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:
“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)
“The common notions that we find in credit around us and infused into our souls by our fathers’ seed, these seem to be the universal and natural ones. Whence it comes to pass that what is off the hinges of custom, people believe to be off the hinges of reason.”
—Michel de Montaigne (1533–1592)
“The only phenomenon with which writing has always been concomitant is the creation of cities and empires, that is the integration of large numbers of individuals into a political system, and their grading into castes or classes.... It seems to have favored the exploitation of human beings rather than their enlightenment.”
—Claude Lévi-Strauss (b. 1908)
“They will visit you at your convenience, whether you are lonesome or not, on rainy days or fair. They propose themselves as either transient acquaintances or permanent friends. They will stay as long as you like, departing or returning as you wish. Their friendship entails no obligation. Best of all, and not always true of our merely human friends, they have Cleopatra’s infinite variety.”
—Clifton Fadiman (b. 1904)
“There is no intrinsic worth in money but what is alterable with the times, and whether a guinea goes for twenty pounds or for a shilling, it is ... the labour of the poor and not the high and low value that is set on gold or silver, which all the comforts of life must arise from.”
—Bernard Mandeville (1670–1733)