The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

On a Definition of Ordinal Numbers

1962 ◽  
Vol 69 (5) ◽  
pp. 381-386 ◽  
Author(s):  
Maurice Sion ◽  
Richard Willmott
Keyword(s):  
Ordinal Numbers ◽  
Definition Of

Short definitions of the ordinals

1966 ◽  
Vol 31 (3) ◽  
pp. 409-414 ◽  
Author(s):  
Kenneth R. Brown ◽  
Hao Wang
Keyword(s):  
Von Neumann ◽  
Ordinal Numbers

In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms.The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions.

On a Definition of Ordinal Numbers

1962 ◽  
Vol 69 (5) ◽  
pp. 381 ◽  
Author(s):  
Maurice Sion ◽  
Richard Willmott
Keyword(s):  
Ordinal Numbers ◽  
Definition Of

scholarly journals The Ordinal Numbers in Hesiod’s Myth of the Races

2020 ◽  
Vol 14 (1) ◽  
pp. 57-81
Author(s):  
Paulo Alexandre Lima
Keyword(s):  
Chronological Order ◽  
Ordinal Numbers ◽  
And Function ◽  
Meaning And Function

To understand the meaning and function of the ordinal numbers in the myth of the races it is essential to have a full grasp of how the myth is composed and its structure is supposed to be perceived by a listener or reader. There is a general silence among Hesiod scholars about the meaning and function of the ordinal numbers in the myth. A tacit agreement may be inferred from such a silence: the ordinal numbers are implicitly taken to merely express the chronological order of the races. In this article, we examine each and every one of the ordinal numbers that appear in Hesiod’s myth. We demonstrate that the ordinal numbers preserve their hierarchical dimension even in the cases in which this appears to be less convincing.

On Common Multiples of Transfinite Numbers

1960 ◽  
Vol 3 (1) ◽  
pp. 31-33 ◽  
Author(s):  
Elias Zakon
Keyword(s):  
Present Note ◽  
Ordinal Numbers

In his well known monograph [l] (p. 81) H. Bachmann indicates that two ordinal numbers > 1 always have a common left multiple > 1, but not always have a right multiple (RM). The monograph does not, however, contain any further analysis of right multiples. The purpose of the present note is to supplement this by formulating the following propositions which, despite their simplicity, seem not yet to be known.

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