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:
“In a famous Middletown study of Muncie, Indiana, in 1924, mothers were asked to rank the qualities they most desire in their children. At the top of the list were conformity and strict obedience. More than fifty years later, when the Middletown survey was replicated, mothers placed autonomy and independence first. The healthiest parenting probably promotes a balance of these qualities in children.”
—Richard Louv (20th century)
“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)
“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)