Related papers: Reflective Cardinals
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…
In this paper, we show that marked quantales have a reflection into quantales. To obtain the reflection we construct free quantales over marked quantales using appropriate lower sets. A marked quantale is a posemigroup in which certain…
In this paper we discuss large cardinals and compactness theorems in abelian group theory. More specifically, we generalize two classical compactness results for free abelian groups to the broader context of direct sums of cyclic groups.
The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more…
NF set theory using intuitionistic logic is called iNF. We develop the theories of finite sets and their power sets and mappings, finite cardinals and their ordering, cardinal exponentiation, addition, and multiplication. We follow Rosser…
In this note we provide some applications of L\"owenheim-Skolem cardinals introduced in \cite{U}.
We study several intertwined hierarchies between $\kappa$-Ramsey cardinals and measurable cardinals to illuminate the structure of the large cardinal hierarchy in this region. In particular, we study baby versions of measurability…
This paper enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: {\sf All $x$ are $y$} and {\sf Some $x$ are…
We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic.
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…
This paper introduces the concept of a productive notion of big Ramsey degree and showcases its versatility through a handful of applications. The main focus is notably providing sufficient conditions for the existence of a finite canonical…
We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to…
The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this…
We show that induction over $\Delta(\mathbb R)$-definable well-founded classes is equivalent to the reflection principle which asserts that any true formula of first order set theory with real parameters holds in some transitive set. The…
We define the concept of a logic frame, which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive…
We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of all non exceptional irreducible complex…
Zermelo's Axiom of Separation is: Exist x: Forall y: (y in x <==> y in a & E(y)) with definite(E) and parameter a. Thoralf Skolem suggested to characterize the terminus "definite" by "the property E should be representable by a FOL…
We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal…
We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly…
Bounded stationary reflection at a cardinal $\lambda$ is the assertion that every stationary subset of $\lambda$ reflects but there is a stationary subset of $\lambda$ that does not reflect at arbitrarily high cofinalities. We produce a…