For example, suppose that X is the set of all non-empty subsets of the real numbers.

But some subsets of the real numbers do not have least elements.

One might say, " Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering.

For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.

Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

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.

* There exists a model of ZF ¬ C in which there is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is sequentially continuous at a, i. e., for any sequence

The object of study is the real numbers.

The graph of a function | graph of the absolute value function for real numbers

Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

I am not making a clinical judgment here, for such personal tragedies are real and are commonplace in the analyst's consulting room, but literature makes a different claim upon our sympathies than tragedy in life.

In a real sense they are admittedly conservative, but their conservatism incorporates a traditionalized embodiment of the original `` radicalism '' of 1776.

One is that there sometimes are real although inadequate compensations in growing old.

Everyone knows that private detectives in real life are not like Sam Spade and Pat Novak, but the real and the imaginary musician are closely linked.

The facts, he adds, are hidden from public view by squeamish objections to calling bad conditions by their right name and by insistence on token integration rather than on real improvement of the schools, regardless of the color of their students.

As the civic temper rises, the more naive citizens begin to play soldier -- but the guns are real.

Assessors in Rhode Island are charged not only with placing a valuation upon real and personal property, but they also have the responsibility to raise by a tax `` a sum not less than nor more than '' a specified amount as ordered by a city council or financial town meeting.

The real question that follows is -- how are those four years used and what is their value as training??

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

These are real analytic periodic functions with period T.

Specifically, it will be asked whether the `` real '' questions people ask are not the `` ultimate '' questions that social science finds itself impotent in the face of.

But these are dreamed in original action, in some particular continuity which we don't remember having seen in real life.

If the photographically realistic continuity of dreams, however bizarre their combinations, denies that it is purely a composition of the brain, it must be compounded from views of diverse realities, although some of them may never be encountered in what we are pleased to call the real life.

Where schools, fire and police protection, and similar municipal services are of equal quality in city and country, real estate taxes are usually about the same.

Fundamentally, both these types, the amateur and the collector, are uncritical and many of them don't distinguish well between real folklore and bogus material.

True, it is no longer cricket for the butler to be the killer in mystery fiction, but we are dealing here with actual people in real life and not imaginary characters and situations.

American Catholic colleges and universities are, in a very real sense, the product of `` private enterprise '' -- the `` private enterprise '' of religious communities.

Their names are Mantle and Maris, their team is the Yankees, and their threat is real.

In our disbelief we think that we can no longer even use the word and so are unable to even name the elemental power which is so vividly real in this play.

We are left helpless to cope with it because we do not dare speak of it as anything real for fear that to do so would imply a commitment to that which has already been discredited and proved false.

Every dual number has the form z = a + bε with a and b uniquely determined real numbers.

Further note, that unlike the mathematical concept of a fluid parcel which can be uniquely identified — as well as exclusively distinguished from its direct neighbouring parcels — in a real fluid such a parcel would not always consist of the same particles.

Almost uniquely for a live British TV show at the time of its creation, The Big Breakfast was broadcast entirely from a real house.

He was a poet of considerable range, and wrote on historical and political subjects, as well as producing elegies on a number of his patrons, bitter satires on Cromwellian planters, religious poems of real feeling and, almost uniquely amongst Gaelic poets, at least two epithalamia.

Ideas about Shakespeare that many people think of as typically post-Romantic were frequently expressed in the 18th and even in the 17th century: he was described as a genius who needed no learning, as deeply original, and as creating uniquely " real " and individual characters ( see Timeline of Shakespeare criticism ).

In 1947 critic Alain Locke described him as " a real and rare genius, combining folk quality with artistic maturity so uniquely as almost to defy classification.

This ambiguity occurs because for real values of the elliptic modulus, the quarter periods and thus the nome are uniquely determined.

Although traditional ghost stories stick to one of three different kinds of ghosts, Gardner uniquely populates this novel with all three: real supernatural entities, " psychological " ghosts that originate in the imaginings of a character, and supernatural seeming occurrences that have natural explanations.

" Dubbing Nas " the real king of rap ", David Samuels of Slate described its recordings as " quasi-legendary " and called The Lost Tapes " a long-awaited step forward from an artist who was uniquely burdened by the success of his first record ", writing that it " isn't as good as Illmatic.

The Monarchist League of Canada is a national, non-partisan, non-profit organization whose mission is " to promote the full expression and a better understanding of the history and real benefits of a uniquely Canadian constitutional monarchy ".

Kite buggying also uniquely offers the pilot the possibility of getting real air time as buggies are sometimes launched into the air by the traction kite.