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Ordinal number

The index (position) to an element in a set or sequence.

Are ordinals just indices to some table or sequence? For example lets consider the sequence $A = {2,4,7,14,25,3,1,6,3}$. If I say that I want the third ordinal of that sequence is the result then = $14$ ? Since we $a_4 = 14$ ?

Cardinal number

Is a natural number $\mathbb{N}$ that measure the size of sets (that is not well-ordered?). So If I say that the cardinal of the set $A$ is $9$, it means that there are $9$ elements of the set $A$ above?

Im just wondering if I got these definitions right. I just want a general simple explanation without many other concepts. Thanks.

  • See Ordinal number : "an ordinal number is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order." – Mauro ALLEGRANZA Apr 19 '18 at 13:39
  • Thus, yes: in the list $A = (2,4,7,14,25,3,1,6,3)$ we have that the first element of $A$ is $2$, the second element of $A$ is $4$, and so on. – Mauro ALLEGRANZA Apr 19 '18 at 13:40
  • If we consider instead the set $A = { 2,4,7,14,25,3,1,6,3 }$ we have no first, second, etc.elemnts, but only nine distinct elements. See Cardinal number : "cardinal numbers are used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set." – Mauro ALLEGRANZA Apr 19 '18 at 13:42
  • @MauroALLEGRANZA: Ok good. –  Apr 19 '18 at 13:45

1 Answers1

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You have it pretty much right. Better on cardinals than on ordinals. The number $3$ is an ordinal when you think of it in position in the sequence $1,2,3, \ldots$ and a cardinal when you say there are $3$ elements in the set $\{5,7,11\}$. You wouldn't usually say an ordinal is an index into a sequence, but your intuition is right even though that usage is not standard.

Ethan Bolker
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