-stable theory every type-definable group is definable. The two main ingredients in the proof are: \begin{enumerate} \item Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from \cite{BenYaacov:TopometricSpacesAndPerturbations}, allowing us to prove the theorem in case the metric is invariant under the group action; and \item Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones. \end{enumerate}
We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable $P$-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which $\mathfrak{a}=\mathfrak{u}=\mathfrak{i}=\aleph_1<2^{\aleph_0}=\aleph_2$, each of $\mathfrak{a}$, $\mathfrak{u}$, $\mathfrak{i}$ has a $Π^1_1$ witness and there is a $Δ^1_3$ well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-o…
We prove the following continuous analogue of Vaught's Two-Cardinal Theorem: if for some $κ>λ\geq \aleph_0$, a continuous theory $T$ has a model with density character $κ$ which has a definable subset of density character $λ$, then $T$ has a model with density character $\aleph_1$ which has a separable definable subset. We also show that if we assume that $T$ is $ω$-stable, then if $T$ has a model of density character $\aleph_1$ with a separable definable set, then for any uncountable $κ$ we can find a model of $T$ with density character $κ$ which has a separable definable subset. In order to …
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 $\ZFC$ in which $\NS$ is $\aleph_2$-saturated and $Δ_1$-definable with $ω_1$ as a parameter which answers a question of Sy-David Friedman and Liuzhen Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with $\NS$ saturated and $Δ_1$-definable with a ladder system $\vec{C}$ and a full Suslin tree $T$ as parameters. Both results rely on a new coding technique whose presentation is the main goal of this article.
Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …
In this note we answer the following question of Grinblat: Is it consistent that for some set X, cov(NULL restriction X)=lambda is a weakly inaccessible cardinal (so X not null of course) while cov(meagre) is small, say it is aleph_1.
An omega-coloring is a pair <f,B> where f:[B]^{2} ---> omega. The set B is the field of f and denoted Fld(f). Let f,g be omega-colorings. We say that f realizes the coloring g if there is a one-one function k:Fld(g) ---> Fld(f) such that for all {x,y}, {u,v} in dom(g) we have f({k(x),k(y)}) not= f({k(u),k(v)}) => g({x,y}) not= g({u,v}). We write f~g if f realizes g and g realizes f. We call the ~-classes of omega-colorings with finite fields identities. We say that an identity I is of size r if |Fld(f)|=r for some/all f in I. For a cardinal kappa and f:[kappa]^2 ---> omega we define I(f) to be…
We prove the consistency of the statement $\mathfrak{u}_{\aleph_ω}<2^{\aleph_ω}$. We show that the consistency strength of this statement is exactly a measurable cardinal $μ$ so that $o(μ)=μ^{++}$.
It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to indescribables but are much weaker in consistency strength. Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.
It is consistent that there is a set mapping from the four-tuples of omega_n into the finite subsets with no free subsets of size t_n for some natural number t_n. For any n< omega it is consistent that there is a set mapping from the pairs of omega_n into the finite subsets with no infinite free sets.
We show that the theory of De Morgan algebras has a model completion and axiomatise it. Then we prove that it is $\aleph_0$-categorical and describe definable and algebraic closures in that theory. We also obtain similar results for Boole-De Morgan algebras.
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.
We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire-space $ω_1^{ω_1}$. First, we will show that none of these classes have the Baire property (unless they are empty). Moreover, under $(V=L)$, (a) the class of Aronszajn and Suslin trees is $Π_1^1$-complete, (b) the class of special Aronszajn trees is $Σ_1^1$-complete, and (c) the class of Kurepa trees is $Π^1_2$-complete. We achieve these results by finding nicely definable reductions that map subsets $X$ of $ω_1$ to trees $T_X$ so that $T_X$ is in a given tree-c…
We prove the consistency of ``CH + 2^{aleph_1} is arbitrarily large + 2^{aleph_1} not-> (omega_1 x omega)^2_2''. If fact, we can get 2^{aleph_1} not-> [omega_1 x omega]^2_{aleph_0}. In addition to this theorem, we give generalizations to other cardinals.
One aim of this work is to get a universe in which weak versions of Martin axioms holds for some forcing notions of cardinality aleph_0, aleph_1 and aleph_2 while on aleph_2 club, the ``small'' brother of diamond, holds. As a consequence we get the consistency of ``there is no Gross space''. Another aim is to present a case of ``Historic iteration''.
We give an example of a finite rank, in fact aleph-1 categorical theory where the CBP (canonical base property) does not hold. We include a "group-like" example. We also prove, in a finite Morley rank context, if all definable Galois groups are "rigid" then the theory has the CBP (in a strong sense).
We show that it is consistent, relative to the consistency of a strongly inaccessible cardinal, that an instance of the generalized Borel Conjecture introduced in [8] holds while the classical Borel Conjecture fails.
The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_…
We prove that Arhangelskii's problem has a consistent positive answer: if V\models CH, then for some aleph_1-complete aleph_2-c.c. forcing notion P of cardinality aleph_2 we have that P forces ``CH and there is a Lindelof regular topological space of size aleph_2 with clopen basis with every point of pseudo-character aleph_0 (i.e. each singleton is the intersection of countably many open sets)''. Also, we prove the consistency of: CH+ 2^{aleph_1} > \aleph_2 + "there is no space as above with aleph_2 points" (starting with a weakly compact cardinal).
We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]
We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} $\mathcal{L}_{ω_1,ω}$-sentence $ψ$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $ψ$ is consistently equal to $[\aleph_0,\aleph_{ω_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $ψ$ is consistently equal to $[\aleph_1,\aleph_{ω_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathc…
We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to the situation of lambda a singular strong limit with 2^lambda>lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{lambda^+}_lambda. The second result is a consistency result (assuming a supercompact cardinal exists) showing that a natural guessing fails. A result of Shelah in math.LO/9808140 shows that if 2^lambda=lambd…
We give some general criteria, when kappa-complete forcing preserves largeness properties -- like kappa-presaturation of normal ideals on lambda (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength ``NS_lambda is aleph_1-preserving'', for lambda > aleph_2 .
We prove the consistency of: if B_1, B_2 are Boolean algebra satisfying the c.c.c. and the 2^{aleph_0}-c.c. respectively then B_1 x B_2 satisfies the 2^{aleph_0}-c.c.
Let $κ$ be any regular cardinal. Assuming the existence of a huge cardinal above $κ$, we prove the consistency of $\binom{κ^{++}}{κ^+}\rightarrow\binomτ{κ^+}$ for every ordinal $τ<κ^{++}$. Likewise, we prove a full amenable relation for $(\aleph_2,\aleph_1)$ with respect to collections which are strongly closed under countable intersections.
We throw some light on the question: is there a MAD family (= a family of infinite subsets of N, the intersection of any two is finite) which is completely separable (i.e. any X subseteq N is included in a finite union of members of the family or include a member of the family). We prove that it is hard to prove the consistency of the negation: (a) if 2^{aleph_0} < aleph_omega, then there is such a family (b) if there is no such families then some situation related to pcf holds whose consistency is large.
We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbfΔ^1_2$ maximal independent set. As a main application we get the consistency of $\mathfrak{r} = \mathfrak{u} = \mathfrak{i} = ω_2$ together with the existence of a $Δ^1_2$ ultrafilter, a $Π^1_1$ maximal independent family and a $Δ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\mathfrak{d} \leq \mathfrak{i}_{cl}$.
We prove a revised version of Laver's indestructibility theorem which slightly improves over the classical result. An application yields the consistency of $(κ^+,κ)\notcc(\aleph\_1,\aleph\_0)$ when $κ$ is supercompact. The actual proofs show that $ω\_1$-regressive Kurepa-trees are consistent above a supercompact cardinal even though ${\rm MM}$ destroys them on all regular cardinals. This rather paradoxical fact contradicts the common intuition.
We study the relationship between a $κ$-Souslin tree $T$ and its reduced powers $T^θ/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $θ$, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph_6$-Souslin tree $T$ and a sequence of uniform ultrafilters $\langle \mathcal U_n\mid n<6\rangle$ such that $ T^{\aleph_n}/\mathcal U_n$ is $\aleph_6$-Aronszajn iff $n<6$ is not a prime number.
We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and forcing axioms imply that there is a homogeneous closed unbounded subset of $ω_1$ for every colouring of the finite sets of countable ordinals that is definable by a $Σ_1$-formula that only uses the cardinal $ω_1$ and real numbers as parameters. Moreover, it is shown that certain large cardinal properties cause analogous partition properties to hold at the given …
A subgroup G \leq Z^omega exhibits the Specker phenomenon if every homomorphism G \to Z maps almost all unit vectors to 0. We give several combinatorial characterizations of the cardinal se, the size of the smallest G \leq Z^omega exhibiting the Specker phenomenon. We also prove the consistency of b < e, where b is the unbounding number and e the evasion number introduced recently by Blass. Finally we show that e \geq \min { aleph_0--e , cov(M) }, where aleph_0--e is the aleph_0--evasion number (the uniformity of the evasion ideal) and cov(M) is the covering number of the meager ideal. All the…
We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $κ$, let {\sf BC}$_κ$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_κ$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (…
Let k be a definable L-cardinal. Then there is a set of reals X, class-generic over L, such that L(X) and L have the same cardinals, X has size k in L(X) and some pi-1-2 formula defines X in all set-generic extensions of L(X). Two corollaries, both assuming the consistency of an inaccessible: It is consistent for the Perfect Set Property to hold for boldface sigma-1-2 sets, yet fail for some lightface pi-1-2 set. It is consistent that the Perfect Set Property holds for boldface sigma-1-2 sets yet some lightface pi-1-2 wellordering of some set of reals has length aleph-1000.
An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compac…
We prove the consistency of the following statement: for some kappa<2^{aleph_0}, there is a kappa-complete ideal on kappa such that the Boolean algebra P(kappa)/I is sigma-centered and there are Q-sets of reals.
The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_ω$ with the coun…
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true $Π_n$-sentences or equivalently the $Σ_n$-soundness of the theory, and the other is $n$-consistency the restriction of $ω$-consistency to the $Σ_n$-formulas. It is also shown that Rosser's Incompleteness Theorem does not generally hold for definable non-recursively enumerable theories, whence Godel-Rosser's Incompleteness Theorem is optimal in a sense. Though the …
The bounded proper forcing axiom BPFA is the statement that for any family of aleph_1 many maximal antichains of a proper forcing notion, each of size aleph_1, there is a directed set meeting all these antichains. A regular cardinal kappa is called {Sigma}_1-reflecting, if for any regular cardinal chi, for all formulas phi, ``H(chi) models `phi ' '' implies ``exists delta < kappa, H(delta) models `phi ' '' We show that BPFA is equivalent to the statement that two nonisomorphic models of size aleph_1 cannot be made isomorphic by a proper forcing notion, and we show that the consistency strength…
With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević's principle $\square(κ)$ implies an indexed version of $\square(κ,λ)$, we show that for all infinite, regular cardinals $λ<κ$, the principle $\square(κ)$ implies the existence of a $κ$-Aronszajn tree containing a $λ$-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with asc…
Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<ł<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $ł$ and its existence is not a result of …
This is a revised version (of late 2020) of [Sh:700], which is arXiv:math/0012170 . First point is noting that the proof of Theorem 4.3 in [Sh:700], which says that the proof giving the consistency $ \mathfrak{b} = \mathfrak{d} = \mathfrak{u} < \mathfrak{a} $ also gives $ \mathfrak{s} = \mathfrak{d} $. The proof uses a measurable cardinal and a c.c.c. forcing so it gives large $ \mathfrak{d} $ and assumes a large cardinal. Second point is adding to the results of \S2,\S3 which say that (in \S3 with no large cardinals) we can force $ {\aleph_1} < \mathfrak{b} = \mathfrak{d} < \mathfrak{a}$. We …
