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:
“The Indian’s intercourse with Nature is at least such as admits of the greatest independence of each.”
—Henry David Thoreau (1817–1862)
“We commonly say that the rich man can speak the truth, can afford honesty, can afford independence of opinion and action;—and that is the theory of nobility. But it is the rich man in a true sense, that is to say, not the man of large income and large expenditure, but solely the man whose outlay is less than his income and is steadily kept so.”
—Ralph Waldo Emerson (1803–1882)
“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)