Cardinal-numbers sentence example

cardinal-numbers
  • The operations of addition and multiplication of two given cardinal numbers can be defined by taking two classes a and 13, satisfying the conditions (1) that their cardinal numbers are respectively the given numbers, and (2) that they contain no member in common, and then by defining by reference to a and (3 two other suitable classes whose cardinal numbers are defined to be respectively the required sum and product of the cardinal numbers in question.
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  • Thus, corresponding to the cardinal numbers 2, 3, 4.
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  • Also in addition to the cardinal numbers there are the ordinal numbers: the fifth apple and the tenth pear claim thought.
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  • A critical defence of them would require a volume.1 Cardinal Numbers.
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  • If s is any class and zero is a member of it, also if when x is a cardinal number and a member of s, also x-}-I is a member of s, then the whole class of cardinal numbers is contained in s.
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  • For there is no self-contained science of cardinal numbers.
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  • Such a number is a "one-many" relation which relates n signed real numbers (or n algebraic complex numbers when they are already defined by this procedure) to the n cardinal numbers I, 2..
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  • Ordinal and Cardinal Numbers 1.2 8.
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  • One difference between the treatment of ordinal and of cardinal numbers may be noted.
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  • In using an ordinal we direct our attention to a term of a series, while in using a cardinal we direct our attention to the interval between two terms. The total number in the series is the sum of the two cardinal numbers obtained by counting up to any interval from the beginning and from the end respectively; but if we take the ordinal numbers from the beginning and from the end we count one term twice over.
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  • The system which is now almost universally in use amongst civilized nations for representing cardinal numbers is the Hindu, sometimes incorrectly called the Arabic, system.
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  • It is therefore remarkable that it should now only be used for ordinal purposes, while the Hindu system, which is ordinal in its nature, since a single series is constantly repeated, is used almost exclusively for cardinal numbers.
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  • and cardinal numbers by the Roman I, II, III,.
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  • for the individual objects cease to have an intelligible meaning, and measurement is effected by the cardinal numbers I, II, III,..
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  • These cardinal numbers have now, however, come to denote individual points in the line of measurement, i.e.
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  • Thus, while arithmetic may be defined as that branch of deductive reasoning concerning classes and relations which is concerned with the establishment of propositions concerning cardinal numbers, it must be added that the introduction of cardinal numbers makes no great break in this general science.
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  • A set of cardinal numbers have an order of magnitude, often called the order of the set because of its insistent obviousness to us; but, if they are the numbers drawn in a lottery, their time-order of occurrence in that drawing also ranges them in an order of some importance.
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  • They are excellent principles of the highest value, but they are in no sense the necessary premisses which must be proved before any other propositions of cardinal numbers can be established.
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  • For example, the application of the theory of cardinal numbers to classes of physical entities involves in practice some process of counting.
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  • Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers.
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