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I have an assignment about the foundations of mathematics.

I am trying to compile a list where I get common construction of integers and a small writing about the constructor and their explanation.

For example, I have the ordered pair construction by Dedekind, and his explanation via his principle of “generality”, where he writes that definitions appearing in a restricted form can be extended to avoid arbitrariness.

What about the original constructors of other constructions, where can I read about their commentaries?

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  • $\begingroup$ E.Bloch, The Real Numbers and Real Analysis and F.Waismann, Introduction to Mathematical Thinking $\endgroup$
    – Mauro ALLEGRANZA
    Commented Mar 5 at 11:12
  • $\begingroup$ The ordered pair construction is for rational numbers, not integers. $\endgroup$
    – Conifold
    Commented Mar 5 at 12:14
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    $\begingroup$ There is also an ordered pair construction of integers from natural numbers $\endgroup$
    – J. W. Tanner
    Commented Mar 5 at 19:19
  • $\begingroup$ @MauroALLEGRANZA in “introduction to mathematical thinking” I was reading the chapter on construction of integers and the author mentions a hueristic principle “principle of permanence of forms” and says mathematicians want to preserve existing structure as much as possible. Is this still true? How do I know I’m not reading stuff that is outdated printed in 1952 $\endgroup$
    – Bamboozle
    Commented Mar 6 at 4:45
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    $\begingroup$ @Conifold ... Sometimes we construct $\mathbb Z$ by taking ordered pairs in $\mathbb N$ with equivalence relation $(x,y) \sim (u,v) \Longleftrightarrow x+v = u+y$ $\endgroup$
    – Gerald Edgar
    Commented Mar 6 at 18:24

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Yes, there is a good resource: it is the book called Numbers, by 8 authors (Springer, 1991). The first chapter of this book written by K. Mainzer is dedicated to the history of constructions of integers and rational numbers.

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  • $\begingroup$ Great I’ll check it out thanks so much $\endgroup$
    – Bamboozle
    Commented Mar 6 at 21:32

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