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 is no dignity quite so impressive, and no independence quite so important, as living within your means.”
—Calvin Coolidge (1872–1933)
“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’re not out to benefit society, to remold existence, to make industry safe for anyone except ourselves, to give any small peoples except ourselves their rights. We’re not out for submerged tenths, we’re not going to suffer over how the other half lives. We’re out for Mary’s job and Luella’s art, and Barbara’s independence and the rest of our individual careers and desires.”
—Anne O’Hagan (1869–?)