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20 votes

How was Fourier analysis important to the development of set theory?

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 ...
Conifold's user avatar
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17 votes

How did Russell arrive at the paradox demonstrating the inconsistency of naive set theory?

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 ...
Conifold's user avatar
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15 votes
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Was the continuum hypothesis commonly believed to be independent before Cohen's proof?

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 ...
Conifold's user avatar
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14 votes
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What was the motivation for the choice of the subset symbol?

That Schröder was influenced by Peano is very unlikely. When introducing his inclusion symbol in Vorlesungen, Schröder explicitly cites $<$ for numbers in arithmetic as motivation, and his version ...
Conifold's user avatar
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13 votes

When did set theory throw off theology?

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. ...
Alexandre Eremenko's user avatar
13 votes
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Was Paul Cohen a student or assistant of Gödel?

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 ...
Conifold's user avatar
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12 votes
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Did du Bois-Reymond invent the diagonal argument before Cantor?

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 ...
Conifold's user avatar
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12 votes
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Origin of Q for the set of rational numbers?

tl;dr Dedekind was the first to use a letter (R) for sets of rational numbers in 1872, then, starting from 1895, Peano began to use the letter r (lowercase) to denote the same set (and, from 1889, R ...
user6530's user avatar
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12 votes
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Was Gottlob Frege hospitalized by Russell's Paradox?

I am almost sure that this is a fiction added for dramatic purposes. First, neither Monk (who writes specifically about this episode and who has a certain flair for the dramatic, and so would ...
Nagase's user avatar
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11 votes
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What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

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 ...
Conifold's user avatar
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9 votes

Was Hilbert ambivalent about set theory?

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 ...
Conifold's user avatar
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9 votes
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What are the origins of Galileo's paradox?

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 ...
Conifold's user avatar
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9 votes
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Kronecker vs Cantor — who won?

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 ...
mobileink's user avatar
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9 votes
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Did Cantor coin the terms countable and uncountable? If not, who did?

Georg Cantor published a 47-page booklet, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre," Leipzig: Teubner 1883. The title translates to "Foundations of a general set theory"...
njuffa's user avatar
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8 votes

Kronecker vs Cantor — who won?

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 ...
Jonathan Cast's user avatar
8 votes

History of the definition of Injective & Surjective Function

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. ...
nwr's user avatar
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8 votes

Did Poincaré say that set theory is a disease?

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 ...
Conifold's user avatar
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8 votes

What brought about the need for real analysis and formal logic in recent years?

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")....
Alexandre Eremenko's user avatar
8 votes

How many important logicians did NOT receive doctoral degrees?

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 ...
J Marcos's user avatar
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8 votes
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How did ZFC become the standard foundations of mathematics?

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 ...
Conifold's user avatar
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8 votes
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When and why was the concept of "having a least upper bound" dubbed "completeness", as in Axiom of Completeness?

According to Burn, Irrational numbers in English language textbooks, 1890–1915: Constructions and postulates for the completeness of the real numbers, "completeness" was first used by ...
Conifold's user avatar
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7 votes
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Why was Kronecker dissatisfied with Cantor's submitted paper?

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" ...
Mauro ALLEGRANZA's user avatar
7 votes

When did set theory throw off theology?

Cantor's metaphysical views are probably largely irrelevant compared to his lasting contribution to set-theoretic foundations commonly used today. Nevertheless there may be an interesting issue here ...
Mikhail Katz's user avatar
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7 votes
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What were the not-so-convincing reasons for using the word "power" for power sets?

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 ...
Conifold's user avatar
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7 votes
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Why were equivalence classes named classes rather than sets?

One detailed account of the history of equivalence relations and associated terminology is Equivalence: An Attempt at a History of the Idea by Ashgari largely based on Fowler's posts on the Historia ...
Conifold's user avatar
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7 votes
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Cantor, set theory and foundations

Cantor was interested in, and developed, a theory of ordinal numbers and cardinalities. This was a significant accomplishment, in part because it overcame traditional resistance to working with ...
Mikhail Katz's user avatar
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6 votes

What brought about the need for real analysis and formal logic in recent years?

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, ...
Francois Ziegler's user avatar
6 votes
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Were $\sigma$-algebras defined for probability?

No and no, I am afraid. Those things are not specific to probability, we do them with logical connectives, which parallel set operations, and with areas and volumes, just as well as with probabilities....
Conifold's user avatar
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6 votes

In which article/book chapter did Cantor, Hilbert, and Poincaré formally define or directly discusse the term “potential infinity”?

You have misinterpreted the article you refer to; nowhere does it say that "Cantor claimed that there would only be potential infinity, not actual infinity". In fact, it says the opposite: ...
Spencer's user avatar
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6 votes

Was such advanced content on set theory taught in American elementary schools back in 1965?

Yes. I was there. I was at an American school in São Paulo, Brazil. Apparently, it was easier to introduce the new ideas in such foreign settings (a private school, no school boards to satisfy, ...
Fernando's user avatar

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