# Tag Info

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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 ...

18

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 ...

18

It was Fourier series rather than Fourier transform. Considering that the sets where Fourier series converge can be very intricate it is not that surprising that they led Cantor to develop set theory for subsets of real numbers. But at some point he took a turn into the abstract (for which he is best known today) that was not really motivated by the initial ...

17

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 ...

16

Your guess is right. Russell's paradox broke only what people considered at that time as foundations of set theory. More specifically, the logical foundation system built by Frege. Of course this was very disturbing, because many people understood that logic and set theory is really the foundation of the whole mathematics. However it did not "invalidate" any ...

15

This morning [13 November 2014] I spent several hours going through many Cantor-related papers and books that I have, and I am now nearly convinced that Galileo probably had no influence on Cantor and Galileo had very little if any influence on other mathematicians. For instance, I believe there was no German translation of Galileo's "Two New Sciences&...

14

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, ...

14

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 ...

12

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 ...

12

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 ...

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

Georg Cantor discovered it. You can see at least : The Early Development of Set Theory : in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind, Cantor asked the question whether the infinite sets $\mathbb N$ of the natural numbers and $\mathbb R$ of real numbers can be placed in one-to-...

11

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. Edit. But there were ...

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

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 what ...

9

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.

9

The answer depends on what "this" means. According to Mancosu's Measuring the Size of Infinite Collections of Natural Numbers (reprinted in his book Abstraction and Infinity): It is actually unclear when the paradox, in the numerical form I just gave, appears. In the Greek tradition we have paradoxes that are related, but are not identical, to it; in this ...

9

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. ...

9

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 ...

8

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 ...

8

No, Cantor did not "win", for a very simple reason: the race is not over. Cantor may be in the lead, but there is no reason to think that Kronecker ( or somebody else ) will not be in the lead 100 or 200 years from now. Also, it is incorrect to say that X is the foundation for mathematics. There are multiple competitors for that title, and the best bet ( ...

8

Very much so. Today someone arguing Kronecker's position would be regarded as a crank (sort of like a finitist). Kronecker was arguing for way more than just "constructive mathematics"; he believed things like: There is not set containing all the subsets of an infinite set (like the integers or the rationals). The nested interval property (that all ...

8

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 ...

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.

8

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 ...

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Thoralf Skolem could perhaps be counted as a quasi example of that. He did not enroll as a PhD candidate before becoming a docent and a member of the Norwegian Academy of Science and Letters, and it seems that he still needed some convincing from his colleagues in order to actually submit a thesis, several years later. When he finally defended it, his formal ...

8

First, "foundations" are not what they once used to be. The idea of "one true logic" and "one true mathematics" justifiable from self-evident truths does not have much currency these days. So interest in real foundations, and belief in their existence or necessity, has been consistently waning, see Azzouni, Is there still a ...

7

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 ...

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....

7

As best as I can tell, the term Potenzbegriff (powerclass, later variation Potenzmenge, powerset) was introduced by Bernstein in the late 1890s (Cantor did not use it in his papers). In his Habilitation dissertation Untersuchungen aus der Mengenlehre (1901, published 1905) he states in the introduction (my translation): "The introduction of the concept of ...

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