I mean this definition:

A sequence $(u_n)_{n\in \mathbb{N}}$ converges to a limit $l$ if and only if:

$$\forall \epsilon>0 ~~\exists N \in \mathbb{N} ~~ \forall n \ge N ~~\vert u_n -l \vert < \epsilon$$

Cauchy gave literally in 1823 a definition of the limit (that can be applied to a sequence) before: "If the values assigned successively to the same variable approach indefinitely a fixed value, so that they differ from it as little as one wishes, than the latter (value) is called their limit."

The Universalis encyclopedia gives the first limit notion to Benjamin Robins in "A Discourse Concerning the Nature and Certainty of Sir Isaac Newton's Method of Fluxions and Prime and Ultimate Ratios" (1735): https://www.universalis.fr/encyclopedie/limite/

Then several mathematicians had the notion of limit in their mind according to the same encyclopedia and Walter Felscher in "Bolzano, Cauchy, Epsilon, Delta" (2000): https://www.karlin.mff.cuni.cz/~spurny/doc/ma1/jirkavesely/Felscher.pdf, like D'Alembert, Gauss, Bolzano, Cauchy, Hausdorff and Cartan.

It seems that the "invention" of quantifiers are attributed to Gottlob Frege in his "Begriffsschrift": https://plato.stanford.edu/entries/frege-logic/.

But I don't find the details of my question.

  • $\begingroup$ Weierstrass spelled it out in words in his lectures of 1850-s. The earliest surviving record are Schwarz's notes of 1861, see Sinkevich, On the history of epsilontics. He did not use the symbols, those were inserted in textbooks much later, after Russell-Whitehead's Principia Mathematica (1910-13). $\endgroup$
    – Conifold
    Dec 28, 2023 at 7:27
  • $\begingroup$ I believe Russel-Whitehead did not have $\forall x$. Their quantifiers were like $(x)$ and $(\exists x)$. $\endgroup$ Dec 28, 2023 at 16:28
  • $\begingroup$ Is your question about the symbols? Or the concepts of quantifiers? $\endgroup$
    – Lee Mosher
    Dec 30, 2023 at 4:56
  • $\begingroup$ Thanks for the comments, it helps. I would precise my question by the use of symbols of quantifiers rather than quantifiers. $\endgroup$
    – someone
    Dec 30, 2023 at 15:44


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