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These and axioms
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 Dianetic " axioms " can be found in Hubbard books such as Scientology 0-8: The Book of Basics and Advanced Procedures and Axioms.
These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of consistency and completeness of number theory.
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 and are
These are just another couple of characters to roll.
These men are not callous.
These things are important to almost all Persians and perhaps most important to the most ordinary.
These are traversed by another line of vaults, and thus rooms, arched on all four sides, are formed.
These songs ( practically all Persian music, for that matter ) are limited to a range of two octaves.
These are not, however, differences in federal structure.
These my grandmother left in their places ( they are still there, more persistent and longer-lived than the generations of man ) and planted others like them, that flourished without careful tending.
These are like the initial ways in which the world forces itself upon the self and thrusts the self into decision and choice.
These are suggestive of Selma Lagerlof.
These assumptions lead to an organization with one man at the top, six directly under him, six under each of these, and so on until there are six levels of personnel.
These ways are absolutely irreconcilable because they offer two different recipes for man's redemption from chaos.
These responses are explicable in terms of characteristics inherent in the crisis.
These discoveries are of present interest chiefly to the scientific community ; ;
These are personality factors ; ;
These are, if the research is done with subtlety and skill, researchable topics, but the research is missing.
These moments are historical events in the lives of individual authors with which the student of comparative literature must be frequently concerned.
These lives are in themselves outside of the moral order and are unburdened with moral responsibility.
These conceptions and the manner in which they were transposed into poetry or engendered by poetic form are intrinsic to western life from the time of Aeschylus to that of Shakespeare.
These biographical analogies are obvious, and far too much time has been spent speculating on their possible implications.
These women whose organization grew out of the old suffrage movement are dedicated to Thomas Jefferson's dictum that one must cherish the people's spirit but `` Keep alive their attention ''.
These are, of course, the same people whose support he has only now rejected to seek the independent vote.
These are New York, Pennsylvania, Michigan, Maryland, Missouri, New Jersey, Illinois and Minnesota.
These men are spenders.

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