# 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 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 ...
297 views

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

The image attached is a comic strip from "Peanuts," dating back to 1965. It portrays Sally, the younger sister of Charlie Brown, throwing a tantrum over learning set theory in elementary ...
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### Who is William Bigelow Easton?

William Bigelow Easton is the full name of the American mathematician who is the eponym of Easton's theorem, a far-reaching application of the method of forcing in set theory. Despite the many ...
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### Who first proved that empty set is subset of all sets?

Who is the mathematician who proved that empty set is subset of all sets and made it known to most mathematicians? I looked into the ripple effects in the mathematical world that would occur if the ...
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### Equicardinality of $\mathbb{R}$ and $\mathbb{R}^2$ via interleaving decimal expansions

As Fernando Q. Gouvêa notes in his paper, Was Cantor Surprised? (Amer. Math. Monthly 118 (March 2011), 198–209) Cantor initially tried to prove that $(0,1]$ and $(0,1] \times (0,1]$ have the same ...
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### When and why was the concept of "having a least upper bound" dubbed "completeness", as in Axiom of Completeness?

The Axiom of Completeness states that any non-empty set with an upper bound has a least upper bound. When and why was this concept of least upper bound dubbed "completeness"? It's true, of ...
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### What were Cantor’s “real numbers of higher type”?

In the preamble to “Essays on the Theory of Numbers”, Dedekind makes passing reference to a theory (expounded in Cantor’s “Ueber die Ausdenung eines Satzes aus der Theorie der trigonometrischen Reihen”...
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In the article Quine’s New Foundations of The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Thomas Forster writes: In [1944] Hailperin gave the first of a number of finite ...
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### What was the motivation for the choice of the subset symbol?

I gather that the symbols $\subset$ and $\supset$ were introduced by Ernst Schröder in his 1890 Vorlesungen über die Algebra der Logik. This account also appears—attributed to good old Cajori—in an ...
1 vote
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### Was "potency set" used for power set?

Cross posted at Math Overflow For historical reasons, the English term "power set" in set theory is a translation of the German "Potenzmenge", which is still in use in German ...
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### Who proposed terminating decimals as a major set and why are them important in France?

After looking at some school sources in French, it is common to provide the various number sets in the following order \mathbb{N}\subset \mathbb{Z}\subset\mathbb{D}\subset\mathbb{Q}\subset\mathbb{R}\...
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### Who was the first to use bijections?

I know that Bourbaki were the first who used the word 'bijection', but one-to one functions were for sure used before them. So do you aware of the earliest examples of one-to-one correspondences?
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### Was Gottlob Frege hospitalized by Russell's Paradox?

After receiving the infamous letter from Bertrand Russell, Gottlob Frege allegedly had a mental breakdown and had to be hospitalized. I've seen various informal references to this, such as: Russell's ...
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### Cantor, set theory and foundations

Did Georg Cantor ever think that set theory could serve as a foundational system for all of mathematics? He died in 1918, but Zermelo set theory (just Z, no ZF or ZFC yet) was described in a paper by ...
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### Was there a view of a figure as a set of points before set theory?

A circle can be written as the solution set of an equation $x^2+y^2=1$. I have heard that Fermat associated figures with equations in the 17th century. Did people see figures as sets of points before ...
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### What are Gödel's property theories

The opening sentence of Roger Myhill's article Paradoxes, in Synthese 60 (1984), 129-143, is: “Gödel said to me more than once "There never were any set-theoretic paradoxes, but the property-...
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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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### 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 ...
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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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