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Similarly, a class of serial relations, called well-ordered serial relations, can be defined, such that their corresponding relation-numbers include the ordinary finite ordinals, but also include relation-numbers which have many properties like those of the finite ordinals, though the fields of the relations belonging to them are not finite.

21are used sometimes as ordinals, i.e.

22are used sometimes as ordinals, i.e.

22Here also it can be seen that the science of the finite ordinals is a particular subdivision of the general theory of classes and relations.

11Contrasting the above definitions of number, cardinal and ordinals, with the alternative theory that number is an ultimate idea incapable of definition, we notice that our procedure exacts a greater attention, combined with a smaller credulity; for every idea, assumed as ultimate, demands a separate act of faith.

11In the case of lists and schedules the numbers are only ordinals; but in the case of mathematical or statistical tables they are usually regarded as cardinals,though, when they represent values of a continuous quantity, they must be regarded as ordinals (§§ 26, 93).

11Thus (writing ordinals in light type, and cardinals in heavy type) 9 comes after 4, and therefore 9 is greater than 4.

11The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals.

12The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals.

12clock-faces, milestones and chemists' prescriptions), they are still used for ordinals.

00This is almost always true of proofs that use countable ordinals.

00Now comes the moment to admit that my ` applications ' of countable ordinals were, in a sense, a con.

00R. Williams: " Proof theoretic ordinals and hierarchies " .

00larger ordinals correspond to more sophisticate descriptions of arithmetic.

00Additional time ordinals can also be expressed using numbers.

00For example, Lisu has up to three preceding and six following day ordinals, also three preceding and three following year ordinals.

00transfinite ordinals is one of the things to be explained.

00Here also it can be seen that the science of the finite ordinals is a particular subdivision of the general theory of classes and relations.

00Owing to the correspondence between the finite cardinals and the finite ordinals, the propositions of cardinal arithmetic and ordinal arithmetic correspond point by point.

00Similarly, a class of serial relations, called well-ordered serial relations, can be defined, such that their corresponding relation-numbers include the ordinary finite ordinals, but also include relation-numbers which have many properties like those of the finite ordinals, though the fields of the relations belonging to them are not finite.

00The arithmetic of the infinite cardinals does not correspond to that of the infinite ordinals.

00It will suffice to mention here that Peano's fourth premiss of arithmetic does not hold for infinite cardinals or for infinite ordinals.

00Contrasting the above definitions of number, cardinal and ordinals, with the alternative theory that number is an ultimate idea incapable of definition, we notice that our procedure exacts a greater attention, combined with a smaller credulity; for every idea, assumed as ultimate, demands a separate act of faith.

00In the case of lists and schedules the numbers are only ordinals; but in the case of mathematical or statistical tables they are usually regarded as cardinals,though, when they represent values of a continuous quantity, they must be regarded as ordinals (§§ 26, 93).

00clock-faces, milestones and chemists' prescriptions), they are still used for ordinals.

00Thus (writing ordinals in light type, and cardinals in heavy type) 9 comes after 4, and therefore 9 is greater than 4.

00Our ability to think about transfinite ordinals is one of the things to be explained.

00

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