Stronger Axioms
The axiom of constructibility and the generalized continuum hypothesis both imply the axiom of choice, but are strictly stronger than it.
In class theories such as Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, there is a possible axiom called the axiom of global choice which is stronger than the axiom of choice for sets because it also applies to proper classes. And the axiom of global choice follows from the axiom of limitation of size.
Read more about this topic: Axiom Of Choice
Famous quotes containing the words stronger and/or axioms:
“Human life in common is only made possible when a majority comes together which is stronger than any separate individual and which remains united against all separate individuals. The power of this community is then set up as “right” in opposition to the power of the individual, which is condemned as “brute force.””
—Sigmund Freud (1856–1939)
“The axioms of physics translate the laws of ethics. Thus, “the whole is greater than its part;” “reaction is equal to action;” “the smallest weight may be made to lift the greatest, the difference of weight being compensated by time;” and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (1803–1882)