38

An immediate motivation of Cantor to work on what became set theory was his earlier work on trigonometric series. To solve a problem in that domain he considered the set (a closed set) of zeros of such a function, then the derived set of this set, the derived set of this set and so on. This is all still classical, but then had to go a step beyond that to ...


16

Actually Cantor was working on a specific problem from the theory of trigonometric series, the so-called uniqueness problem (I cannot be more specific until MathJax is introduced to this site). This problem led him to consideration of arbitrary sets on the real line. I mean more complicated sets than finite sets or finite union of intervals. At that time ...


15

Zeno (around 500 BC) raised this paradox to argue against the notion of "plurality", arguing that a belief in the existence of many things rather than only one leads to absurd conclusions: If there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited. See section 2.2 of Zeno's paradoxes in the ...


12

According to Cohen himself, no. Most people, apparently, did not bother to believe one way or the other, and those who did (perhaps with notable exceptions of Lusin and Sierpinski) relied on Gödel's semi-philosophical remarks in a popular paper, and word of mouth that he already partially solved the problem (originating from him, presumably). This is ...


11

Aristotle gave the first systematic rebuttal of Zeno, in particular he wrote in Physics: "…a line cannot be composed of points, the line being continuous and the point indivisible". According to Aristotle, a line can be composed only of smaller, indefinitely divisible lines, and not of points without magnitude. This was the mainstream view until the "...


11

Russell was not the first to discover "his" paradox. By June 1901 when he arrived at it (it was not published until the first edition of Principia in 1903), it was already known for a while to the Hilbert's circle at Göttingen. Russell did not belong to that circle, so his rediscovery was independent, but he was familiar with Schröder's Algebra of Logic (...


11

Vitali's construction is probably the first example "unpleasant" in the modern sense, but well-ordering of the continuum was unpleasant enough to some at the time of Zermelo's proof in 1904. Accepting that the continuum can be dissociated into points (contra Aristotle) was recent and hard enough, that it could be well-ordered strained credulity further. ...


10

This paper is one in a series of papers in which du Bois Reymond studied functions on the positive real line ordered by the "order of infinity" (order of growth at infinity), what he later called infinitary pantachy. This was motivated by attempts to find "ideal boundary" between converging and diverging series in terms of the growth of their terms by ...


10

For modern times: Richard Dedekind, Was sind und was sollen die Zahlen? (1888), page 20: §71. Definition. A system $N$ [Ein system $N$] is said to be simply infinite when there exists a similar transformation $\phi$ of $N$ in itself such that $N$ appears as chain of an element not contained in $\phi(N)$. We call this element, which we shall denote in ...


10

George Boole introduced the concept of empty set, or "nothing" as he called it, as the complement to the "universe" in his Mathematical Analysis of Logic (1847). His notations for them were somewhat boring, $0$ and $1$ respectively. Cantor wrote in 1880 "for the absence of points we choose the letter $O$". Frege, the founder of mathematical logic, ...


9

As I understand, people did not think much about comparing the size of infinite sets before Cantor, so for most of them the question you are asking had no meaning, and they did not think about it. Perhaps few people felt intuitively that a countable set is "smaller" than a continuum, but Cantor was the first to state and prove this.


8

No, he was not, as one can see from the full passage from Hilbert's lecture On the Infinite delivered June 4, 1925, before a congress of the Westphalian Mathematical Society in Munster, and published in Mathematische Annalen vol. 95 (1926): "In summary, let us return to our main theme and draw some conclusions from all our thinking about the infinite. Our ...


8

It was never really connected. Cantor's mathematical papers contain nothing of the sort you cite. That Cantor himself was interested in theology is no more relevant than Newton's interest in the same. But you do not ask "when Mechanics threw of theology?" They were never related. But one person can write on several different subjects.


7

About logic, the answer is : NO, but ... The development of medieval logic is wide area of study; see at least SEP's entries : Medieval Theories of the Syllogism Medieval Theories of Consequence Medieval Theories of Modality Medieval Theories: Properties of Terms and more ..., as well as : Dov Gabbay & John Woods (editors), Handbook of the History of ...


7

Latin loan words in German usually retained their "c" until about the end of the 19th century. The modern spellings like Kontinuum (for Continuum) and Zentrum (for Centrum) result from a fairly recent spelling reform.


7

Georg Cantor was the first; see at least: "Beiträge zur Begründung der transfiniten Mengenlehre (1)" (1895), Mathematische Annalen, 46, Engl.transl.P.E.B.Jourdain (1915, also Dover reprint): By an "aggregate" (Menge) we are to understand any collection into a whole $M$ of definite and separate objects $m$ of our intuition or our thought. These objects ...


7

No, he was not. Cohen wrote his own account of the history of forcing, The Discovery of Forcing (Rocky Mountain J. Math. 32 (4) (2002), 1071-1100), where he addresses his relationship with Gödel in detail. In particular, he calls him "one of my heroes" (along with Skolem), and mentions some conversations "years later, after my own proof". Nonetheless, it ...


7

As I explained in my answer to your other question, mathematics was always done using ordinary (non-formalized) logic. Attempts to formalize logic begin with Aristotle. (This is called "formal logic"). At the later time, the idea arose to use mathematics to formalize logic. One of the early proponents of this idea was Leibniz, and it achieved further ...


6

According to this page on "earliest know uses of some mathematical words", the terms injective, surjective, and bijective were first introduced in Bourbaki's Théorie des ensembles, of 1954, page 80. The authors' motivations were to standardise terminology, stating : Standard terms are badly needed for “one-to-one,” “onto” and “one-to-one onto”; will ...


5

Cantor did not invent the term, it goes back to antiquity. "Latin was a lingua franca, the learned language for scientific and political affairs, for more than a thousand years, being eventually replaced by French in the 18th century and English in the late 19th". In Latin, French and English continuum starts with "c". Cantor saw himself as confronting the ...


5

Did Bolzano conclude that the infinity of real numbers is of a different type from the infinity of integers ? Yes and no ... See: Guillermina Waldegg, Bolzano‘s Approach to the Paradoxes of Infinity: Implications for Teaching (2005): [page 569] The comparison of infinite sets and the creation of an ‘order’ between them was one of Bolzano’s main ...


5

Regarding: how did Russell come up with that formal presentation? Do we have any ideas of how it came to be? you can see Russell's Paradox and Russell's Paradox. For details, see : Ivor Grattan-Guinness, How Bertrand Russell discovered his paradox (1978) Ivor Grattan-Guinness, Bertrand Russell on his paradox and the multiplicative axiom (1972) ...


5

I would say that the result is pretty definite. Gray does not simply deny that the quote is genuine, he quotes the scholarship of Moore (Zermelo' s Axiom of Choice), Cassinet-Guillemot (L'Axiome du choix) and Dauben (Georg Cantor), all of whom neglect to mention it. Moreover, the sources that do mention it, like Skolem and Kline, trace it to Poincare's ...


5

The site has legends that identify him: 4th of back row in 1968, 3rd of front row in 1979.


5

We didn't seem to have a "proof theory" where we all agreed what constituted a proof or what was considered a correct / incorrect proof. Yes we did. (The Greeks theorized proof by contradiction, excluded middle, contraposition, etc.) A short answer to the title question (with its emphasis on analysis) is that new “needs” arose from very concrete problems ...


4

I believe the credit mainly goes to Bertrand Russell and Émile Borel. José Ferreirós [2] writes (see below) that Bertrand Russell [4] (1903) was largely responsible for rephrasing Cantor's 1892 diagonalization result in terms of the collection of all subsets of a set, and then emphasizing the importance of this rephrasing. However, Russell was not the only ...


4

About Kronecker's views on "foundational" issues, you can see: Anne Troelstra & Dirk van Dalen, Constructivism in mathematics: An Introduction. Volume 1 (1988), page 17. There is no "explicit" philosophy of mathematics in Kronecker's works. He may be regarded as a "constructivist", or perhaps as a precursor of the finitist approach. In his essay "...


4

In MathWords we find EMPTY SET. A JSTOR search found the term used--without explanation--in J. E. McAtee "Modular Invariants of a Quadratic Form for a Prime Power Modulus," American Journal of Mathematics, 41, (1919), p. 237. The term began to become common in the 1930s, although in these early days it was nothing like as common as null set.   ...


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