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 a trivial cardinal arithmetical reason, then $ł$ ``resembles'' $\aleph_1$ --- a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the non-existence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).
We prove in the theory "ZFC + there is no inner model with a Woodin cardinal" that there is a universal weasel. This shows in particular that one doesn't have to assume the existence of large cardinals in V to prove the existence of some such weasel.
Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $λ,$ there is no universal graph of size $λ$.
From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.
The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu^{++} graphs on mu^+ that taken jointly are universal for the graphs on mu^+, while 2^{mu^+}>> mu^{++} . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal kappa exists. The result on the exist…
We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $κ$ with the property that the collection of all initial segments of the wellordering is definable by a $Σ_1$-formula with parameter $κ$. A short argument shows that the existence of a measurable cardinal $δ$ implies that such wellorderings do not exist at $δ$-inaccessible cardinals of cofinality not equal to $δ$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a…
In this paper we investigate some properties of first order theories which prevent them from having universal models under certain cardinal arithmetic assumptions. Our results give a new syntactical condition, oak property, which is a sufficient condition for a theory not to have universal models in cardinality lambda when certain cardinal arithmetic assumptions implying the failure of GCH (and close to the failure of SCH) hold.
We consider a weak version of Schindler's remarkable cardinals that may fail to be $Σ_2$-reflecting. We show that the $Σ_2$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and we show that the existence of a non-$Σ_2$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an $ω$-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + "Ord is not $Δ_2$-Mahlo" and $\text{gVP}({\bfΠ}_1)$ + "there is no proper clas…
The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $κ$ is supercompact and the cardinals $θ< κ$, $λ> κ$ are regular, as an application of a more general method we obtain a forcing extension in which $\textrm{cf}(κ) = θ$, the Singular Cardinal Hypothesis fails at $κ$ and there exists a universal graph in cardinality $λ\in (κ,2^κ)$.
An ideal $I$ on a cardinal $κ$ is called \emph{rigid} if all automorphisms of $P(κ)/I$ are trivial. An ideal is called \emph{$μ$-minimal} if whenever $G\subseteq P(κ)/I$ is generic and $X\in P(μ)^{V[G]}\setminus V$, it follows that $V[X]=V[G]$. We prove that the existence of a rigid saturated $μ$-minimal ideal on $μ^+$, where $μ$ is a regular cardinal, is consistent relative to the existence of large cardinals. The existence of such an ideal implies that GCH fails. However, we show that the existence of a rigid saturated ideal on $μ^+$, where $μ$ is an \emph{uncountable} regular cardinal, is c…
In this paper we present a few properties of $K$-partitions, which are partitions of Baire spaces such that all subfamilies of such a partition sum to a set with the Baire property. Among the result proven we have general existence result that state that the existence of any $K$-partition implies the existence of $K$-partition of a metrizable space as well as existence of $K$-partition of a compact space implies the existence of $K$-partition of a completely metrizable space. We also prove some connections between existence of $K$-partitions and existence of precipitous ideals as well as measu…
We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a proof-theoretic bound on definable countable ordinals whose existence is provable from the existence of second order indescribable cardinals over ZF.
We deal with the existence of universal members in a given cardinality for several classes. First we deal with classes of Abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality. Second, we then deal with (variants of) the oak property (from a work of Dzamonja and the author), a property of complete first order theories, sufficient for the non-existence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups (naturally interpreted, so for qu…
In other work we have outlined how, building on ideas of Welch and Roberts, one can motivate believing in the existence of supercompact cardinals. After making this observation we strove to formulate a justification for large-cardinal axioms of greater strength, and arrived at a motivation for a new large-cardinal property, which we define here and prove to be equivalent to the property of being a Vopěnka scheme cardinal. Making use of this result, one can also show that a theory $B_0(V_0)$ described in a previous paper of Victoria Marshall implies the existence of a Vopěnka scheme cardinal $κ…
Starting from a supercompact cardinal we build a model in which $2^{\aleph_{ω_1}}=2^{\aleph_{ω_1+1}}=\aleph_{ω_1+3}$ but there is a jointly universal family of size $\aleph_{ω_1+2}$ of graphs on $\aleph_{ω_1+1}$. The same technique will work for any uncountable cardinal in place of $ω_1$.
We determine the large cardinal consistency strength of the existence of a $λ$-supercompact cardinal $κ$ such that GCH fails at $λ$. Indeed, we show that the existence of a $λ$-supercompact cardinal $κ$ such that $2^λ\geq θ$ is equiconsistent with the existence of a $λ$-supercompact cardinal that is also $θ$-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.
Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $δ$, $δ$ is strong limit, $2^δ=δ^{+3}$ and the tree property at $δ^{++}$ holds. This answers a question of Friedman, Honzik and Stejskalova [8]. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of $ZFC$ in which the tree property holds at all regular even cardinals. The result answers questions of Friedman-Halilovic [5] and Friedman-Honzik [6].
This paper continues the study of the Ramsey-like large cardinals. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only $α$-many iterations for some countable ordinal $α$. Here we study such $α$-iterable cardinals. We show that the $α$-iterable cardinals form a strict hierarchy for $α\leqω_1$, that they are downward absolute to $L$ for $α<ω_1^L…
The study of inner models was initiated by Gödel's analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin's proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel's proof of levels of projective determinacy from Woodin cardinals with a measurable cardinal on top. First Woodin and later N…
The Axiom of Full Reflection at a measurable cardinal has been conjectured to be equiconsitent with the existence of a coherent sequence of measures with a repeat point. However we prove that the Axiom of Full Reflection at a measurable cardinal is surprisingly equiconsistent just with the existence of a measurable cardinal. We also generalize the result to larger cardinals as strong or supercompact. Hence we can conclude that the Axiom of Full Reflection at large cardinals weaker than measurable, e.g. as n-Mahlo, does push the consistency strength up, but does not push the consistency strengt…
We recently formulated a new large-cardinal axiom of strength intermediate between a totally indescribable cardinal and an $ω$-Erdős cardinal, positing the existence of what we called an "extremely reflective cardinal", and we showed that the property of being extremely reflective was in fact equivalent to the property of being remarkable, and we sought to argue that this axiom should be seen as intrinsically justified. This built on related earlier work in which the notion of an $α$-reflective cardinal was formulated. Then Welch and Roberts put forward a family of reflection principles, Welch…
The existence of End Elementary Extensions of models M of ZFC is related to the ordinal height of M, according to classical results due to Keisler, Morley and Silver. In this paper, we further investigate the connection between the height of M and the existence of End Elementary Extensions of M. In particular, we prove that the theory `ZFC + GCH + there exist measurable cardinals + all inaccessible non weakly compact cardinals are possible heights of models with no End Elementary Extensions' is consistent relative to the theory `ZFC + GCH + there exist measurable cardinals + the weakly compact…
Let λbe a cardinal with λ=λ^{\aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality λand characteristic p such that the automorphism group of K_0 is a free group of cardinality 2^λand the automorphism group of K_1 is a free abelian group of cardinality 2^λ. This partially answers a question from [8] and complements results from [15], [16] and [17]. The methods developed in the proof of the above statement also allow us to show that the above cardinal arithmetic assumption is consistently not necessary for the existence of such fields and that th…
We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH around lambda) that if K is categorical in lambda, lambda^+, LS(K) <= lambda and 1 <= I(lambda^{++},K)< 2^{lambda^{++}} then K has a model in lambda^{+++} .
Starting from the existence of many supercompact cardinals, we construct a model of ZFC in which the tree property holds at a countable segment of successor of singular cardinals.
The notion of commutation of operations in universal algebra leads to the concept of centralizer clone and gives rise to a well-known class of problems that we call centralizer problems, in which one seeks to determine whether a given set of operations arises as a centralizer or, equivalently, coincides with its own double centralizer. Centralizer clones and centralizer problems in universal algebra have been studied by several authors, with early contributions by Cohn, Kuznecov, Danil'čenko, and Harnau. In this paper, we work within a generalized setting of infinitary universal algebra relati…
We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness under proper forcings. In particular, said absoluteness does not imply Pi^1_1 determinacy.
We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving embedding preserves the cofinality and continuum functions.
Local sentences were introduced by J.-P. Ressayre who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact (consistency) strength was left open in [O. Finkel and J.-P. Ressayre, Stretchings, Journal of Symbolic Logic, Volume 61 (2), 1996, p. 563-585 ]. Here, we solve this problem, using a combinatorial result of J. H. Schmerl. In fact, we show…
Assuming the existence of a strong cardinal $κ$ and a measurable cardinal above it, we force a generic extension in which $κ$ is a singular strong limit cardinal of any prescribed cofinality, and such that the tree property holds at $κ^{++}$.
We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong (0<n<omega+1), hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we refer to them as mangroves and prove that their existence is equi…
It is shown that if T is stable unsuperstable, and aleph_1< lambda =cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)< mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g. aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega. These results are generalized to kappa =cf(kappa) < kappa (T) in the place of aleph_0. Also: if there is a universal model in lambda >|T|, T stable and kappa < kappa (T) then there is a universal tree of height kappa +1 in cardinality lambda .
We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the "high-enough" threshold: $\mathbf{Theorem}$ Let $ψ$ be a universal $\mathbb{L}_{ω_1, ω}$ sentence. If $ψ$ is categorical in some $λ\ge \beth_{\beth_{ω_1}}$, then $ψ$ is categorical in all $λ' \ge \beth_{\beth_{ω_1}}$. As a byproduct of the proof, we show that a conj…
We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures (WVP), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular we show that WVP for $Σ_2$-definable classes is equivalent to the existence of a strong cardinal. The main theorem shows, more generally, that WVP for $Σ_n$-definable classes is equivalent to the existence of a $Σ_n$-strong cardinal. Hence, WVP is equivalent to the existence of a $Σ_n$-strong cardinal, all $n <ω$.
Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained that way (`Unfoldable cardinals') behave as a `boundary' between properties consistent with `V=L' and existence of indiscernibles. We also provide an `embedding characterisation' of the unfoldable cardinals and study their preservation and destruction by various different forcings.
We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal $\mathbb{L}_{ω_1, ω}$ sentence categorical on an end segment of cardinals below $\beth_ω$ must be categorical also everywhere above $\beth_ω$. This is done without any additional model-theoretic hypotheses (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.
We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author.
Assuming the existence of a strong cardinal $κ$, a weakly compact cardinal $λ$ above it and $γ> λ,$ we force a generic extension in which $κ$ is a singular strong limit cardinal of any given cofinality $δ$, $2^κ\geq γ$ and such that the tree property holds at $κ^{++}$.
Let T be an abelian group and lambda an uncountable regular cardinal. We consider the question of whether there is a lambda-universal group G^* among all torsion-free abelian groups G of cardinality less than or equal to lambda satisfying Ext(G,T)=0. Here G^* is said to be lambda-universal for T if, whenever a torsion-free abelian group G of cardinality less than or equal to lambda satisfies Ext(G,T)=0, then there is an embedding of G into G^*. For large classes of abelian groups T and cardinals lambda it is shown that the answer is consistently no. In particular, for T torsion, this solves a …
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 $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …
We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
An abelian group is said to be aleph_1-free if all its countable subgroups are free. Our main result is: If R is a ring with R^+ free and |R|<lambda <= 2^{aleph_0}, then there exists an aleph_1-free abelian group G of cardinality lambda with End(G)=R . A corollary to this theorem is: Indecomposable aleph_1-free abelian groups of cardinality aleph_1 do exist.
We study $Σ_1(ω_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $Σ_1$-formula with parameter $ω_1$) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is $Σ_1(ω_1)$-definable, the set of all stationary subsets of $ω_1$ is not $Σ_1(ω_1)$-definable and the complement of every $Σ_1(ω_1)$-definable Bernstein subset of ${}^{ω_1}ω_1$ is not $Σ_1(ω_1)$-definable. In contrast, we show that the existence of a Woodin cardinal is compatible …
If M is a proper class inner model of ZFC and omega_2^M=omega_2, then every sound mouse projecting to omega and not past 0-pistol belongs to M. In fact, under the assumption that 0-pistol does not belong to M, K^M \| omega_2 is universal for all countable mice in V. Similarly, if M is a proper class inner model of ZFC, delta>omega_1 is regular, (delta^+)^M = delta^+, and in V there is no proper class inner model with a Woodin cardinal, then K^M \| delta is universal for all mice in V of cardinality less than delta.
We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct implication and consistency strength.
A characteristic result is that if 2^{aleph_0}< mu < mu^+< lambda = cf(lambda)< mu^{aleph_0}, then among the separable reduced p-groups of cardinality lambda which are (< lambda)-stable there is no universal one.
We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on t…
We begin with the existence of groups with trivial duals for cardinals aleph_n (n in omega). Then we derive results about strongly aleph_n-free abelian groups of cardinality aleph_n (n in omega) with prescribed free, countable endomorphism ring. Finally we use combinatorial results of [Sh:108], [Sh:141] to give similar answers for cardinals >aleph_omega. As in Magidor and Shelah [MgSh:204], a paper concerned with the existence of kappa-free, non-free abelian groups of cardinality kappa, the induction argument breaks down at aleph_omega. Recall that aleph_omega is the first singular cardinal an…
Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel's constructible universe $L$, in which the transfer principles $(\aleph_2, \aleph_0) \to (\aleph_3, \aleph_1)$ and $(\aleph_3, \aleph_1) \to (\aleph_2, \aleph_0)$ fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.
Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inaccessible in HOD.
We consider a class K of structures e.g. trees with omega +1 levels, metric spaces and mainly, classes of Abelian groups like the one mentioned in the title and the class of reduced separable (Abelian) p-groups. We say M in K is universal for K if any member N of K of cardinality not bigger than the cardinality of M can be embedded into M . This is a natural, often raised, problem. We try to draw consequences of cardinal arithmetic to non-existence of universal members for such natural classes.
The purpose of this paper is to provide an introductory overview of the large cardinal hierarchy in set theory. By a large cardinal, we mean any cardinal $κ$ whose existence is strong enough of an assumption to prove the consistency of ZFC. We assume basic familiarity with set theory, model theory, and the ZFC axioms, though certain concepts will be reviewed as necessary. We attempt to clarify the vast landscape of the higher infinite, motivating the definitions of some (but certainly not all) of the known large cardinals. We also discuss connections with the Continuum Hypothesis and provide s…
