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: Inductive Set (axiom Of Infinity)
Famous quotes containing the word independence:
“We must have constantly present in our minds the difference between independence and liberty. Liberty is a right of doing whatever the laws permit, and if a citizen could do what they forbid he would no longer be possessed of liberty.”
—Charles Louis de Secondat Montesquieu (1689–1755)
“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)
“Hail, Columbia! happy land!
Hail, ye heroes! heaven-born band!
Who fought and bled in Freedom’s cause,
Who fought and bled in Freedom’s cause,
And when the storm of war was gone,
Enjoyed the peace your valor won.
Let independence be our boast,
Ever mindful what it cost;”
—Joseph Hopkinson (1770–1842)