These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.

These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

These Dianetic " axioms " can be found in Hubbard books such as Scientology 0-8: The Book of Basics and Advanced Procedures and Axioms.

These conditions are examples of separation axioms.

These assumptions are the elementary theorems of the particular theory, and can be thought of as the axioms of that field.

These axioms are still in common use to this day.

These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T < sub > 5 </ sub > spaces, and perfectly normal Hausdorff spaces, or T < sub > 6 </ sub > spaces.

These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.

These conditions are examples of separation axioms.

These operations then satisfy all of the axioms for meets, joins, and complements in a Boolean algebra.

These do not provide a resolution to Hilbert's second question, however, because someone who doubts the consistency of Peano arithmetic is unlikely to accept the axioms of set theory ( which is much stronger ) to prove its consistency.

These three axioms are another way of saying that the binary operation is bilinear.

These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra.

These basic beliefs cannot, by definition, be proven ( in the logical sense ) within the worldview precisely because they are axioms, and are typically argued from rather than argued for.

These are not a minimal set of axioms but rather the complete set of possible single folds.

These axioms provide semantics by allowing systems to infer additional information based on the data explicitly provided.

These two properties are called Weyl's axioms.

These are essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC ; the axiom for Reinhardt cardinals is stronger, but is not consistent with the axiom of choice.

These conditions are separation axioms that are somewhat stronger than the more familiar Hausdorff axiom T < sub > 2 </ sub >.

These axioms are sound for standard second-order semantics.

These symbols act on certain rules of formation and are developed by inference from a set of axioms.

These expressions are called axioms, in the case of those previously supposed to be true, or theorems, in the case of those derived.

These axioms have been relatively useful for mathematical treatments of Schwarz-type QFTs, although it isn't clear that they capture the whole structure of Witten-type QFTs.

These axioms differ basically in whether or not they study a TQFT defined on a single fixed n-dimensional Riemannian / Lorentzian spacetime M or a TQFT defined on all n-dimensional spacetimes at once.

These illiterate boors conscripted from villages all across the Czarina's empire had, Suvorov may have told Lewis, just two things a commander could count on: physical fitness and personal courage.

These new poems have only a few direct references to jazz and jazz musicians, but they show changes in Patchen's approach to his poetry, for he has tried to enter into and understand the emotional attitude of the jazz musician.

These have never gone out of style in Scandinavian homes and now seem to be reappearing here and there in shops which specialize in handicrafts.

These gentlemen already have done the party harm by their seeming reluctance to vote aid for the depressed areas and by their criticism of Mr. Kennedy for talking about a recession and unemployment.

These proposals would reduce the amount of tax that DuPont stockholders might have to pay -- from an estimated 1.1 billion dollars under present law to as little as 192 million dollars.

These arrangements would have been impossible if the business community was truly interested in the welfare of its employes.

These forty-seven special purpose governments have the authority to levy taxes, to borrow money, own property, sue and be sued, and in general to exercise normal corporate powers.

These registries are sponsored by 18 national medical, dental, and veterinary societies and have as their mission the assembling of selected cases of interest to military medicine and of establishing through the mechanism of follow-up of living patients the natural history of various diseases of military-medical importance.

These thermometers have now been sent to the United Kingdom for calibration at the National Physical Laboratory.

These provisions are designed to reflect the differences in wealth and population among the States, with the objective that a vocationally handicapped person have access to needed services regardless of whether he resides in a State with a low or high per capita income or a sparsely or thickly populated State.

These could be costly and have a serious effect both at home and abroad.

These destructive forces also have a seriously adverse effect upon the watersheds and their life-supporting waterflows, and upon the other renewable forest resources.

These specialists, I learned, have done a great deal of work to improve the size and health of the plants and the resulting flowers.

These engines have a cylinder diameter of 3-7/8 inches and a stroke length of 3 inches.

`` These recent steps do not offer the possibility of extension to the great number of senior engineers who have displayed technical competence.

These have to do with property rights, municipal official attitudes and a host of others.

These vehicles, with large sensitive areas, have collected data for long enough times to give reliable impact rates for the periods of exposure.

These dosages could have been increased by increasing the source strength which was small in this case.

These have been labeled Regions 1, 2, and 4, respectively, in Fig. 1.

These drugs have no effect on the iodide concentrating mechanism, but they inhibit organification.

These groups have varied widely from mere families, primitive, totemic groups, and small modern cults and sects, to the memberships of great denominations, and great, widely dispersed world religions.

These things have been disseminated by other means, but always in the wake of extensive publication of analytic results.

These children have been described as those who were trying to say something to adults who did not understand.

These have an upper-middle-class leadership and a middle-class membership, with rare exceptions, where working-class parents are active in local P.-T.A. matters.

These must have been for local calls strictly, as in May 1900 the `` only long distance telephone '' in town was transferred from C. B. Carleton's to Young's shoe store.