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: Axiom Of Infinity
Famous quotes containing the words extracting, natural, numbers, infinite and/or set:
“Life is a means of extracting fiction.”
—Robert Stone (b. 1937)
“The natural role of twentieth-century man is anxiety.”
—Norman Mailer (b. 1923)
“The principle of majority rule is the mildest form in which the force of numbers can be exercised. It is a pacific substitute for civil war in which the opposing armies are counted and the victory is awarded to the larger before any blood is shed. Except in the sacred tests of democracy and in the incantations of the orators, we hardly take the trouble to pretend that the rule of the majority is not at bottom a rule of force.”
—Walter Lippmann (1889–1974)
“Gratiano speaks an infinite deal of nothing, more than any man in all Venice. His reasons are as two grains of wheat hid in two bushels of chaff; you shall seek all day ere you find them, and when you have them, they are not worth the search.”
—William Shakespeare (1564–1616)
“The time is out of joint. O cursèd spite
That ever I was born to set it right!”
—William Shakespeare (1564–1616)