I'm currently working on a thesis about Zermelo's axioms. In my first chapter I'm giving an introduction to the numerical treatment that Cantor gave to infinity.

When I was writing something about transfinite numbers as an expansion of natural numbers, I realized that I needed to include some reference to the formalization of numbers. I know that von Neumann gave the formal definition of natural number, but I couldn't find who formalized Integers ($\mathbb{Z}$).

Does anyone know who formalized integers, rationals, and reals? I know the construction as the palm of my hand but I don't know who did the great job.

Thanks in advance and may the health be with you!

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    $\begingroup$ Peirce, Dedekind and Peano (who cited Dedekind) axiomatized natural numbers in 1880s, see Peano axioms. So did Frege, but his approach was not followed. The extension from natural numbers to integers, rationals and reals was known by then from the work of Weierstrass, Dedekind and Cantor. $\endgroup$ – Conifold Aug 22 '20 at 7:17
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    $\begingroup$ Regarding the reals, three competing models were proposed in the late nineteenth century - the Weierstrass-Heine model using infinite decimal expansions; the Cantor-Heine-Meray model using Cauchy sequences; and the Dedekind model using his eponymous cuts. $\endgroup$ – Nick Aug 22 '20 at 16:45

There's two notions of the formalisations of integers that I know of. The better known tradition follows Dedekind and Peano, culminating in Peanos axioms (this actually was developed under the impetus of then newly developing set theory. However, it turns out that it's possible to lift Peanos axioms into the context of topos theory, a rival to set theory whose underlying logic is intuitionistic rather than classical - in that context it's called the Natural Numbers Object).

The less well-known tradition, but older one, is one instigated by Gauss: his Gaussian integers. The modern interpretation is constructing a ring of integers for every number field.


@Bachfold: it would be bad practise in a journal article. However, this is not a journal. Moreover, we live in the age of search engines and SE is hosted on the net and one can assume that if they are browsing SE then they also have access to the net. Why can't I then assume that if they have the wit to read what I wrote and are interested in what I have written and want to know more, or verify its accuracy that they simply can't type 'Gaussian Integers' or 'Natural Numbers Object' into a search engine? For example, the top three results after typing the former into Google are articles on such from NLab, Wikipedia and Springers Encyclopedia of Mathematics - any of these three will do as standard references. It seems to me that readers should put in a little work themselves - especially when writers are unpaid for their time and trouble.

  • $\begingroup$ You seem to be talking about natural numbers but the question seems to be about integers. Also, not giving references is bad practice on a history QA site, in my opinion. $\endgroup$ – Michael Bächtold Aug 30 '20 at 9:08
  • $\begingroup$ In mathematics it is often the case that something is named after a person that did not actually come up with the concept. So citing a Wikipedia or nlab article on Gaussian integers will certainly not serve to convince the reader that Gauss introduced the concept. $\endgroup$ – Michael Bächtold Sep 5 '20 at 10:29
  • $\begingroup$ Also, using an @ in your answers does not notify the person you are trying to talk to. That only works in comments. But maybe you're not really interested in addressing the people you @? $\endgroup$ – Michael Bächtold Sep 5 '20 at 10:41

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