# Can the so-called completeness of real numbers be understood as closure under limits in the real number system?

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

• 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. Aug 19, 2023 at 9:30
• It seems your question is missing from this post (although you included it on MSE). Aug 19, 2023 at 13:07
• 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. Aug 19, 2023 at 13:33
• 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. Aug 19, 2023 at 15:34
• 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. Aug 20, 2023 at 0:08