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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Zermelo's or Fraenkel's early consideration of something equivalent to countable Replacement
I have now claimed a few times on the internet, based on something (sensible!) I read, that at some point in the 1920s, that Zermelo at one point considered as a set theoretic axiom (schema) something ...
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What problem was Borel trying to solve?
I may be mistaken but I understood that Borel developed his sigma algebra before Lebesgue developed his measure. If correct, then Borel can't have been trying to find a collection of sets in $\mathbb{...
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Did Cantor coin the terms countable and uncountable? If not, who did?
I just assumed that he did since we owe the proof that $\mathbb{R}$ is uncountable to him, but a search of my favorite source for answering such questions -- https://mathshistory.st-andrews.ac.uk/...
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The definition of 'countable' and 'finite' set
I have an equivalent definition of finite and countable sets that seems 'less' based on set theory than the usual definitions. I am wondering if anyone has encountered these definitions anywhere, esp....
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where does trace $\sigma$-algebra come from
I came across the term trace $\sigma$-algebra in the context of conditional probability. Does anyone know where it comes from? i found a reference to trace of a matrix here: https://mathshistory.st-...
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When did Zermelo and Fraenkel publish their axioms?
I have googled the heck out of this but cannot find a reference to the year Z&F published their axioms. I'd expected to see an article reference but none that I could find.
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Definition of ordinal multiplication
The ordinal multiplication $\cdot$ can be defined recursively via ordinal addition $+$ for any ordinal $\alpha$ as follows:
$\alpha\cdot 0=0$.
$\alpha\cdot (\beta+1)=\alpha\cdot \beta+\alpha$ for any ...
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Origin of Q for the set of rational numbers?
It seems many sources$^1$ attribute the use of the letter "Q" to represent the rationals to the N. Bourbaki group (in the 1930's); however, the Wikipedia entry on rational numbers claims ...
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Origin / first use of $\mathbb{Z}$ (blackboard bold Z)?
I'm aware that the choice of "Z" comes from German zahlen (for "numbers"); however, I was curious to know when the dedicated font, which I believe is called "blackboard bold&...
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Where can I find the actual first order formulation of ZFC Axioms?
I'm currently writing my thesis about ZFC origins, and I need a source from where to know who gives the actual first order formulation of ZFC. I got to the point where Bernays writes about it and ...
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How was Fourier analysis important to the development of set theory?
I recently read the following quote (unfortunately, I copied it down without attribution):
You may be surprised to know that Fourier analysis played a role in the early development of set theory. In ...
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What role has Whitehead's Conjecture played in the thinking on the plurality of set theories?
I am curious about the history of the Whitehead's Conjecture, as this was the first natural mathematical statement, in the sense that mathematicians were actually interested in the answer, that was ...
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How did ZFC become the standard foundations of mathematics?
I would like to hear about the historical and technical reasons for why Zermelo-Fraenkel set theory with the axiom of Choice became the dominant standard for the foundations of mathematics.
The system ...
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In which article/book chapter did Cantor, Hilbert, and Poincaré formally define or directly discusse the term “potential infinity”?
Some media sources say that "Cantor claimed that there would only be potential infinity, not actual infinity"
In addition, the following link claims that Hilbert, Poincaré, and Cantor were ...
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Has Cantor's irregular enumeration of rationals ever been discussed?
Enumeration of all positive fractions recently has gained renewed interest (see the list below).
By translation invariance we can be sure that in all intervals (n, n+1] of the real axis, there are the ...
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Kreisel on "set" as a coherent idea
In his paper Two notes on the foundations of set-theory, Kreisel begins:
Recall that, naively, sets present themselves in a number of distinct contexts. [...] One may therefore doubt whether any ...
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Were $\sigma$-algebras defined for probability?
If you want a crash course in $\sigma$-algebras and probability spaces, you should absolutely read this amazing answer by @Sycorax on Cross Validated. Sycorax says something in particular though that ...
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Why were equivalence classes named classes rather than sets?
If $R\subseteq A\times A$ is an equivalence relation (i.e., a relation that is reflexive on $A$, symmetric, and transitive), then for each element $x\in A$, the subset $[x]_R=\{y\in A: \langle x,y\...
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History of exponential notation for the set of functions between two sets
It's well-known that if $A$ and $B$ are two sets, then the set of all functions from $A$ to $B$ can be denoted by $B^A$: explanations of this particular notation can be found in many places:
https://...
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How many important logicians did NOT receive doctoral degrees?
I can think of three. Saul Kripke quite famously could only be begrudged to finish his undergraduate degree at Harvard before being hired as a full professor. Donald Martin (the set theorist of Martin'...
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What were the not-so-convincing reasons for using the word "power" for power sets?
A footnote of Enderton's Elements of Set Theory (1977, page 4) for the definition of power set states that
the reasons for using the word "power" in this context are not very convincing, but the ...
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Translations of "Sur le théorème de Zorn"?
Are there any translations of the following into English, German, or Russian?
Nicolas Bourbaki, Sur le théorème de Zorn, Archiv der Mathematik, Volume 2, pages 434–437, November 1949.
Any help is ...
user10553
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Was the continuum hypothesis commonly believed to be independent before Cohen's proof?
Someone drew my attention to the Russian translation of Rademacher and Toeplitz's Von Zahlen und Figuren (The Enjoyment of Mathematics in the English translation). In the chapter on set theory the ...
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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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Cantor's fortune
Wiki says that his transfinite numbers met opposition:
Henri Poincaré referred to his ideas as a "grave disease" infecting
the discipline of mathematics, and Leopold Kronecker's public
...
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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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Who decided on the convention for ordinal multiplication?
Ordinal multiplication has always seemed backwards to me. $\alpha \times \beta$ is defined as the concatenation of $\beta$ copies of $\alpha$, not the other way 'round as one might expect.
Does this ...
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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 (...
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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 ...
user6103
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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 ...
user5699
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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 are the origins of Galileo's paradox?
Galileo's paradox is the observation that the natural numbers can be put into one to one correspondence with the square numbers, showing that an infinite set can be bijected to a proper subset of ...
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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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