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Someone suggested (please see the comments below) that I post this question on hsm.stackexchange. There is a connection to the history of mathematics in this, regarding the relationship between the completeness of real numbers and the development of calculus, as well as the discovery of the incompleteness of rational numbers. Can the so-called completeness of real numbers be understood as closure under limits in the real number system?


comment: As for the mathematical analysis (your last comment), it is not that much of an achievement to prove that Q is incomplete as to prove that R is complete. In other words, you can do calculus on it and you do not need to introduce infinitesimals (which was very important milestone to achieve as it made calculus rigorous). – Stinking Bishop

@ StinkingBishop Your point is: it wasn't the development of calculus that led humanity to discover the incompleteness of rational numbers. Humanity had already realized that rational numbers were "incomplete" long before, but they didn't know that real numbers were complete. Even in the time of Pythagoras, they only became aware of the incompleteness of rational numbers; nobody had rigorously proven the completeness of real numbers. The discovery of the square root of 2 highlighted the problem but didn't solve it. The square root of 2 alone couldn't prove the completeness of real numbers.– bokabokaboka

@StinkingBishop To make calculus more rigorous, humans rigorously proved the completeness of real numbers based on the understanding that rational numbers were already incomplete. This resolved the concerns of mathematicians in the field of mathematical analysis and made calculus perfect. – bokabokaboka

Yes, that is my understanding (although the history is always messier as it happens than when we try to summarise it.) You may want to post this also on History of Science and Mathematics StackExchange. – Stinking Bishop

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    $\begingroup$ What exactly is the question here relevant to history of mathematics? I would suggest significantly tightening the above text and clarifying the historical question that you are seeking an answer for. $\endgroup$
    – njuffa
    Aug 19, 2023 at 9:30
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    $\begingroup$ It seems your question is missing from this post (although you included it on MSE). $\endgroup$ Aug 19, 2023 at 13:07
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    $\begingroup$ It would be better to try to link everything together into connected prose rather then just quoting chunks of some else's comments to a different question on another site. $\endgroup$
    – mdewey
    Aug 19, 2023 at 13:33
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    $\begingroup$ If by "closure under limits" you mean that all Cauchy sequences converge then yes, this is equivalent to the least upper bound property (given algebraic and order axioms for real numbers), as proved by Weierstrass. But historically, it is not so much that "humans rigorously proved the completeness of real numbers", but rather that it fell out of genetic constructions of real numbers by Dedekind, Weierstrass and Cantor in 19th century. Before them, there was no clear concept of real numbers whose completeness was to be proved. $\endgroup$
    – Conifold
    Aug 19, 2023 at 15:34
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    $\begingroup$ The square root of 2 is constructible, you will not get all real numbers out of quadratic irrationalities. Even adjoining all algebraic numbers, which one could get by intersecting algebraic curves as Descartes did, would not get you real numbers. The supremum property was not focused on because constructive approaches dominant since Euclid did not need it, nor was a set of numbers that would have it even defined until 19th century. History does not follow textbook presentations. $\endgroup$
    – Conifold
    Aug 20, 2023 at 0:08

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Concerning the "connection to the history of mathematics" that you mentioned: There has been a fairly clear concept of what we call a real number only since Simon Stevin. In particular, the ancient Greeks did not possess such a concept and therefore could not possibly have asked related questions such as the one you ask. Stevin's proposal was to represent every number (whether rational or irrational) by an unending decimal; see this publication for more details.

Until the middle of the 19th century, the dominant view was to stay away from (what mathematicians thought were contradictory) "infinite wholes" as Leibniz called them; Cauchy expressed similar sentiments. Absent an entity that we call today the "set of real numbers", the issue of what we call completeness did not arise until the end of the 19th century. At First time the real numbers were axiomatized as the "unique complete ordered field" one finds that Hilbert first axiomatized the set of real numbers by means of an axiom system that included the axiom of completeness in 1900.

Cauchy could conceivably have asked himself why exactly it was that what we call a Cauchy sequence always converges to a real number. For whatever reason, he didn't, and simply assumed this to be true (as in his proof of the intermediate value theorem).

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  • $\begingroup$ @Mikhall Katz Inspired by your response, I went to explore the history of irrational numbers. Can I interpret it in this way:In the time of Pythagoras, humans indeed discovered that there were numbers that couldn't be expressed as ratios of integers (i.e., irrational numbers). However, they didn't have a clear definition or understanding of irrational numbers at that time. Using square roots or nth roots doesn't provide a complete definition of irrational numbers because these methods only involve algebraic numbers. $\endgroup$ Aug 23, 2023 at 15:12
  • $\begingroup$ It wasn't until later, with the contributions of mathematicians like Richard Dedekind, Giuseppe Peano, and Karl Weierstrass, that a clear and rigorous definition of real numbers was established. This definition confirmed that irrational numbers are spread throughout the real number line. The development of limit theory further allowed humans to differentiate between algebraic and transcendental numbers, providing a more complete understanding of irrational numbers. $\endgroup$ Aug 23, 2023 at 15:13
  • $\begingroup$ This is how humans achieved a complete understanding of irrational numbers. From this, it is evident that the mere consideration of whether operations like square roots are closed or not cannot fundamentally define the completeness of real numbers! $\endgroup$ Aug 23, 2023 at 15:17
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    $\begingroup$ @boka, It is certainly true that the consideration of square roots (or any roots) is insufficient to give an account of the real numbers. However, as I mentioned in my answer, there is a perfectly rigorous alternative to the more abstract constructions developed by Cantor and Dedekind. This is the idea of using unending decimals, which was familiar to Cauchy. Cauchy historian Laugwitz, in particular, pointed out already in the 1980s that Cauchy had a perfectly reasonable idea of number, based on this idea (which is essentially due to Stevin). $\endgroup$ Aug 23, 2023 at 15:26

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