Stronger Forms of The Negation of AC
Now, consider stronger forms of the negation of AC. For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets. Note that strengthened negations may be compatible with weakened forms of AC. For example, ZF + DC + BP is consistent, if ZF is.
It is also consistent with ZF + DC that every set of reals is Lebesgue measurable; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the perfect set property (all three of these results are refuted by AC itself). ZF + DC + AD is consistent provided that a sufficiently strong large cardinal axiom is consistent (the existence of infinitely many Woodin cardinals).
Read more about this topic: Axiom Of Choice
Famous quotes containing the words stronger, forms and/or negation:
“I think it is a matter of love: the more you love a memory, the stronger and stranger it is.”
—Vladimir Nabokov (1899–1977)
“That food has always been, and will continue to be, the basis for one of our greater snobbisms does not explain the fact that the attitude toward the food choice of others is becoming more and more heatedly exclusive until it may well turn into one of those forms of bigotry against which gallant little committees are constantly planning campaigns in the cause of justice and decency.”
—Cornelia Otis Skinner (1901–1979)
“I am firmly opposed to the government entering into any business the major purpose of which is competition with our citizens ... for the Federal Government deliberately to go out to build up and expand ... a power and manufacturing business is to break down the initiative and enterprise of the American people; it is the destruction of equality of opportunity amongst our people, it is the negation of the ideals upon which our civilization has been based.”
—Herbert Hoover (1874–1964)