# 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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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 ...
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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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### 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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### 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 ...