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Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.
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Thus and axiom
Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.
Thus the axiom of choice is not generally available in constructive set theory.
Thus Post in his 1936 was also discounting Kurt Gödel's suggestion to Church in 1934 – 5 that the thesis might be expressed as an axiom or set of axioms.
Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members.
Thus, what the axiom schema is really saying is that, given a set A and a predicate P, we can find a subset B of A whose members are precisely the members of A that satisfy P. By the axiom of extensionality this set is unique.
Thus, if one class is " small enough " to be a set, and there is a bijection from that class to a second class, the axiom states that the second class is also a set.
Thus, the essence of the axiom is that every set has a power set.
Thus the essence of the axiom is:
Thus were true prior to the assignment, by the assignment axiom, then would be true subsequent to which.
Thus this axiom guarantees the existence of an infinite tower of inaccessible cardinals ( and may occasionally be referred to as the inaccessible cardinal axiom ).
Thus a subgroup of a group is a subset on which the binary product and the unary operation of inversion satisfy the closure axiom.
Thus, Adam Smith's single axiom, broadly interpreted ... is sufficient to characterize a major portion of the human social and cultural enterprise.
Thus, allowing arbitrary sets X and Y ( rather than just Polish spaces ) would make the axiom of uniformization equivalent to AC.
Thus and is
Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.
Thus jazz is transmuted into something holy, the sacred road to integration of being.
Thus the fictional detective is much more than a simple businessman.
Thus the cocktail party would appear to be the ideal system, but there is one weakness.
Thus in both types attention is focused on the community itself, and its phenomenological life.
Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.
Thus human perception and human volition is the immanent cause of all social change and this most truly when the change reaches the civilizational level.
Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.
Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.
Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.
Thus science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.
Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.
Thus there is a clearer division of authority, administrative and legislative.
Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.
Thus, the need for the B-70 as a strategic weapon system is doubtful.
Thus technical efficiency is achieved at the expense of actual experience.
Thus, there is an added incentive to stay on the job.
Thus, the Span of its ossification was shortened and the center's ability to `` catch up '' in ossification is demonstrated.
Thus T is not diagonalizable.
Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.
Thus, the study of the solutions to the equation Af is reduced to the study of the space of solutions of a differential equation of the form Af.
Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.
Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.
Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.
Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.
Thus and elementary
Thus, in general, elementary light optical effects, light scatter, and electron scatter determine the obtainable resolution limit.
Thus language — at least the supranational language — played an elementary role.
Charge quantization is the principle that the charge of any object is an integer multiple of the elementary charge e. Thus, e. g., an object's charge can be exactly 0 e, or exactly 1 e, − 1 e, 2 e, etc., but not, say, e, or − 3. 8 e, etc.
Thus, there is a one-to-one correspondence between the elementary excitations of a Fermi gas system and a Fermi liquid system.
Thus " elementary " refers to " element " ( i. e. state-of-matter ) as well as to something that is fundamental or innate ( physical ).
Thus, for example: Weighted averages of elementary aggregate indexes ( e. g. for men's shirts, raincoats, women's dresses, etc.
Thus the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of algebraic fractions, that produces a single rational function with a numerator and denominator usually of high degree.
Thus the amount of time taken and the number of elementary operations performed by the algorithm differ by at most a constant factor.
Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations.
Thus, in terms of instruction, the school provided higher education in traditional subjects and elementary education in modern subjects, an anomaly that persisted well into the 20th century.
Thus the head is related to the Empyreal, the chest to the ethereal heaven and the belly to the elementary substance.
Thus language — at least the supranational language — played an elementary role.
Thus, the life cycle of C. pneumoniae is divided between the elementary body, which is able to infect new hosts but can not replicate, and the reticulate body, which replicates but is not able to cause new infection.
Thus computing the Gamma function becomes a matter of evaluating only a small number of elementary functions and multiplying by stored constants.
* Given any ( set-size ) generic extension V of V, L ( R ) is an elementary submodel of L ( R ) as calculated in V. Thus the theory of L ( R ) cannot be changed by forcing.
Thus, the life cycle of C. psittaci is divided between the elementary body which is able to infect new hosts, but can not replicate, and the reticulate body, which replicates, but is not able to cause new infection.
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