Extracting The Natural Numbers From The Infinite Set
The infinite set I is a superset of the natural numbers. To show that the natural numbers themselves constitute a set, the axiom schema of specification can be applied to remove unwanted elements, leaving the set N of all natural numbers. This set is unique by the axiom of extensionality.
To extract the natural numbers, we need a definition of which sets are natural numbers. The natural numbers can be defined in a way which does not assume any axioms except the axiom of extensionality and the axiom of induction—a natural number is either zero or a successor and each of its elements is either zero or a successor of another of its elements. In formal language, the definition says:
Or, even more formally:
Here, denotes the logical constant "false", so is a formula that holds only if n is the empty set.
Read more about this topic: Inductive Set (axiom Of Infinity)
Famous quotes containing the words extracting, natural, numbers, infinite and/or set:
“He had been eight years upon a project for extracting sunbeams out of cucumbers, which were to be put into vials hermetically sealed, and let out to warm the air in raw, inclement summers.”
—Jonathan Swift (1667–1745)
“It is natural to indulge in the illusions of hope. We are apt to shut our eyes to that siren until she “allures” us to our death.”
—Gertrude Stein (1874–1946)
“I’m not even thinking straight any more. Numbers buzz in my head like wasps.”
—Kurt Neumann (1906–1958)
“Nothing has shown more fully the prodigious ignorance of human ideas and their littleness, than the discovery of [Sir William] Herschell, that what used to be called the Milky Way is a portion of perhaps an infinite multitude of worlds!”
—Horace Walpole (1717–1797)
“God is the immemorial refuge of the incompetent, the helpless, the miserable. They find not only sanctuary in His arms, but also a kind of superiority, soothing to their macerated egos: He will set them above their betters.”
—H.L. (Henry Lewis)