mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable chain condition...

explicit treatment of proper classes. There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence...

Martin's axiom has generalizations called the proper forcing axiom and Martin's maximum. Sheldon W. Davis has suggested in his book that Martin's axiom is motivated...

ideals, and partition relations, iterated forcing and Axiom A, proper forcing and the proper forcing axiom, chromatic number of graphs, a thin very-tall...

algorithm, a fast algorithm for computing the discrete Fourier transform Proper forcing axiom Probability of False Alarm, a synonym for false positive rate which...

arithmetic. Non-logical axioms may also be called "postulates", "assumptions" or "proper axioms". In most cases, a non-logical axiom is simply a formal logical...

unfoldable implies the consistency of a slightly weaker version of the proper forcing axiom. Assuming V = L, the least unfoldable cardinal is greater than the...

and from the axiom of determinacy. The full OCA is consistent with (but independent of) ZFC, and follows from the proper forcing axiom. OCA implies that...

of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings for which a forcing axiom is consistent...

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection...

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962...

3.  A proper forcing is a forcing notion that does not collapse any stationary set 4.  The proper forcing axiom asserts that if P is proper and Dα is...

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty...

axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of class construction, or axiom...

assuming the consistency of a suitable large cardinal: Proper forcing axiom Open coloring axiom Martin's maximum Existence of 0# Singular cardinals hypothesis...

mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique...

to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated...

of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins...

isomorphic. Baumgartner's axiom is a consequence of the proper forcing axiom. It is consistent with a combination of ZFC, Martin's axiom, and the negation of...

this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto V. Von Neumann's axiom implies...

specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets...

Principle (NAP), also called the Non-Aggression Axiom, the non-coercion principle, the non-initiation of force and the zero aggression principle, is a concept...

than the axiom of choice by using forcing to construct a model that satisfies the axiom of choice and all the axioms of NBG except the axiom of global...

axiom (usually together with the negation of the continuum hypothesis), Martin's maximum ◊ and ♣ Axiom of constructibility (V=L) proper forcing axiom...

forcing (see Laver forcing), adding a real, was iterated with countable support iteration. This method was later used by Shelah to introduce proper and...

theory admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its class existence axioms. This causes...

well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets...

into [X] by mapping a set m to {m} × X, and so by the axiom of limitation of size, [X] is a proper class. The definition does work however in type theory...

special, a proposition sometimes abbreviated by EATS. The stronger proper forcing axiom implies the stronger statement that for any two Aronszajn trees there...

schools (see Motivations and epistemic status below). A large cardinal axiom is an axiom stating that there exists a cardinal (or perhaps many of them) with...