We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that $\mathrm {WVP}$ for $\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that $\mathrm {WVP}$ for $\Sigma _n$ -definable classes is equivalent to the existence of (...) a $\Sigma _n$ -strong cardinal (Definition 5.1). Hence, $\mathrm {WVP}$ is equivalent to the existence of a $\Sigma _n$ -strong cardinal for all $n<\omega $. (shrink)

In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (where (...) T is $\mathsf {ZFC}$ +Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of $\mathsf {ZFC}$. To this end, we take into account several strategies, and assess their prospects in the light of $\mathsf {MV}$ ’s evidential framework. (shrink)

We show that there are universes of sets which contain descending ϵ-sequences of length α for every ordinal α, though they do not contain any ϵ-cycle. It is also shown that there is no set universe containing a descending ϵ-sequence of length On. MSC: 03E30; 03E65.

An ℵ1-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ℵ1-Souslin tree. In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a λ+-Souslin tree. This class includes Prikry, Magidor, and Radin forcing.

In this paper, we study a stability transfer theorem in d-tame metric abstract elementary classes, in a similar way as in 2, but using superstability-like assumptions which involves a new independence notion instead of ℵ0-locality.

We show that both Rado's Conjecture and strong Chang's Conjecture imply that there are no special ℵ2-Aronszajn trees if the Continuum Hypothesis fails. We give similar result for trees of higher heights and we also investigate the influence of Rado's Conjecture on square sequences.

We show that both Rado's Conjecture and strong Chang's Conjecture imply that there are no special ℵ2-Aronszajn trees if the Continuum Hypothesis fails. We give similar result for trees of higher heights and we also investigate the influence of Rado's Conjecture on square sequences.

In this paper we study iterated circular multisets in a coalgebraic framework. We will produce two essentially different universes of such sets. The unisets of the first universe will be shown to be precisely the sets of the Scott universe. The unisets of the second universe will be precisely the sets of the AFA-universe. We will have a closer look into the connection of the iterated circular multisets and arbitrary trees. RID=""ID="" Mathematics Subject Classification (2000): 03B45, 03E65, 03E70, 18A15, (...) 18A22, 18B05, 68Q85 RID=""ID="" Key words or phrases: Multiset – Non-wellfounded set – Scott-universe – AFA – Coalgebra – Modal logic – Graded modalities. (shrink)

We answer some questions about graphs that are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the class Mod(T) (...) of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ℵ1, ℵ2, ℵω, ℶω, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs. (shrink)

We develop an untyped framework for the multiverse of set theory. $\mathsf {ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf {Uni}(\mathcal {U})$ and $\mathsf {Mod}(\mathcal {U, \sigma })$, expressing that $\mathcal {U}$ is a universe and that $\sigma $ is true in the universe $\mathcal {U}$, respectively. Here $\sigma $ ranges over the augmented language, leading to liar-style phenomena that are analyzed. The framework is both compatible with a broad range of multiverse conceptions and suggests its (...) own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf {ZF}$, and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins’ multiverse theory. (shrink)

After discussing the limitations inherent to all set-theoretic reflection principles akin to those studied by A. Lévy et. al. in the 1960s, we introduce new principles of reflection based on the general notion of Structural Reflection and argue that they are in strong agreement with the conception of reflection implicit in Cantor’s original idea of the unknowability of the Absolute, which was subsequently developed in the works of Ackermann, Lévy, Gödel, Reinhardt, and others. We then present a comprehensive survey of (...) results showing that different forms of the new principle of Structural Reflection are equivalent to well-known large cardinal axioms covering all regions of the large-cardinal hierarchy, thereby justifying the naturalness of the latter. (shrink)