The Axiom of Dependent Choice and No Infinite Descending Sequence of Sets Implies Regularity
Let the non-empty set S be a counter-example to the axiom of regularity; that is, every element of S has a non-empty intersection with S. We define a binary relation R on S by, which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (an) in S satisfying anRan+1 for all n in N. As this is an infinite descending chain, we arrive at a contradiction and so, no such S exists.
Read more about this topic: Axiom Of Regularity
Famous quotes containing the words sequence, implies, sets, axiom, descending, infinite, choice, dependent and/or regularity:
“It isn’t that you subordinate your ideas to the force of the facts in autobiography but that you construct a sequence of stories to bind up the facts with a persuasive hypothesis that unravels your history’s meaning.”
—Philip Roth (b. 1933)
“Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.”
—David Hume (1711–1776)
“It is odd but agitation or contest of any kind gives a rebound to my spirits and sets me up for a time.”
—George Gordon Noel Byron (1788–1824)
““You are bothered, I suppose, by the idea that you can’t possibly believe in miracles and mysteries, and therefore can’t make a good wife for Hazard. You might just as well make yourself unhappy by doubting whether you would make a good wife to me because you can’t believe the first axiom in Euclid. There is no science which does not begin by requiring you to believe the incredible.””
—Henry Brooks Adams (1838–1918)
“There is, however, this consolation to the most way-worn traveler, upon the dustiest road, that the path his feet describe is so perfectly symbolical of human life,—now climbing the hills, now descending into the vales. From the summits he beholds the heavens and the horizon, from the vales he looks up to the heights again. He is treading his old lessons still, and though he may be very weary and travel-worn, it is yet sincere experience.”
—Henry David Thoreau (1817–1862)
“This moment exhibits infinite space, but there is a space also wherein all moments are infinitely exhibited, and the everlasting duration of infinite space is another region and room of joys.”
—Thomas Traherne (1636–1674)
“... given a choice between hearing my daughter say “I’m pregnant” or “I used a condom,” most mothers would get up in the middle of the night and buy them herself.”
—Joycelyn Elders (b. 1933)
“It is possible to make friends with our children—but probably not while they are children.... Friendship is a relationship of mutual dependence-interdependence. A family is a relationship in which some of the participants are dependent on others. It is the job of parents to provide for their children. It is not appropriate for adults to enter into parenthood recognizing they have made a decision to accept dependents and then try to pretend that their children are not dependent on them.”
—Donald C. Medeiros (20th century)
“The man of business ... goes on Sunday to the church with the regularity of the village blacksmith, there to renounce and abjure before his God the line of conduct which he intends to pursue with all his might during the following week.”
—George Bernard Shaw (1856–1950)