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:
“I saw the man my friend ... wants pardoned, Thomas Flinton. He is a bright, good-looking fellow.... Of his innocence all are confident. The governor strikes me as a man seeking popularity, who lacks the independence and manhood to do right at the risk of losing popularity. Afraid of what will be said. He is prejudiced against the Irish and Democrats.”
—Rutherford Birchard Hayes (1822–1893)
“In England the judges should have independence to protect the people against the crown. Here the judges should not be independent of the people, but be appointed for not more than seven years. The people would always re-elect the good judges.”
—Andrew Jackson (1767–1845)
“Independence I have long considered as the grand blessing of life, the basis of every virtue; and independence I will ever secure by contracting my wants, though I were to live on a barren heath.”
—Mary Wollstonecraft (1759–1797)