Dorset-born broadcaster Ralph Wightman wrote of the fair that " For many years time was dated in this part of Dorset by Shroton Fair.

For finite sets X, the axiom of choice follows from the other axioms of set theory.

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

For example, Hacking writes " And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption.

For example, it is immediately proven from the axioms that the identity element of a group is unique.

For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools.

For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself.

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms.

For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations.

* complete: For each statement in Presburger arithmetic, either it is possible to deduce it from the axioms or it is possible to deduce its negation.

For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.

For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity ( or the weaker property, truth ).

For more on this issue, see History of the separation axioms.

For example, in the " game " of Euclidean geometry ( which is seen as consisting of some strings called " axioms ", and some " rules of inference " to generate new strings from given ones ), one can prove that the Pythagorean theorem holds ( that is, you can generate the string corresponding to the Pythagorean theorem ).

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms.

: For the separation axioms in topology, see separation axiom.

For more on this issue, see History of the separation axioms.

For example, Euclidean geometry without the parallel postulate is incomplete ; it is not possible to prove or disprove the parallel postulate from the remaining axioms.

For example, one might take all true statements about the natural numbers to be axioms ( and no false statements ), which gives the theory known as " true arithmetic ".

" For Frye critical integrity means that " the axioms and postulates of criticism.

# Structures such as fields have some axioms that hold only for nonzero members of S. For an algebraic structure to be a variety, its operations must be defined for all members of S ; there can be no partial operations.

For example, one may replace the three axioms FALSE, NOT-1 ', and NOT-2 ' with the two axioms

For example, he suggests that the properties described as natural phenomena by Schenker be considered axioms and he also formulated a system to compose twelve-tone music that was " equally intricate and fruitful.

For the definition of a specific bargaining solution it is usual to follow Nash's proposal, setting out the axioms this solution should satisfy.

For example, the treatment of the projective plane starting from axioms of incidence is actually a broader theory ( with more models ) than is found by starting with a vector space of dimension three.

For over a hundred years Southerners have felt that the North was picking on them.

For example, there are persons who are in physical science, in the field of mineralogy, trained in crystallography, who use only X-rays, applying only the powder technique of X-ray diffraction, to clay minerals only, and who have spent the last fifteen years concentrating on the montmorillonites ; ;

William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.

For some happy reason Doric, Ionic, and Corinthian have always stuck in my mind.

For a while there was such shrill girlish commotion I couldn't have made myself heard if I'd had the equivalent of the message to Garcia.

For some compulsive reason which would have fascinated Dr. Freud, Communists of all shapes and sizes almost invariably impute to others the very motives which they harbor themselves.

For added comfort some of the Italian designed sandals have foam padded cushioning.

For something, clearly, has gone very, very seriously wrong in Soviet-Chinese relations, which were never easy, and have now deteriorated.

For instance -- what about all those people Harold Rhodes went toward unhesitatingly, as if this were the one moment they would ever have together, their one chance of knowing each other??

For those communities which have financial difficulties in effecting adjustments, there are a number of alternatives any one of which alone, or in combination with others, would minimize if not even eliminate the problem.

For example, it probably will be necessary for the Corps to have authority to pay medical expenses of volunteers.

For the making of selections on the basis of excellence requires that any foundation making the selections shall have available the judgments of a corps of advisors whose judgments are known to be good: such judgments can be known to be good only by the records of those selected, by records made subsequent to their selection over considerable periods of time.

For the 1960-1961 season, The Carleton Players have announced Blood Wedding by Federico Garcia Lorca, The Knight Of The Burning Pestle by Beaumont and Fletcher and A Moon For The Misbegotten by Eugene O'Neill, with a pre-season production of The Glass Menagerie by Tennessee Williams.

For the aged horses: Mr. Budlong, p, 2:00.2, Lottie Thomas, p, 2:04.2, Mighty Signal 2:03, Clever Braden, p, 2:01.1, and Glow Star, p, 2:02.3 have been in 2:35 ; ;

For almost 3,000 years Europe and Asia have rubbed shoulders in its streets.

For site planning work, it is best to have a qualified and experienced park planner to carry through the study.

For proper accreditation of schools, teachers in any course must have a degree at least one level above that for which the student is a candidate.

For the last two years, this frontier of the arts has produced a number of so-called `` non-dramas '' which have left indelible, bittersweet impressions on the psyche of this veteran theatregoer.

For an experiment to qualify as a binomial experiment, it must have four properties: ( 1 ) there must be a fixed number of trials, ( 2 ) each trial must result in a `` success '' or a `` failure '' ( a binomial trial ), ( 3 ) all trials must have identical probabilities of success, ( 4 ) the trials must be independent of each other.

For example, in accordance with the fashion of the times, most transitional societies have announced economic development plans of varying numbers of years ; ;

For one thing, although considerable numbers of men have been trained, bureaucracies are still deficient in many respects ; ;

For a number of years Kentucky, Louisiana and several other states have been building state-sponsored vocational education schools that serve nearby school districts in several counties.

For example, the huge postwar demand on the part of veterans for housing under the VA home loan guaranty program seems to have largely exhausted itself.