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(originally asked at M.SE: https://math.stackexchange.com/questions/4094361/first-time-the-reals-were-axiomatized-as-the-unique-complete-ordered-field)

I'm looking for historical references on the history of the axiomatization of real numbers. I found out that Tarski had an axiomatization of the reals as in https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals. However, the usual approach to the reals is as the "unique complete ordered field", which is somewhat different.

Who first described the reals as the "unique complete ordered field"?

Any other axiomatizations which have appeared throughout history are also welcome.

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  • $\begingroup$ The first characterization of the reals as a complete, ordered field, unique up to isomorphism, was give by Dedekind. $\endgroup$
    – nwr
    Commented Apr 8, 2021 at 18:00
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    $\begingroup$ I agree with Nick, at least at the informal level, see SEP, Formal Axiomatics:"What is Dedekind’s theory of real numbers?—It is best understood as the theory of complete (or continuous) ordered fields." It was formalized as axiomatic theory by Hilbert in Über den Zahlbegriff (1900), where "the axioms for an ordered field are formulated and “completed” by the requirement of continuity via the Archimedean axiom and the axiom of completeness." $\endgroup$
    – Conifold
    Commented Apr 8, 2021 at 19:58
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    $\begingroup$ Related Who first used the Completeness Axiom for real numbers? $\endgroup$
    – Conifold
    Commented Apr 8, 2021 at 19:59

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After Hilbert published a paper on complete ordered field axioms "Über den Zahlbegriff" in 1900, a major paper that laid the foundation of abstract field theory was "Algebraische Theorie der Körper" published by Ernst Steinitz in 1910. It contains axioms and proofs for field theory that are (very) closed to modern algebra texts. The topics covered include abstract field axioms, isomorphism, integral domain, prime field, quotient field, finite field (with characteristics of p), function field, field extension, algebraic complete field, order field (in order type and set), and more. This paper also greatly influenced Emmy Noether for her subsequent work in modern algebra.

Dedekind defined reals based on rationals known as Dedekind cut in his 1872's paper "Continuity and Irrational Numbers".

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    $\begingroup$ Hilbert published complete ordered field axioms already in 1900. $\endgroup$
    – Conifold
    Commented Apr 8, 2021 at 20:03

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