Questions tagged [set-theory]
For questions about the mathematical branch that is based on the study of sets, i.e. collections of objects.
77
questions
39
votes
3
answers
3k
views
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 ...
- 782
27
votes
2
answers
1k
views
Did Galileo's writings on infinity influence Cantor?
To what extent was Cantor motivated by Galileo's paradox? More generally, to what extent were late 19th century mathematicians motivated by, or even aware of, Galileo's paradox?
This is an issue I've ...
- 3,047
22
votes
4
answers
2k
views
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 ...
17
votes
1
answer
2k
views
How did mathematicians notate the empty set before $\varnothing$?
Recently, I learned that $\emptyset$ or $\varnothing$ is a relatively new notation for the empty set and was created in 1939. I know $\{\}$ is also used along with $\{\cdot\}$ to denote empty sets. ...
- 273
15
votes
1
answer
699
views
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 ...
- 1,305
14
votes
0
answers
573
views
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 ...
13
votes
4
answers
2k
views
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 ...
- 259
13
votes
3
answers
636
views
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 ...
- 1,397
12
votes
2
answers
632
views
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 ...
user106
12
votes
2
answers
1k
views
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'...
- 121
11
votes
2
answers
752
views
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 ...
- 73.6k
11
votes
1
answer
1k
views
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 ...
- 235
11
votes
1
answer
416
views
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 “...
- 6,689
11
votes
1
answer
306
views
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 $[x]_R=\{y\in A: \langle x,y\...
- 213
10
votes
4
answers
930
views
How much of mathematics did Russell's paradox really break?
According to my quite likely wildly oversimplified understanding, a revolution occurred in the foundations of mathematics when Cantor's formulation of set theory was found to be inconsistent due to ...
- 314
10
votes
1
answer
332
views
Was the continuum hypothesis commonly believed to be independent before Cohen's proof?
Someone drew my attention to the Russian translation of Rademacher and Toeplitz's Von Zahlen und Figuren (The Enjoyment of Mathematics in the English translation). In the chapter on set theory the ...
- 305
10
votes
1
answer
2k
views
Origin of Q for the set of rational numbers?
It seems many sources$^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 ...
- 474
9
votes
1
answer
359
views
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 ...
- 573
9
votes
1
answer
1k
views
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 ...
- 1,305
8
votes
2
answers
296
views
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 ...
- 2,972
8
votes
2
answers
2k
views
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 ...
- 89
8
votes
1
answer
373
views
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 ...
- 565
8
votes
1
answer
297
views
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 ...
8
votes
1
answer
317
views
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&...
- 474
7
votes
2
answers
2k
views
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
7
votes
1
answer
111
views
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.
...
- 337
6
votes
1
answer
452
views
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 ...
- 275
6
votes
1
answer
184
views
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 ...
6
votes
1
answer
394
views
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 ...
- 565
6
votes
4
answers
674
views
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 ...
5
votes
3
answers
761
views
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 ...
- 201
5
votes
4
answers
2k
views
Kronecker vs Cantor — who won?
Now set theory is taught even to kids and it is the foundation of mathematics. Can we say that Cantor won?
- 153
5
votes
1
answer
431
views
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 ...
- 205
5
votes
0
answers
488
views
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 ...
- 151
4
votes
2
answers
592
views
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 ...
- 143
4
votes
2
answers
547
views
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 ...
4
votes
1
answer
151
views
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/...
- 315
4
votes
1
answer
367
views
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 (...
4
votes
2
answers
403
views
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 ...
- 6,689
4
votes
2
answers
519
views
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 ...
- 151
4
votes
1
answer
388
views
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 ...
- 229
4
votes
1
answer
1k
views
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 ...
- 320
4
votes
0
answers
174
views
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}\...
- 2,688
4
votes
0
answers
143
views
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 ...
- 3,613
4
votes
0
answers
89
views
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 ...
- 137
4
votes
0
answers
222
views
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 ...
- 680
3
votes
2
answers
443
views
Who discovered the difference between the infinities?
As we know, there is a difference between the (infinite) size (or cardinality) of the integer numbers and the size of the reals ($\aleph_0$ and $\mathfrak c=2^{\aleph_0}$).
Who discovered it first?
- 909
3
votes
1
answer
666
views
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 ...
- 275
3
votes
1
answer
1k
views
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 ...
- 173
3
votes
1
answer
251
views
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 ...
- 131