5 research outputs found
Cardinal Well-foundedness and Choice
We consider several notions of well-foundedness of cardinals in the absence
of the Axiom of Choice. Some of these have been conflated by some authors, but
we separate them carefully. We then consider implications among these, and also
between these and other consequences of Choice. For instance, we show that the
Partition Principle implies that all of our versions of well-foundedness are
equivalent. We also show that one version, concerning surjections, implies the
Dual Cantor-Schr\"oder-Bernstein theorem. It has been conjectured that
well-foundedness, in one form or another, actually implies the Axiom of Choice,
but this conjecture remains unresolved.Comment: 20 pages, 5 figure
Flow: the Axiom of Choice is independent from the Partition Principle
We introduce a general theory of functions called Flow. We prove ZF, non-well
founded ZF and ZFC can be immersed within Flow as a natural consequence from
our framework. The existence of strongly inaccessible cardinals is entailed
from our axioms. And our first important application is the introduction of a
model of Zermelo-Fraenkel set theory where the Partition Principle (PP) holds
but not the Axiom of Choice (AC). So, Flow allows us to answer to the oldest
open problem in set theory: if PP entails AC.Comment: 37 pages, 4 Figure
Cantor, Choice, and Paradox
I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that is bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory