Intuitions
Forcing is equivalent to the method of Boolean-valued models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.
Intuitively, forcing consists of expanding the set theoretical universe V to a larger universe V*. In this bigger universe, for example, one might have lots of new subsets of ω = {0,1,2,…} that were not there in the old universe, and thereby violate the continuum hypothesis. While impossible on the face of it, this is just another version of Cantor's paradox about infinity. In principle, one could consider
identify with, and then introduce an expanded membership relation involving the "new" sets of the form . Forcing is a more elaborate version of this idea, reducing the expansion to the existence of one new set, and allowing for fine control over the properties of the expanded universe.
Cohen's original technique, now called ramified forcing, is slightly different from the unramified forcing expounded here.
Read more about this topic: Forcing (mathematics)
Famous quotes containing the word intuitions:
“Doubts of all things earthly, and intuitions of some things heavenly; this combination makes neither believer nor infidel, but makes a man who regards them both with equal eye.”
—Herman Melville (1819–1891)
“He is a strong man who can hold down his opinion. A man cannot utter two or three sentences, without disclosing to intelligent ears precisely where he stands in life and thought, namely, whether in the kingdom of the senses and the understanding, or, in that of ideas and imagination, in the realm of intuitions and duty.”
—Ralph Waldo Emerson (1803–1882)
“Certainly the philosopher of ‘possible worlds’ must take care that his technical apparatus not push him to ask questions whose meaningfulness is not supported by our original intuitions of possibility that gave the apparatus its point.”
—Saul Kripke (b. 1940)