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Questions tagged [set-theory]

For questions about the mathematical branch that is based on the study of sets, i.e. collections of objects.

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

I can't seem to find a clear, definitive, non-circular answer on this. For centuries and centuries, we've been doing mathematics in one form or another, be it geometry and pictures, or inventing ...
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Lengths as equivalence classes

From Wikipedia on cardinal numbers: The oldest definition of the cardinality of a set $X$ (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the class $[X]$ of all sets ...
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Can we identify Paul Benacerraf in these photos

This question is about Paul Benacerraf, who worked on the philosophy of mathematics, and wrote the 1965 essay What numbers could not be (see: Benacerraf's identification problem). He was at Princeton ...
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Was Paul Cohen a student or assistant of Gödel?

In The Man Who Loved Only Numbers, a biography about Paul Erdős, by Paul Hoffman, the author claims that Paul Cohen was "Gödel's former assistant" (p 225). However, I can't find any other sources ...
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Who was (were) the first mathematician(s) who did not doubt the empty set?

Today there is no doubt that the empty set for the whole of mathematics is as reasonable and useful as zero for arithmetic. This however was not always the case, and surprisingly even Zermelo, who ...
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When did set theory throw off theology?

"The general set theory [...] definitely belongs to metaphysics. You can easily convince yourself when examining the categories of cardinal numbers and the order type, these basic notions of set ...
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Did Zermelo in 1914 miss the fact that there are only countably many finite strings?

Did Zermelo in 1914 miss the fact, known in 1905 already, that there are only countably many finite strings, for instance polynomials with defined parameters? He wrote: Every real or complex number $\...
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Endorsers of the method of well-ordering reported by Hausdorff?

In the course of preparing a lecture for graduate students about errors of early set theory I have come across this nice example reported by Hausdorff which violates Peano's successor axiom but ...
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Details of Hilbert's proof of the continuum hypothesis [closed]

Toward the end of his paper on the infinite Hilbert sketched his proof of the continuum hypothesis. [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925)] Can experts on Hilbert give ...
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Was Hilbert ambivalent about set theory?

There is the well-known quote of Hilbert: "No one shall drive us from the paradise which Cantor has created for us."[D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167] On the ...
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Is there a formal distinction between potential and actual infinities?

In modern set theory the difference between actual infinity and potential infinity is often not understood or even denied. Some decades back however mathematicians like Hilbert or Poincaré, let alone ...
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Why was Kronecker dissatisfied with Cantor's submitted paper?

It is said here that "In 1874 Cantor published an article in Crelle's Journal which marks the birth of set theory. A follow-up paper was submitted by Cantor to Crelle's Journal in 1878 but already set ...
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Set Theory, onto and into their relation to spoken language definitions

Does anyone know how the definitions for onto and into map to the spoken language definitions of the words? I compared the Bourbaki definitions to these words and have a suspicion that the German ...
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Did Cantor knew the work of Paul du Bois-Reymond (the original inventor of the diagonal argument proof method in mathematics)? [duplicate]

In the wikipedia page about Cantor's diagonal argument, it says: Historically, the diagonal argument first appeared in the work of Paul du Bois-Reymond in 1875. However, the diagonal argument is ...
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What was Zermelo's _reception_ of von Neumann's definition of ordinal numbers?

Redactional note Like suggested by one of this question's (or so I understand the meta.discussion), this question should be given some too-many-subquestions-redacting, in particular since the easiest ...
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Where was it first proved that the cardinality of the continuum equals the cardinality of the power set of the naturals?

I can find many resources regarding Cantor's proof of the uncountability of the continuum. But was he the first to prove that $\mathfrak{c} = | \mathfrak{P} (\mathbf{N})|$? Which paper?
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Did Kronecker say that set theory is not mathematics?

I have frequently come across Kronecker's statement about set theory: "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there." It ...
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Where did Borel stress that $\mathbb{Q}$ being effectively enumerable by $\mathbb{N}$ is not about its size?

In the book 'Practical Foundations of Mathematics' (Paul Taylor), available online, one reads: As Emile Borel stressed in 1908, the important observation about $\mathbb{Q}$ [that there is a ...
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Did Poincaré say that set theory is a disease?

This question has been discussed on several sites including MathOverflow but with not definite result. Presumely HSE is best suited. Jeremy Gray denies that Poincare said, "Later generations will ...
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Has Bolzano's opinion about the irrelevance of an infinite bijection ever found supporters?

From the only reason that two sets, $A$ and $B$, are corresponding to each other by the fact that for every part $a$ being in $A$ there is a part $b$ being in $B$ such that all pairs $(a + b)$ formed ...
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What were the inconsistent classes identified by Ernst Schröder?

From G.H.Moore’s Zermelo’s Axiom of Choice : In 1890 Schröder had already introduced a notion of consistent and inconsistent classes in his Algebra der Logik, well before the modern discovery of ...
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What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

I read that Zermelo formulated AC in 1904 in order to formally prove the well-ordering theorem. Vitali’s 1905 proof of the existence of a non-measurable set of real numbers appears to the first “...
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How did Russell arrive at the paradox demonstrating the inconsistency of naive set theory?

As described here, we know that: In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted ...
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Did Bolzano conclude that $| \mathbb R | \ne | \mathbb N|$?

Boyer's History of Mathematics includes a brief account of Bolzano's contributions in the chapter on Gauss and Cauchy. Building on Galileo's "paradox" on the one-to-one correspondence between ...
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History of the definition of Injective & Surjective Function

I'm a college student, just beginning to study Elementary Set Theory. In studying about the definition of injective and surjective function, out of curiosity, it came to my mind a question about the ...
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Did du Bois-Reymond invent the diagonal argument before Cantor?

The Wiki article on Cantor's diagonal argument mentions that the first use of a diagonal argument was in the work of Paul du Bois-Reymond in 1875. This would be one year after Cantor's first proof of ...
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History of $0 \in \mathbb N$.

There are different conventions around whether $0 \in \mathbb N$. I know that $\mathbb N = \{0,1,2,3, \cdots\}$ is called the Bourbakian definition of $\mathbb N$. But did this convention really ...
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First use of curly braces to denote a set?

I was wondering who was the first person to Use curly braces to represent a finite set. Exempli gratia, $\{1,2,3\}$. Use set builder notation. Such as $\{2n:n \in \mathbb{Z}\}$ to represent the even ...
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Who was the first to mention the nine problem?

When Cantor published his seminal diagonal argument, he used binary code and did not distinguish the real numbers 0.1000… and 0.0111… Later, Koenig found a way to save the diagonal argument in binary ...
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What did Dedekind's The Nature and Meaning of Numbers contribute to the founding of Set Theory?

As best as I can tell Dedekind's paper was published in 1887 already several years after Cantor's flurry of papers on Set Theory between 1879-1883. With this in mind my central questions are: 1) What ...
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How did mathematicians notate the empty set before $\varnothing$?

I recently learned that $\emptyset$ or $\varnothing$ is a relatively new notation for the empty set created in 1939. I know $\{\}$ is also used along with $\{\cdot\}$ to denote empty sets. What was ...
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Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?

Kontinuum is German for continuum, but Cantor used $\mathfrak{c}$. Revision. J.W.Perry questions whether or not Cantor ever in fact used the symbol $\mathfrak{c}$. I must admit I just assumed that he ...
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Why is ZFC used more widely than NBG?

In studying the solutions proposed for Russell's Paradox on naive set theory (mainly the corresponding entry in the Stanford Encyclopedia of Philosophy) I came across the following considerations: von ...
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What was the answer to this paradox before Cantor?

I do not remember the name/source of this paradox,but I remember I have discussed this with mathematicians and non mathematicians at least 5 times. It goes like this: "Every point of a line has ...
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Historical Instances of Set Theory

Context: I've been reading a lot about Set Theory lately, and how it suddenly sprung onto the mathematical scene in the late 1800's, thanks largely to Cantor. But it seems strange to me that no one ...
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Did Dedekind prove this lemma about posets (or an equivalent)?

Here's an easy lemma: Any poset $(S, \preceq)$ is order-isomorphic to a subset of the powerset $\mathcal{P}(S)$ ordered by set-inclusion. I seem to recall having seen this attributed to Dedekind. ...
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How much did scholasticism contribute to logic and mathematics?

I occasionally encounter mentions suggesting that late middle ages weren't as barren intellectually as commonly believed. For example, Occam and Scotus are credited with developing modal and ...
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Were transcendental numbers considered rare, pre-Cantor?

Because the real numbers are uncountable and the real algebraic numbers are countable, there are uncountably many transcendental numbers. So there are far more transcendentals than rationals. With the ...
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What motivated Cantor to invent set theory?

I can't imagine mathematics without sets, but the question "what was mathematics like before there were sets" is not answerable, I think. Instead, a good answer to the title question should cover a ...