Questions tagged [set-theory]
For questions about the mathematical branch that is based on the study of sets, i.e. collections of objects.
99 questions
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Did anyone seek to prove consistency by reducing math to a single theory, then prove that theory consistent?
For some reason I was led to believe that one of the responses to the discovery that naive set theory was inconsistent (Russell's Paradox) was a two-fold project to prove the rest of mathematics ...
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A question on a quote by Jerry L. Bona
According to this web site, Jerry L. Bona stated
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"
What is a precise ...
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Reconciling two different published versions of a letter from Dedekind to Weber
In Dedekind's Gesammelte Werke, in a commentary on "Zweite Definition (1889.3.9) des Endlichen und Unendlichen" (p. 450), Noether quotes from a letter Dedekind wrote to Weber, starting at
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Who is the first to use $\{\,\}$ to denote the empty set?
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Many people use $\{\,\}$ to denote the empty set (see wikipedia), but who was the first to do so? From the link, Frege uses $\{\,\}$ to denote the empty set, but I don't find such uses in ...
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Was there a special reason for using $j$ as the marker for nontrivial elementary embeddings?
In set theory, there are these functions called "nontrivial elementary embeddings" which can be used as introduction parameters/"generators" of various large cardinals and similar ...
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How and when did the extensionality principle for sets become so canonical?
Zermelo had the Axiom of Extensionality in his 1908 article which was translated to Investigations in the foundations of set theory I in Jean van Heijenoorts collection From Frege to Gödel: A Source ...
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Did Lebesgue consider the axiom of choice false?
The Wikipedia article on Tarski's theorem states that
Tarski told Jan Mycielski (2006) that when he tried to publish the theorem in Comptes Rendus de l'Académie des Sciences de Paris, Fréchet and ...
user17348
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Reference request on Georg Cantor
Georg Cantor is reported to have stated "Das Wesen der Mathematik liegt in ihrer Freiheit." Where did he state this, if at all?
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Resources on the history of the well-ordering principle of $\mathbb{Z}^{+}$
Can you recommend a book or paper on the origin of what we nowadays call the well-ordering principle of $\mathbb{Z}^{+}$?
I have several doubts regarding the provenance of this important principle. ...
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Is this a misstatement of Euclid in Halmos' Naive Set Theory book?
I was reading Halmos’ Naive Set Theory. On page 1, he writes,
One thing that the development will not include is a definition of
sets. The situation is analogous to the familiar axiomatic approach to
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What notation, if any, did Principia use for power set?
I have read in Florian Cajori, A history of Mathematical Notation (https://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf) to check notations for the power set ...
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When was $\mathcal{P}$ first used for power set?
Crossposted with Mathoverflow: https://mathoverflow.net/questions/476307/when-was-the-proof-wiki-mathcalp-first-used-for-power-set
The $\mathcal{P}$, as in https://proofwiki.org/wiki/Symbols:Fonts/...
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Criticisms of the Dedekind's definition of infinite sets for violating Euclid's Common Notion 5
Dedekind's definition of the infinite set says in its essence that part may be equal to the whole contradicting Euclid's Common Notion 5 stating that "the whole is greater than the part."
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Who first proved the "Fundamental Theorem of Well-Ordered Sets"
This theorem is also known as the "Comparability theorem for well-orderings", https://proofwiki.org/wiki/Fundamental_Theorem_of_Well-Ordering.
It states that two well-ordered sets are either ...
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How and when did the set brackets disappear from standard accounts of classical set theory?
On page 287 of the article On the Axiom of Extensionality, Part II, The Journal of Symbolic Logic, Vol. 24, No. 4 (Dec., 1959), pp. 287-
300, the author R. O. Gandy writes:
"But, in the absence ...