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:
“Our treatment of both older people and children reflects the value we place on independence and autonomy. We do our best to make our children independent from birth. We leave them all alone in rooms with the lights out and tell them, “Go to sleep by yourselves.” And the old people we respect most are the ones who will fight for their independence, who would sooner starve to death than ask for help.”
—Margaret Mead (1901–1978)
“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)
“The Indian’s intercourse with Nature is at least such as admits of the greatest independence of each.”
—Henry David Thoreau (1817–1862)