Universal Set - Set Theories With A Universal Set

Set Theories With A Universal Set

There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V does exist (and is true). In these theories, Zermelo's axiom of separation does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way.

The most widely studied set theory with a universal set is Willard Van Orman Quine’s New Foundations. Alonzo Church and Arnold Oberschelp also published work on such set theories. Church speculated that his theory might be extended in a manner consistent with Quine’s, but this is not possible for Oberschelp’s, since in it the singleton function is provably a set, which leads immediately to paradox in New Foundations.

Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative hierarchy, do not allow for the existence of a universal set.

Read more about this topic:  Universal Set

Famous quotes containing the words set, theories and/or universal:

    it pleaseth me when I see through the meadows
    The tents and pavilions set up, and great joy have I
    When I see o’er the campana knights armed and horses arrayed.

    And it pleaseth me when the scouts set in flight the folk with
    their goods;
    And it pleaseth me when I see coming together after them an host of
    armed men.
    Bertrans De Born (fl. 12th century)

    Philosophers of science constantly discuss theories and representation of reality, but say almost nothing about experiment, technology, or the use of knowledge to alter the world. This is odd, because ‘experimental method’ used to be just another name for scientific method.... I hope [to] initiate a Back-to-Bacon movement, in which we attend more seriously to experimental science. Experimentation has a life of its own.
    Ian Hacking (b. 1936)

    Music is the sound of the universal laws promulgated. It is the only assured tone. There are in it such strains as far surpass any man’s faith in the loftiness of his destiny. Things are to be learned which it will be worth the while to learn.
    Henry David Thoreau (1817–1862)