Graph theory, a branch of mathematics, studies the relationships between entities using
vertices and edges. Uncertain Graph Theory has emerged within this field to model the …

We show that there is a β-model of second-order arithmetic in which the choice scheme
holds, but the dependent choice scheme fails for a Π 2 1-assertion, confirming a conjecture …

We make use of generalized iterations of Jensen forcing to define a cardinal-preserving
generic model of ZF for any $ n\ge 1$ and each of the following four Choice hypotheses:(1) …

This dissertation has two major threads, one is mathematical, namely descriptive set theory,
the other is philosophical, namely generalisation in mathematics. Descriptive set theory is …

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 …

The forcing theorem is the most fundamental result about set forcing, stating that the forcing
relation for any set forcing is definable and that the truth lemma holds, that is everything that …

Models of set theory are defined, in which nonconstructible reals first appear on a given
level of the projective hierarchy. Our main results are as follows. Suppose that n≥ 2. Then …

We study elementary theories of well-pointed toposes and pretoposes, regarded as category-
theoretic or “structural” set theories in the spirit of Lawvere's “Elementary Theory of the …

In this article we introduce and study hyperclass-forcing (where the conditions of the forcing
notion are themselves classes) in the context of an extension of Morse-Kelley class theory …

We introduce the-definable universal finite sequence and prove that it exhibits the universal
extension property amongst the countable models of set theory under end-extension. That …