Counting number

References

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5. ^ . PMC 11475258. PMID 38895756.
6. ^ says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."
   Bluman (2010), p. 1: "Numbers make up the foundation of mathematics."
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8. ^ one of our main conceptualizations of natural numbers (is) that of cardinality, or measuring how many elements there are in a set".
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10. ^ ... 'sameness of size', defined in terms of one-to-one correspondence, is logically prior to the notion of counting or, indeed, to the notion of 'number' ..."
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12. ^ "
13. ^ by an arbitrary well-ordered set ⁠${\displaystyle \langle A,\leq \rangle }$⁠. If ${\displaystyle X}$ is any set and ${\displaystyle I}$ is an ideal of ⁠${\displaystyle A}$⁠, then a function ${\displaystyle s_{I}}$ defined on ${\displaystyle I}$ with values in ${\displaystyle X}$ is called a sequence of type ⁠${\displaystyle I}$⁠. It is therefore a subset of ${\displaystyle X}$ indexed by elements of ⁠${\displaystyle I}$⁠; i.e., ⁠${\displaystyle s_{i}=\{\langle x,s(x)\rangle \mid x\in I\}}$⁠.
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16. ^ ... Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role – not by being a paradigm of any object which plays it, but by representing the relation that any third member of a progression bears to the rest of the progression."
17. ^ 2020.
18. ^ 2017 – via Google Books.
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45. ^Jahresbericht der Deutschen Mathematiker-Vereinigung [Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived from the original on 9 August 2018;"access to Jahresbericht der Deutschen Mathematiker-Vereinigung". Archived from the original on 20 August 2017.
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47. ^ 95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
48. ^ . Indiana University Press. pp. 43–52. ISBN .
49. ^ 73.
50. ^ 352. doi:10.1002/malq.19930390138. MR 1270381.
51. ^ 293. doi:10.1112/blms/14.4.285. ISSN 0024-6093.
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57. ^ . ISO 80000-2:2019 Quantities and units Part 2: Mathematics. International Organization for Standardization. 24 June 2025.
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