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 say that the hour of death cannot be forecast, but when we say this we imagine that hour as placed in an obscure and distant future. It never occurs to us that it has any connection with the day already begun or that death could arrive this same afternoon, this afternoon which is so certain and which has every hour filled in advance.”
—Marcel Proust (1871–1922)
“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)
“Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
—Willard Van Orman Quine (b. 1908)