Independence
The axiom of infinity cannot be derived from the rest of the axioms of ZFC, if these other axioms are consistent. Nor can it be refuted, if all of ZFC is consistent.
Indeed, using the Von Neumann universe, we can make a model of the axioms where the axiom of infinity is replaced by its negation. It is, the class of hereditarily finite sets, with the inherited element relation.
The cardinality of the set of natural numbers, aleph null, has many of the properties of a large cardinal. Thus the axiom of infinity is sometimes regarded as the first large cardinal axiom, and conversely large cardinal axioms are sometimes called stronger axioms of infinity.
Read more about this topic: Axiom Of Infinity
Famous quotes containing the word independence:
“...there was the annual Fourth of July picketing at Independence Hall in Philadelphia. ...I thought it was ridiculous to have to go there in a skirt. But I did it anyway because it was something that might possibly have an effect. I remember walking around in my little white blouse and skirt and tourists standing there eating their ice cream cones and watching us like the zoo had opened.”
—Martha Shelley, U.S. author and social activist. As quoted in Making History, part 3, by Eric Marcus (1992)
“The Indian’s intercourse with Nature is at least such as admits of the greatest independence of each.”
—Henry David Thoreau (1817–1862)
“It is my duty to prevent, through the independence of Cuba, the U.S.A. from spreading over the West Indies and falling with added weight upon other lands of Our America. All I have done up to now and shall do hereafter is to that end.... I know the Monster, because I have lived in its lair—and my weapon is only the slingshot of David.”
—José Martí (1853–1895)