Connection To Graph Theory
Using the fact that in ZFC, we have (see above), it is not hard to see that the failure of the axiom of symmetry — and thus the success of — is equivalent to the following combinatorial principle for graphs:
-
- The complete graph on can be so directed, that every node leads to at most -many nodes.
- In the case of, this translates to: The complete graph on the unit circle can be so directed, that every node leads to at most countably-many nodes.
Thus in the context of ZFC, the failure of a Freiling axiom is equivalent to the existence of a specific kind of choice function.
Read more about this topic: Freiling's Axiom Of Symmetry
Famous quotes containing the words connection to, connection, graph and/or theory:
“It may comfort you to know that if your child reaches the age of eleven or twelve and you have a good bond or relationship, no matter how dramatic adolescence becomes, you children will probably turn out all right and want some form of connection to you in adulthood.”
—Charlotte Davis Kasl (20th century)
“We should always remember that the work of art is invariably the creation of a new world, so that the first thing we should do is to study that new world as closely as possible, approaching it as something brand new, having no obvious connection with the worlds we already know. When this new world has been closely studied, then and only then let us examine its links with other worlds, other branches of knowledge.”
—Vladimir Nabokov (1899–1977)
“When producers want to know what the public wants, they graph it as curves. When they want to tell the public what to get, they say it in curves.”
—Marshall McLuhan (1911–1980)
“The weakness of the man who, when his theory works out into a flagrant contradiction of the facts, concludes “So much the worse for the facts: let them be altered,” instead of “So much the worse for my theory.””
—George Bernard Shaw (1856–1950)