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 [nineteenth-century] young men who were Puritans in politics were anti-Puritans in literature. They were willing to die for the independence of Poland or the Manchester Fenians; and they relaxed their tension by voluptuous reading in Swinburne.”
—Rebecca West (1892–1983)
“The subject of the novel is reality liberated from soul. The reader in complete independence presented with a structured process: let him evaluate it, not the author. The façade of the novel cannot be other than stone or steel, flashing electrically or dark, but silent.”
—Alfred Döblin (1878–1957)
“... 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–?)