We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, (...) if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (shrink)

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming $O^{Sword}$ does not exist. Namely, if $\kappa$ is a Jónsson cardinal then $\kappa^+ = \kappa^{+K}$ , provided that there is no non-trivial elementary embedding $j:K \longrightarrow K$ . There are a number of related results in ZFC concerning $\cal{P}(\kappa)$ in V and inner models, for $\kappa$ a Jónsson or singular cardinal.

We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinal axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinal properties can be defined in terms of suitable elementary embeddings j:Vγ→Vλ. One key observation is that such embeddings are uniquely determined by the image structures j[Vγ]≺Vλ. These structures will be the prototypes guessing models. We shall show, (...) using guessing models M, how to prove for the ordinal κM=jM) many of the combinatorial properties that we can prove for the cardinal j) using the structure j[Vγ]≺Vj. κM will always be a regular cardinal, but consistently can be a successor and guessing models M with κM=ℵ2 exist assuming the proper forcing axiom. By means of these models we shall introduce a new structural property of models of PFA: the existence of a “Laver function” f:ℵ2→Hℵ2 sharing the same features of the usual Laver functions f:κ→Hκ provided by a supercompact cardinal κ. Further applications of our analysis will be proofs of the singular cardinal hypothesis and of the failure of the square principle assuming the existence of guessing models. In particular the failure of square shows that the existence of guessing models is a very strong assumption in terms of large cardinal strength. (shrink)

We introduce the Easton collapse and show that the two-stage iteration of Easton collapses gives a model in which the successor of a regular cardinal carries a strongly saturated filter. This allows one to get a model in which many successor cardinals carry saturated filters just by iterating Easton collapses.

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s found (...) one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L. One of the nice things about AD is that the theory ZF + AD + V = L appears as a choiceless “completion” of ZF in that any interesting question seems to find an at least attractive answer in that theory. Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension. (shrink)

We present a general construction of a □κ-sequence in Jensen's fine structural extender models. This construction yields a local definition of a canonical □κ-sequence as well as a characterization of those cardinals κ, for which the principle □κ fails. Such cardinals are called subcompact and can be described in terms of elementary embeddings. Our construction is carried out abstractly, making use only of a few fine structural properties of levels of the model, such as solidity and condensation.

The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications.

This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U (n) and symmetric groups $S_{n},\ n\in {\Bbb N}$ . Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular (...) it is still unknown if every group is hyperlinear and/or sofic. (shrink)

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

My ultimate goal in this paper is to illuminate, from a naturalistic point of view, the significance of the application of mathematics in the natural sciences for the practice of contemporary set theory.

This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow’s theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted axiom of choice do not meet this criterion. The (...) technically novel part of this paper is a proposal to use a set-theoretic principle known as the axiom of determinacy ), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of \ from the point of view of the research area in question is its game-theoretic character. (shrink)

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them: If m ≥ 1 and the ultrafilter D is , equation imagem)-regular, then D is κ -decomposable for some κ with λ ≤ κ ≤ 2λ ). If λ is a strong limit cardinal and D is , equation imagem)-regular, then either D is -regular or there are arbitrarily large κ < λ for which D is κ -decomposable ). Suppose that λ is singular, (...) λ < κ, cf κ ≠ cf λ and D is -regular. Then: D is either -regular, or -regular for some λ' < λ . If κ is regular, then D is either -regular, or -regular for every κ' < κ . If either λ is a strong limit cardinal and λ<λ < 2κ, or λ<λ < κ, then D is either λ -decomposable, or -regular for some λ' < λ . If λ is singular, D is -regular and there are arbitrarily large ν < λ for which D is ν -decomposable, then D is κ -decomposable for some κ with λ ≤ κ ≤ λ<μ . D × D' is -regular if and only if there is a ν such that D is -regular and D' is -regular for all ν∼ < ν .We also list some problems, and furnish applications to topological spaces and to extended logics. (shrink)

Reductivist programs in logicand philosophy, especially inthe philosophy of mathematics,are reviewed. The paper argues fora ``methodological realism'' towardsnumbers and sets, but still givesreductionism an important place,albeit in methodology/epistemologyrather than in ontology proper.

We provide an exposition of supercompact Radin forcing and present several methods for iterating Radin forcing.

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement (...) with this feature in the sense that any other $\Sigma _1^2 $ -statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment $V_\lambda $ of the universe of sets (for example, one might take $V_\lambda $ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of $V_\lambda $ . If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies¬CH. (shrink)

We show that strongly compact cardinals and MM are sensitive to $\lambda$-closed forcings for arbitrarily large $\lambda$. This is done by adding ‘regressive' $\lambda$-Kurepa trees in either case. We argue that the destruction of regressive Kurepa trees requires a non-standard application of MM. As a corollary, we find a consistent example of an $\omega_2$-closed poset that is not forcing equivalent to any $\omega_2$-directed-closed poset.

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".

We work under the assumption of the Axiom of Determinacy and associate a measure to each cardinal $\kappa < \aleph_{\varepsilon_0}$ in a recursive definition of a canonical measure assignment. We give algorithmic applications of the existence of such a canonical measure assignment (computation of cofinalities, computation of the Kleinberg sequences associated to the normal ultrafilters on all projective ordinals).

The theory of infinite games with slightly imperfect information has been developed for games with finitely and countably many moves. In this paper, we shift the discussion to games with uncountably many possible moves, introducing the axiom of real Blackwell determinacy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{Bl-AD}_\mathbb{R}}$$\end{document} (as an analogue of the axiom of real determinacy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{AD}_\mathbb{R}}$$\end{document}). We prove that the consistency strength of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} (...) \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{Bl-AD}_\mathbb{R}}$$\end{document} is strictly greater than that of AD. (shrink)

We say that κ is μ-hypermeasurable for a cardinal μ≥κ+ if there is an embedding j:V→M with critical point κ such that HV is included in M and j>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V* where F is realised on all V-regular (...) cardinals and moreover, all F-hypermeasurable cardinals κ, where F>κ+, with a witnessing embedding j such that either j=κ+ or j≥F, are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2α{α+,α++} for every cardinal α below κ ≥F or j=κ+). (shrink)

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one set-forcing ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed (...) from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions. (shrink)

We show that the theory, consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there are sets of (...) reals of every size, but none of size, and therefore, in which the collection axiom fails; there are models of for which the Łoś theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of for which the Gaifman theorem fails, in that there is an embedding of models that is Σ1‐elementary and cofinal, but not elementary; there are elementary embeddings of models whose cofinal restriction is not elementary. Moreover, the collection of formulas that are provably equivalent in to a Σ1‐formula or a Π1‐formula is not closed under bounded quantification. Nevertheless, these deficits of are completely repaired by strengthening it to the theory, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach. (shrink)

We deal with some natural properties of ultrafilters which trivially fail for normal ultrafilters.

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with (...) V = L. (shrink)

Erdős proved that for every infinite \ there is \ with \, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, \ is a finite set of formulas of (...) T, \, and X is an infinite subset of M. Then there is \ with \ and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is \\)-indiscernible. We also consider the definable version of these problems, for example we assume \ is perfect and we find some perfect \ with all distances distinct. Finally we show that Erdős’s theorem requires some use of the axiom of choice. (shrink)

The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α<β→F≤F. The classic example of an Easton function is the continuum function αmaps to2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if we also wish (...) to preserve measurable cardinals, new restrictions must be put on F. We say that κ is F-hypermeasurable iff there is an elementary embedding j:V→M with critical point κ such that H)Vsubset of or equal toM; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V[G] such that if κ, closed under F, is F-hypermeasurable in V and there is a witnessing embedding j such that j≥F, then κ will remain measurable in V[G]. (shrink)

An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa$. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every $n\geq 2$ and $\mu\geq \aleph_n$, we have $(\aleph_n, \mu)$-ITP.

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\aleph_2, \mu)}$$\end{document} -ITP and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\aleph_3, \mu')}$$\end{document} -ITP hold, for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu\geq \aleph_2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu'\geq \aleph_3}$$\end{document}. (shrink)

We prove that if V = L then there is a $\Pi _{1}^{1}$ maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

Set theory deals with the most fundamental existence questions in mathematics—questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what (...) is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo—Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect. (shrink)

Many philosophers have become worried about the use of standard real numbers for the probability function that represents an agent's credences. They point out that real numbers can't capture the distinction between certain extremely unlikely events and genuinely impossible ones—they are both represented by credence 0, which violates a principle known as “regularity.” Following Skyrms 1980 and Lewis 1980, they recommend that we should instead use a much richer set of numbers, called the “hyperreals.” This essay argues that this popular (...) view is the result of two mistakes. The first mistake, which this essay calls the “numerical fallacy,” is to assume that a distinction that isn't represented by different numbers isn't represented at all in a mathematical representation. In this case, the essay claims that although the real numbers do not make all relevant distinctions, the full mathematical structure of a probability function does. The second mistake is that the hyperreals make too many distinctions. They have a much more complex structure than credences in ordinary propositions can have, so they make distinctions that don't exist among credences. While they might be useful for generating certain mathematical models, they will not appear in a faithful mathematical representation of credences of ordinary propositions. (shrink)

In this paper we are concerned about the ways GCH can fail in relation to rank-into-rank hypotheses, i.e., very large cardinals usually denoted by I3, I2, I1 and I0. The main results are a satisfactory analysis of the way the power function can vary on regular cardinals in the presence of rank-into-rank hypotheses and the consistency under I0 of the existence of j:Vλ+1≺Vλ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j : V_{\lambda+1} {\prec} V_{\lambda+1}}$$\end{document} with the failure of GCH (...) at λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda}$$\end{document}. (shrink)

In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimen\-sional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.

Jensen's celebrated Covering Lemma states that if 0# does not exist, then for any uncountable set of ordinals X, there is a Y∈L such that X⊆Y and |X| = |Y|. Working in ZF + AD alone, we establish the following analog: If ℝ# does not exist, then L(ℝ) and V have exactly the same sets of reals and for any set of ordinals X with |X| ≥ΘL(ℝ), there is a Y∈L(ℝ) such that X⊆Y and |X| = |Y|. Here ℝ is (...) the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ. (shrink)

Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) ${\kappa < {\rm cf}(F(\kappa))}$ , (2) ${\kappa < \lambda}$ implies ${F(\kappa) \leq F(\lambda)}$ , and (3) δ is closed under F, then there is a cofinality-preserving forcing extension in which 2 γ = F(γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogous (...) results for supercompact cardinals [Menas in Trans Am Math Soc 223:61–91, (1976)] and strong cardinals [Friedman and Honzik in Ann Pure Appl Logic 154(3):191–208, (2008)], there is no requirement that the function F be locally definable. I deduce a global version of the above result: Assuming GCH, if F is a function satisfying (1) and (2) above, and C is a class of Woodin cardinals, each of which is closed under F, then there is a cofinality-preserving forcing extension in which 2 γ = F(γ) for all regular cardinals γ and each cardinal in C remains Woodin. (shrink)

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ◊ $_{k^ - }^* $ at a proper class of cardinals k. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, (...) specifically, reverse Easton iterations of increasingly directed closed partial orders. (shrink)

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions'', ``substances'' or ``substantial forms''. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances'' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae N of formulae of a normal modal theory Tm based on (...) T, such that the counterpart `i' of a the modal variable `xi' of L(Tm) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T). (shrink)